W14-W15. Differential Calculus

Author

Mohammad Alkousa

Published

December 10, 2025

1. Theory

1.1 Introduction: Tangent Lines

When we look at the graph of a function, we often want to understand how it behaves at a specific point. One of the most important tools for this is the tangent line - a line that just “touches” the curve at a single point and has the same direction as the curve at that point.

1.1.1 The Slope of a Curve

Unlike a straight line which has a constant slope, a curved function has a slope that changes from point to point. To find the slope of a curve at a specific point , we use a limiting process:

  1. Take a nearby point on the curve
  2. Draw the secant line connecting and
  3. Calculate the slope of this secant line:
  4. Let point approach point (i.e., let )
  5. The limiting value of these slopes is the slope of the tangent line

This gives us the formula:

The expression (where ) is called the difference quotient of at with increment . It represents the average rate of change of the function over the interval from to .

Example: For the function , the slope at any point is:

This slope is always negative for , becoming steeper (approaching ) as approaches 0, and becoming flatter (approaching 0) as moves away from 0.

1.1.2 Vertical Tangent Lines

Sometimes a curve can have a vertical tangent line at a point. This occurs when the limit of the difference quotient exists but equals .

Example: The function has a vertical tangent line at because:

However, not all functions with vertical-looking behavior at a point have vertical tangent lines. The function does not have a vertical tangent at because the limit from the left is and from the right is - they don’t match, so the limit doesn’t exist.

1.2 The Derivative at a Point

The concept of the slope of a tangent line leads us to one of the most fundamental ideas in calculus: the derivative.

1.2.1 Definition of the Derivative at a Point

Let be a function defined on an interval containing a point . The derivative of at , denoted , is:

provided this limit exists and is finite.

Alternative notations for the derivative at include:

We can also write this definition using a different variable substitution. If we set , then as , we have , giving:

The derivative is also called the rate of change of with respect to at . This interpretation is crucial in applications - it tells us how fast the function is changing at that particular point.

1.3 The Derivative as a Function

Instead of finding the derivative at just one specific point, we can find the derivative at every point where it exists, creating a new function called the derivative function.

1.3.1 Definition of the Derivative Function

The derivative of with respect to is the function whose value at is:

provided the limit exists.

The domain of consists of all points in the domain of where this limit exists. This domain may be smaller than the domain of itself.

If exists at a particular point , we say that is differentiable at . If exists at every point in its domain, we call differentiable.

Examples of computing derivatives from the definition:

  1. For (where is a constant):
  2. For :
  3. For on :
1.3.2 One-Sided Derivatives and Differentiability on Intervals

A function can have different behaviors when approached from the left versus the right at a point. This leads to the concept of one-sided derivatives.

For a function on a closed interval :

  • The right-hand derivative at is:
  • The left-hand derivative at is:

A function is:

  • Differentiable on an open interval if it has a derivative at each point of the interval
  • Differentiable on a closed interval if it is differentiable on and the right-hand derivative exists at and the left-hand derivative exists at

Important: A function has a derivative at an interior point if and only if both the left-hand and right-hand derivatives exist there and are equal.

Example: The function has no derivative at because:

  • Right-hand derivative:
  • Left-hand derivative:

Since these are different, doesn’t exist. However, when , we have , and when , we have .

1.4 Differentiability and Continuity

There’s an important relationship between differentiability and continuity: differentiable functions are always continuous, but continuous functions are not always differentiable.

1.4.1 Theorem: Differentiability Implies Continuity

Theorem: If has a derivative at , then is continuous at .

Proof: We need to show that . For , we can write:

Taking limits as :

This proves continuity at .

Important: The converse is not true. The function is continuous at but not differentiable there (as shown earlier).

1.4.2 Nowhere Differentiable Functions

It’s possible for a function to be continuous everywhere but differentiable nowhere! A famous example is the Weierstrass function:

where , , and . This function is continuous everywhere but has no derivative at any point - its graph is so jagged and irregular that it has no tangent line anywhere!

1.5 Differentiation Rules

Computing derivatives using the definition (the limit process) can be tedious. Fortunately, we have differentiation rules that allow us to find derivatives quickly and efficiently.

1.5.1 Basic Rules

Constant Multiple Rule: If is differentiable and is a constant, then:

Proof: By definition, .

Sum Rule: If and are both differentiable, then:

Proof: By definition:

Difference Rule: Combining the Sum Rule with the Constant Multiple Rule gives:

These rules extend to sums of more than two functions by mathematical induction.

1.5.2 Product and Quotient Rules

Product Rule: If and are differentiable at , then:

Proof: Starting from the definition:

We add and subtract in the numerator:

Since is differentiable at , it’s continuous there, so . Therefore:

Quotient Rule: If and are differentiable at and , then:

Proof: By definition:

Adding and subtracting in the numerator:

1.6 Derivatives of Elementary Functions

Using the definition of the derivative and the rules above, we can derive formulas for all elementary functions.

1.6.1 Constant and Power Functions

Derivative of a Constant: If (constant), then:

Power Rule: For any real number :

Proof for positive integers: We use the algebraic identity:

From the definition:

(There are terms, each approaching as .)

The proof for general real requires the exponential function derivative, which we’ll see next.

1.6.2 Exponential Functions

Derivative of Exponential Functions: For :

Proof: We use the important limit . From the definition:

Special case: For the natural exponential function:

This is one of the most remarkable properties of - the exponential function is its own derivative!

1.6.3 Trigonometric Functions

Derivative of Sine: Using the angle addition formula and important limits and :

Derivative of Cosine: Similarly:

Other Trigonometric Functions: Using the Quotient Rule:

1.7 The Chain Rule

When we have a composite function (a function inside another function), we need a special rule to find its derivative.

1.7.1 The Chain Rule Theorem

Theorem (The Chain Rule): If is differentiable at and is differentiable at , then the composite function defined by is differentiable at , and:

In other words: the derivative of the outer function (evaluated at the inner function) times the derivative of the inner function.

Alternative notation: If we set , then:

The Chain Rule tells us that the rate of change at for the composite function equals the rate of change of at multiplied by the rate of change of at .

1.7.2 Chain Rule for Elementary Functions

Applying the Chain Rule to our elementary functions (where is a differentiable function):

  • (Power Rule with Chain Rule)
  • for

Examples:

1.8 Derivatives of Inverse Functions

When a function has an inverse, there’s a beautiful relationship between their derivatives.

1.8.1 The Inverse Function Rule

Theorem: Let be a one-to-one function on an interval . If exists and for all , then is differentiable at every point in its domain (the range of ).

Moreover, if and , then:

Proof sketch: From , differentiating both sides using the Chain Rule:

Therefore:

Example: Let for , and . We can find without finding a formula for :

1.8.2 Derivative of the Natural Logarithm

Since has derivative , its inverse has derivative:

Therefore:

By the Chain Rule, for a positive differentiable function :

Examples:

1.8.3 Derivatives of Inverse Trigonometric Functions

Derivative of : Since is the inverse of for :

Therefore:

With Chain Rule: for .

Derivative of : Since :

Derivative of : Since :

Therefore:

Other inverse trigonometric functions:

  • for
  • for
1.9 Logarithmic Differentiation

For functions involving products, quotients, and powers, taking the logarithm before differentiating can greatly simplify the calculation. This technique is called logarithmic differentiation.

Example: To find the derivative of for :

  1. Take the natural logarithm of both sides:
  2. Differentiate both sides (using ):
  3. Solve for :
1.9.1 Derivatives of the Form

For functions where both the base and exponent depend on , we write and then differentiate:

Examples:

  1. For (where ): , so:
  2. For : , so:
  3. For : , so:
1.10 Higher-Order Derivatives

If the derivative of a function is itself differentiable, we can take its derivative to get the second derivative , and continue this process to get higher-order derivatives.

1.10.1 Notation for Higher-Order Derivatives

The -th derivative of is denoted in several ways:

For the first few derivatives:

  • First derivative:
  • Second derivative:
  • Third derivative:
  • Fourth derivative:
1.10.2 Leibniz Formula for the -th Derivative of a Product

There’s a formula analogous to the binomial theorem for the -th derivative of a product:

where is the binomial coefficient.

1.10.3 Examples of Higher-Order Derivatives

Example 1: For :

We can see a pattern and prove by induction that:

Example 2: For :

The pattern repeats with period 4. By induction, we can prove:

Similarly, .

Other useful formulas:

  • for
  • for
1.11 Indeterminate Forms and L’Hôpital’s Rule

When evaluating limits, we often encounter expressions that don’t have an obvious value. These are called indeterminate forms.

1.11.1 Indeterminate Forms

The common indeterminate forms are:

Each requires special techniques to evaluate. The most powerful technique is L’Hôpital’s Rule.

1.11.2 L’Hôpital’s Rule

Theorem (L’Hôpital’s Rule): Suppose and are differentiable on an open interval containing (except possibly at ), and on (except possibly at ). If:

or

Then:

provided the limit on the right side exists (or is ).

Important notes:

  1. L’Hôpital’s Rule says: differentiate the numerator and denominator separately, then take the limit
  2. The rule applies to one-sided limits and limits at infinity (replace with , , , or )
  3. Only use when you have an indeterminate form! For example, (not indeterminate), so L’Hôpital’s Rule doesn’t apply

Why it works (special case): When and both derivatives are continuous at with :

1.11.3 Examples Using L’Hôpital’s Rule

Type examples:

Type example:

Type (convert to or ):

Type (combine into a single fraction):

1.11.4 Indeterminate Powers: , ,

For limits of the form where we get an indeterminate power, we either:

  1. Take the logarithm: , then find
  2. Rewrite as an exponential:

Both lead to evaluating the indeterminate product (type or ).

Example 1: (type )

Let , then .

Therefore: .

Example 2: (type )

We write . We know (from earlier example), so:

1.11.5 When L’Hôpital’s Rule Doesn’t Help

Sometimes L’Hôpital’s Rule leads to an infinite loop or more complicated expressions. In such cases, use algebraic manipulation instead.

Example 1:

L’Hôpital’s Rule keeps cycling. Instead, divide numerator and denominator by :

Example 2:

L’Hôpital’s Rule keeps producing similar forms. Instead, divide by the dominant term :

As : , . Factor out from numerator:

As : , , . The numerator grows like , so:

1.12 Taylor Series and Taylor’s Theorem

One of the most powerful ideas in calculus is representing functions as infinite series. This allows us to approximate complicated functions with polynomials and use them to calculate limits and analyze function behavior.

1.12.1 Motivation: Approximating Functions with Polynomials

Polynomials are the simplest functions to work with - they only involve addition and multiplication. Can we approximate more complicated functions (like , , ) using polynomials?

Consider approximating near . In physics (e.g., simple pendulum), we often use:

For example, , and - very close!

Where does this approximation come from? It’s the second-degree Taylor polynomial of at .

1.12.2 Power Series Representations

A power series centered at has the form:

Theorem (Term-by-Term Differentiation): If a power series has radius of convergence , then it defines a function:

on the interval . This function has derivatives of all orders, obtained by differentiating term by term:

Important consequence: Within its interval of convergence, a power series sum is infinitely differentiable.

Question: If has derivatives of all orders on an interval , can it be expressed as a power series? If so, what are the coefficients?

1.12.3 Determining the Coefficients

Suppose equals a power series:

with positive radius of convergence. Differentiating term by term:

In general, for the -th derivative:

Evaluating at :

Therefore:

Conclusion: If has a power series representation near , it must be:

1.12.4 Taylor Series and Maclaurin Series

Definition: Let have derivatives of all orders in an interval containing . The Taylor series of centered at is:

When , this is called the Maclaurin series of :

The Taylor polynomial of order is:

Example: For at :

  • , , , , , …
  • General pattern: ,

The Maclaurin series is:

The Taylor polynomials are:

1.12.5 Taylor’s Theorem and the Remainder

Important question: When does the Taylor series actually equal the function? Not always!

Taylor’s Theorem: If has derivatives of all orders on an open interval containing , then for each positive integer and each :

where is the Taylor polynomial of order and the remainder is:

for some between and .

Convergence condition: equals its Taylor series if and only if:

In that case:

Example: For at :

Since for all , we have:

for some between 0 and .

  • If : , so and
  • If : , so and

In both cases, (factorials grow faster than exponentials), so .

Therefore:

Setting :

1.12.6 Frequently Used Taylor/Maclaurin Series

Here are the most important series (all at ):

Exponential and Logarithmic:

Trigonometric:

Hyperbolic:

Inverse Trigonometric:

Powers and Fractions:

where (generalized binomial coefficient).

1.12.7 Using Taylor Series to Find Limits

Taylor series are extremely useful for calculating limits, especially when L’Hôpital’s Rule becomes complicated.

Asymptotic notation:

  • as means is bounded near
  • as means
  • as means

Example 1: Calculate .

Using Taylor series:

Therefore:

So:

Example 2: Calculate (type ).

First:

For the base:

Therefore:

using the limit .

1.13 Applications: Extreme Values of Functions

One of the most important applications of derivatives is finding the maximum and minimum values of functions - crucial in optimization problems across science, engineering, and economics.

1.13.1 Types of Extreme Values

Definitions: Let and . We say is:

  1. An absolute (global) maximum of on if for all
  2. An absolute (global) minimum of on if for all
  3. A local (relative) maximum of if for all in some open interval containing
  4. A local (relative) minimum of if for all in some open interval containing

Absolute extrema are “global” - they’re the highest/lowest values over the entire domain. Local extrema are “neighborhood” - they’re highest/lowest in some small region around the point.

1.13.2 The Extreme Value Theorem

Theorem (Extreme Value Theorem): If is continuous on a closed interval , then attains both an absolute maximum value and an absolute minimum value at some numbers in .

This theorem is intuitively clear but requires deep knowledge of real numbers to prove rigorously.

Important: All three conditions are essential:

  1. Closed interval: On , has no maximum
  2. Bounded interval: On , has no maximum
  3. Continuity: On , has no maximum
1.13.3 Fermat’s Theorem

Theorem (Fermat’s Theorem): If has a local maximum or minimum at an interior point of its domain, and if exists, then .

Proof: Suppose has a local maximum at , so for all near .

From the right: (since numerator and denominator )

From the left: (since numerator and denominator )

Since both limits must equal , we have and , so .

Important notes:

  1. The converse is false: has but no extremum at
  2. Extrema can occur where doesn’t exist: has a minimum at where is undefined
1.13.4 Critical Points

Definition: An interior point of the domain of where or is undefined is called a critical point of .

Key result: If has a local extremum at , then is either:

  • A critical point, or
  • An endpoint of the domain

This gives us a strategy for finding extreme values:

  1. Find all critical points (where or is undefined)
  2. Evaluate at all critical points and endpoints
  3. The largest value is the absolute maximum, the smallest is the absolute minimum

Example: For on :

Critical points:

Evaluate: , , ,

Absolute max: ; Absolute min:

1.14 The Mean Value Theorem

The Mean Value Theorem is one of the most important theoretical tools in calculus, with far-reaching consequences.

1.14.1 Rolle’s Theorem

Theorem (Rolle’s Theorem): Suppose is continuous on and differentiable on . If , then there exists at least one such that .

Geometric interpretation: If a differentiable curve starts and ends at the same height, there must be at least one point where the tangent is horizontal.

Proof: By the Extreme Value Theorem, attains its maximum and minimum on .

  • If both occur at endpoints, then is constant (since ), so everywhere
  • If either occurs at an interior point , then has a local extremum at , so by Fermat’s Theorem,
1.14.2 The Mean Value Theorem

Theorem (Mean Value Theorem): Suppose is continuous on and differentiable on . Then there exists at least one such that:

Geometric interpretation: There exists a point where the tangent line is parallel to the secant line connecting the endpoints.

Proof: Consider the function:

This measures the vertical distance between and the secant line. Note that , and satisfies the conditions of Rolle’s Theorem. Therefore, there exists with:

1.14.3 Consequences of the Mean Value Theorem

Corollary 1: If for all in , then is constant on .

Proof: For any with , by MVT there exists with:

Corollary 2: If for all in , then for some constant .

Proof: Apply Corollary 1 to .

Corollary 3 (Monotonicity Test):

  1. If for all , then is increasing on
  2. If for all , then is decreasing on

Proof: For in , by MVT:

for some . Since :

  • If , then (increasing)
  • If , then (decreasing)
1.15 The First and Second Derivative Tests

The derivative tests give us practical methods for classifying critical points.

1.15.1 First Derivative Test

Theorem (First Derivative Test): Suppose is a critical point of a continuous function , and is differentiable on an interval containing (except possibly at itself). Moving from left to right across :

  1. If changes from negative to positive at , then has a local minimum at
  2. If changes from positive to negative at , then has a local maximum at
  3. If does not change sign at , then has no local extremum at

Example: For :

Critical points: (derivative undefined) and (derivative zero).

Interval
Sign of
Behavior decreasing decreasing increasing

Therefore: no extremum at (no sign change), local minimum at .

1.15.2 Concavity

Definition: The graph of a differentiable function is:

  1. Concave up (convex) on interval if is increasing on (graph lies above tangent lines)
  2. Concave down on interval if is decreasing on (graph lies below tangent lines)

Second Derivative Test for Concavity:

  1. If on , then is concave up on
  2. If on , then is concave down on

Definition: A point on the curve is an inflection point if is continuous there and the concavity changes (from up to down or vice versa). At an inflection point , either or doesn’t exist.

1.15.3 Second Derivative Test

Theorem (Second Derivative Test): Suppose is continuous near .

  1. If and , then has a local maximum at
  2. If and , then has a local minimum at
  3. If and , the test is inconclusive

Proof of (1): Since and is continuous, on some interval around . This means is decreasing on this interval. Since , we have for slightly less than and for slightly greater than . By the First Derivative Test, has a local maximum at .

Why the test can fail when :

  • : , , but has a local minimum at
  • : , , but has a local maximum at
  • : , , but has no extremum at
1.16 Curve Sketching

Combining our tools, we can sketch accurate graphs of functions.

1.16.1 Curve Sketching Procedure
  1. Domain and symmetry: Find the domain and check for symmetry (even/odd functions)
  2. Asymptotes: Find vertical, horizontal, and oblique asymptotes
  3. Derivatives: Calculate and
  4. Critical points: Find where or is undefined
  5. Monotonicity: Determine where is increasing/decreasing using
  6. Concavity and inflection points: Determine where is concave up/down using , find inflection points
  7. Plot key points: Mark intercepts, critical points, inflection points, and sketch
1.16.2 Example

Sketch .

  1. Domain: (no restrictions). No symmetry.

  2. Asymptotes: So is a horizontal asymptote.

  3. Derivatives:

  4. Critical points:

  5. Monotonicity:

    Interval
    Sign of
    Behavior decreasing increasing decreasing

    Local minimum at : Local maximum at :

  6. Concavity:

    Inflection points at

  7. Sketch: Plot the key points and connect smoothly, respecting the monotonicity and concavity.

2. Definitions

  • Tangent Line: A line that touches a curve at a single point and has the same slope as the curve at that point. For the curve at point , the tangent line has slope (if this limit exists).
  • Difference Quotient: The expression for , which represents the slope of the secant line through points and , and gives the average rate of change of over the interval.
  • Vertical Tangent Line: A curve has a vertical tangent line at if the limit of the difference quotient exists and equals .
  • Derivative at a Point: The derivative of at , denoted , is (if this limit exists and is finite). It represents the instantaneous rate of change of at .
  • Derivative Function: The function whose value at is (wherever this limit exists).
  • Differentiable at a Point: A function is differentiable at if exists.
  • Differentiable on an Interval: A function is differentiable on an open interval if it has a derivative at each point; differentiable on a closed interval if differentiable on and the appropriate one-sided derivatives exist at the endpoints.
  • Right-Hand Derivative: The limit , giving the derivative from the right.
  • Left-Hand Derivative: The limit , giving the derivative from the left.
  • Critical Point: An interior point in the domain of where either or does not exist.
  • Increasing Function: A function is increasing on an interval if whenever in that interval.
  • Decreasing Function: A function is decreasing on an interval if whenever in that interval.
  • Concave Up (Convex): A differentiable function is concave up on an interval if is increasing on that interval, equivalently if the graph lies above all its tangent lines.
  • Concave Down: A differentiable function is concave down on an interval if is decreasing on that interval, equivalently if the graph lies below all its tangent lines.
  • Inflection Point: A point on the curve where is continuous and the concavity changes from up to down or down to up.
  • Absolute (Global) Maximum: A value is an absolute maximum of on domain if for all .
  • Absolute (Global) Minimum: A value is an absolute minimum of on domain if for all .
  • Local (Relative) Maximum: A value is a local maximum if for all in some open interval containing .
  • Local (Relative) Minimum: A value is a local minimum if for all in some open interval containing .
  • Higher-Order Derivatives: The -th derivative is the derivative of , denoted for second derivative, for third, etc.
  • Indeterminate Form: An expression arising in limit evaluation that doesn’t have an obvious value, such as , , , , , , or .
  • Taylor Series: The series generated by an infinitely differentiable function at point .
  • Maclaurin Series: A Taylor series centered at : .
  • Taylor Polynomial: The polynomial , which is the -th partial sum of the Taylor series.
  • Remainder (Taylor’s Theorem): The term for some between and .
  • Logarithmic Differentiation: A technique where we take the natural logarithm of both sides of an equation before differentiating, useful for products, quotients, and variable exponents.
  • Convex Function (without differentiability): A function on interval where for all and with .
  • Strictly Convex Function: A convex function where the inequality is strict whenever and .

3. Formulas

Basic Differentiation Rules:

  • Constant Rule:
  • Power Rule: for any real
  • Constant Multiple Rule:
  • Sum Rule:
  • Difference Rule:
  • Product Rule:
  • Quotient Rule: (where )
  • Chain Rule:

Derivatives of Elementary Functions:

  • Exponential Functions:
    • for
  • Logarithmic Functions:
    • for
    • for
  • Trigonometric Functions:
  • Inverse Trigonometric Functions:
    • for
    • for
    • for all
    • for
    • for
  • Hyperbolic Functions:

Chain Rule Forms (where ):

  • for
  • for
  • for

Inverse Function Rule:

  • where

Higher-Order Derivatives:

  • Leibniz Formula:
  • Common Higher Derivatives:
    • for
    • for
    • for
    • for

L’Hôpital’s Rule:

  • If or both , then: (provided the right side exists or is )

Taylor Series (Maclaurin Series at ):

  • General Formula:
  • Taylor’s Formula: where
  • Common Series:
    • for
    • for
    • for
    • for

Theorems and Tests:

  • Fermat’s Theorem: If has a local extremum at interior point and exists, then
  • Rolle’s Theorem: If is continuous on , differentiable on , and , then with
  • Mean Value Theorem: If is continuous on and differentiable on , then with:
  • Monotonicity Test:
    • on increasing on
    • on decreasing on
  • First Derivative Test: At critical point :
    • changes from to : local minimum
    • changes from to : local maximum
    • doesn’t change sign: no extremum
  • Second Derivative Test: If :
    • : local minimum
    • : local maximum
    • : inconclusive
  • Concavity Test:
    • on concave up on
    • on concave down on

4. Practice

4.1. Limit with Indeterminate Form (Lab 15, Task 1)

Calculate the limit:

Click to see the solution

Key Concept: This is an indeterminate form . Use the identity and then apply L’Hôpital’s Rule or Taylor series.

  1. Rewrite using exponential form:

  2. Find the limit of the exponent:

  3. Use Taylor series or L’Hôpital’s Rule: As : , so

    Also, as

    Therefore:

  4. Compute the limit:

Answer:

4.2. Limit with Indeterminate Form at Infinity (Lab 15, Task 2)

Calculate the limit:

Click to see the solution

Key Concept: This is an indeterminate form . Analyze the behavior of the base and exponent as approaches infinity.

  1. Rewrite using exponential form:
  2. Simplify the base:
  3. Find the limit of the exponent:
  4. Evaluate each term:
    • (by L’Hôpital’s Rule)
    • , so , and
  5. Compute the limit:

Answer:

4.3. Taylor Polynomials for (Lab 15, Task 3)

Find the Taylor Polynomials of orders 1, 2, and 3 generated by at .

Click to see the solution

Key Concept: Use the formula .

  1. Find derivatives:
    • , so
    • , so
    • , so
    • , so
  2. Construct Taylor polynomials:
    • Order 1:
    • Order 2:
    • Order 3:

Answer:

4.4. Taylor Polynomials for (Lab 15, Task 4)

Find the Taylor Polynomials of orders 1, 2, and 3 generated by at .

Click to see the solution

Key Concept: Use the formula .

  1. Find derivatives:
    • , so
    • , so
    • , so
    • , so
  2. Construct Taylor polynomials:
    • Order 1:
    • Order 2:
    • Order 3:

Answer:

4.5. Taylor Series for (Lab 15, Task 5)

Find the Taylor series generated by at .

Click to see the solution

Key Concept: Use the formula .

  1. Find derivatives: Since for all , we have for all .
  2. Construct the Taylor series:
  3. Simplify:

Answer:

4.6. Taylor Series for (Lab 15, Task 6)

Find the Taylor series generated by at .

Click to see the solution

Key Concept: Use the formula for geometric series or find derivatives directly.

  1. Method 1: Using geometric series: Write

    For , we have:

    Therefore:

  2. Method 2: Using derivatives:

    • , so
    • , so
    • , so
    • , so

    Therefore:

Answer: for

4.7. Limit Using Taylor Series (Lab 15, Task 7)

Calculate the limit:

Click to see the solution

Key Concept: Expand each term using Taylor series (Maclaurin series) at and keep terms up to the dominant power.

  1. Expand each function:

    • , so

  2. Compute the numerator:

    We need higher order terms. Let’s be more precise:

    • where
    • Therefore:

  3. Compute the denominator:

  4. Compute the limit:

Answer:

4.8. Analyze Variation and Sketch the Graph (Lab 15, Task 8)

Study the variation and graph of the function:

Click to see the solution

Key Concept: Analyze domain, symmetry, asymptotes, derivatives, and monotonicity to sketch the graph.

  1. Domain: We need , so . Domain:

  2. Symmetry:

    So is an even function.

  3. Asymptotes:

    • As : So is a horizontal asymptote as .
    • By symmetry, is also a horizontal asymptote as .
    • No vertical asymptotes (function is continuous on its domain).

  4. First derivative:

    For : , so , and , so .

    For : and the difference in parentheses is still negative, so .

    Therefore: is decreasing on and increasing on .

  5. Critical points: when , but is not in the domain. No critical points in the domain.

  6. Behavior at endpoints:

    • As :
    • As :

  7. Second derivative (for concavity):

    This is negative for (function is concave down), and positive for (function is concave up).

Answer: The function is even, defined on , decreasing on , increasing on , with horizontal asymptote at both ends, and maximum value at .

4.9. Find the Slope of a Curve at Any Point (Chapter 6, Example 1)

Find the slope of the curve at any point . Where does the slope equal ?

Click to see the solution

Key Concept: Use the definition of the derivative to find the slope at any point.

  1. Apply the definition: We have . The slope at is:
  2. Simplify the numerator:
  3. Cancel and evaluate:
  4. Find where slope equals :

Answer: The slope at is . The slope equals at the points and .

Note: The slope is always negative for . As , the slope approaches (tangent becomes vertical). As , the slope approaches 0 (tangent becomes horizontal).

4.10. Compute Derivatives from the Definition (Chapter 6, Example 2)

By the definition, find the derivative of the following functions:

(a)
(b)
(c)
(d)

Click to see the solution

(a) :

  1. Apply the definition:
  2. Simplify:
  3. Evaluate:

Answer (a):

(b) :

  1. Apply the definition:
  2. Expand:
  3. Simplify:

Answer (b):

(c) :

  1. Apply the definition:
  2. Rationalize:
  3. Evaluate:

Answer (c):

(d) :

  1. Apply the definition:
  2. Combine fractions:
  3. Expand and simplify:
  4. Evaluate:

Answer (d):

4.11. Evaluate a Limit Using Taylor Series (Chapter 6, Example 3)

Calculate (type ).

Click to see the solution

Key Concept: This is type . Expand the base and exponent using Taylor series.

  1. Expand the exponent:

  2. Expand the base:

    Therefore:

  3. Compute the limit:

    Using the limit , we get:

Answer:

4.12. Apply Differentiation Rules (Chapter 6, Example 4)

Find the derivatives of the following functions:

(a)
(b)
(c)
(d)

Click to see the solution

(a) :

Answer (a):

(b) :

  1. Derivative of first term:
  2. Derivative of second term (quotient rule):

Answer (b):

(c) :

  1. First term (product rule):
  2. Second term (power rule):
  3. Third term:

Answer (c):

Note: , so .

(d) :

  1. First term (product rule):
  2. Second term (product rule):

Answer (d):

4.13. Differentiate Composite and Rational Functions (Chapter 6, Example 5)

Find the derivatives of the following functions:

(a)
(b)
(c)
(d)
(e)
(f)

Click to see the solution

(a) :

Using the quotient rule, or first expand numerator and denominator:

Numerator:
Denominator:

After simplification:

Answer (a): or equivalently

(b) :

Using quotient rule:

Answer (b):

(c) :

Using product rule:

Answer (c):

(d) :

Using quotient rule:

Answer (d):

(e) :

Answer (e):

(f) :

Using quotient rule (or chain rule with product):

Answer (f):

4.14. Chain Rule Applications (Chapter 6, Example 6)

Find the derivatives:

(a)
(b)
(c)

Click to see the solution

(a): Using chain rule:

Answer (a):

(b): Using product rule multiple times:

Answer (b): or

(c): Using chain rule:

Answer (c):

4.15. Derivative of Inverse Function (Chapter 6, Example 7)

Let for , and . Find without finding a formula for .

Click to see the solution

Key Concept: Use the inverse function rule:

  1. Identify the values: We have and (since )
  2. Find :
  3. Evaluate :
  4. Apply the inverse function rule:

Answer:

4.16. Logarithmic Differentiation (Chapter 6, Example 8)

Find the derivative of for .

Click to see the solution

Key Concept: Take the logarithm of both sides before differentiating to simplify products and quotients.

  1. Take the natural logarithm:
  2. Differentiate both sides:
  3. Solve for :

Answer:

4.17. Derivatives of Forms (Chapter 6, Example 9)

Find the derivatives:

(a) for
(b) for
(c)

Click to see the solution

Key Concept: For , write and use the chain rule.

(a) :

  1. Rewrite:
  2. Differentiate:

Answer (a):

(b) :

  1. Rewrite:
  2. Differentiate:

Answer (b):

(c) :

  1. Rewrite:
  2. Differentiate the exponent:
  3. Apply chain rule:

Answer (c):

4.18. Find the -th Derivative of (Chapter 6, Example 10)

Find the -th order derivative of and prove your formula by induction.

Click to see the solution

  1. Compute the first few derivatives:

  2. Identify the pattern:

  3. Prove by induction:

    Base case (): ✓

    Inductive step: Assume true for . Show for : ✓

Answer: for all

4.19. L’Hôpital’s Rule Examples (Chapter 6, Example 11)

Calculate the following limits using L’Hôpital’s Rule:

(a)
(b)
(c)
(d)
(e)

Click to see the solution

(a) Type :

Answer (a):

(b) Type :

This is still , apply L’Hôpital again:

Answer (b):

(c) Type :

Answer (c):

(d) Type , convert to :

Answer (d):

(e) Type , combine into single fraction:

Type , apply L’Hôpital:

Still , apply L’Hôpital again:

Answer (e):

4.20. Indeterminate Powers (Chapter 6, Example 12)

Calculate:

(a) (type )
(b) (type )

Click to see the solution

(a) Type :

Let , then .

Therefore:

Answer (a):

(b) Type :

Write . From Example 4.10(d), we know . Therefore:

Answer (b):

4.21. Sum of Series Using (Chapter 6, Example 13)

Given that , find:

(a)
(b)

Click to see the solution

(a) Split the sum:

For the first sum, note that for :

For the second sum:

Therefore:

Answer (a):

(b) Split the sum:

For , write :

Therefore:

Answer (b):

4.22. Find Limits Using Taylor Series (Chapter 6, Example 14)

Calculate .

Click to see the solution

Key Concept: Use Taylor series expansions around and keep terms up to the dominant power.

  1. Expand each function:
    • , so:
  2. Compute numerator:
  3. Compute the limit:

Answer:

4.23. Find Absolute Extrema on a Closed Interval (Chapter 6, Example 15)

(a) Find the absolute maximum and minimum values of on .
(b) Find the absolute maximum and minimum values of on .

Click to see the solution

(a) on :

  1. Find critical points:

    Both are in the interval .

  2. Evaluate at critical points and endpoints:

  3. Identify extrema:

    • Absolute maximum: (at endpoint)
    • Absolute minimum: (at critical point)

Answer (a): Absolute max: at ; Absolute min: at

(b) on :

  1. Find critical points:

    is undefined at , which is in the interval. This is a critical point.

  2. Evaluate at critical point and endpoints:

  3. Identify extrema:

    • Absolute maximum: (at endpoint)
    • Absolute minimum: (at critical point - a cusp)

Answer (b): Absolute max: at ; Absolute min: at

4.24. First Derivative Test (Chapter 6, Example 16)

(a) For , find all critical points and classify them using the First Derivative Test.

(b) For , find all critical points and classify them using the First Derivative Test.

Click to see the solution

(a) :

  1. Find :

  2. Find critical points:

    • undefined

  3. Test sign of on intervals:

    Interval
    Sign of
    Behavior decreasing decreasing increasing

  4. Classify critical points:

    • At : no sign change, so no extremum
    • At : sign changes from to , so local minimum

Answer (a): Critical points: (no extremum), (local minimum with value )

(b) :

  1. Find using product rule:

  2. Find critical points:

    (Note: for any )

  3. Test sign of on intervals:

    Interval
    Sign of
    Behavior increasing decreasing increasing

  4. Classify critical points:

    • At : sign changes from to , so local maximum
    • At : sign changes from to , so local minimum

Answer (b): Local max at with value ; Local min at with value

4.25. Curve Sketching: Example 1 (Chapter 6, Example 17)

Sketch the graph of .

Click to see the solution

  1. Domain: (no restrictions). No symmetry.

  2. Asymptotes:

    Horizontal asymptote: . No vertical asymptotes.

  3. First derivative:

    Critical points:

  4. Monotonicity:

    Interval
    Sign of
    Behavior decreasing increasing decreasing
    • Local minimum at :
    • Local maximum at :

  5. Second derivative:

    Inflection points:

  6. Concavity:

    Interval
    Sign of
    Concavity up down up down

  7. Key points:

    • Intercepts:
    • Inflection points at

Answer: See sketch with horizontal asymptote , local min at , local max at , and three inflection points.

4.26. Curve Sketching: Example 2 (Chapter 6, Example 18)

Sketch the graph of .

Click to see the solution

  1. Domain: . No symmetry.

  2. Limits and asymptotes:

    • (horizontal asymptote as )
    • (horizontal asymptote as )
    • (vertical asymptote at )
    • (approaches 0 from left)

  3. First derivative:

    Since and , we have for all .

    No critical points. is decreasing on and on .

  4. Second derivative:

    Inflection point:

  5. Concavity:

    Interval
    Sign of
    Concavity down up up

Answer: See sketch with vertical asymptote at (approaching from right, 0 from left), horizontal asymptote on both sides, always decreasing, inflection point at .

4.27. Curve Sketching: Example 3 (Chapter 6, Example 19)

Sketch the graph of .

Click to see the solution

  1. Domain: (cube root defined everywhere). No symmetry.

  2. Limits:

    No asymptotes.

  3. First derivative:

    Critical points:

    • undefined or

  4. Monotonicity:

    Interval
    Sign of
    Behavior increasing increasing decreasing decreasing
    • At : no sign change, local minimum (cusp),
    • At : sign changes from to , local maximum,
    • At :

  5. Second derivative:

    Inflection point: has no solution, but changes sign at .

  6. Concavity:

    Interval
    Sign of
    Concavity up down down

Answer: See sketch with cusp at origin , local max at , crosses -axis at and , going to as and to as .

4.28. Find the Sum of Series (Preparing for Final, Series 1)

Find the sum:

Click to see the solution

Key Concept: Use partial fractions and telescoping series.

  1. Factor the denominator:

  2. Use partial fractions:

    Solving: ,

  3. Write partial sum:

  4. Take limit:

Answer:

4.29. Limits Without L’Hôpital - Problem 1 (Preparing for Final, Limits 1)

Evaluate (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Factor numerator and denominator.

  1. Factor:
  2. Simplify:
  3. Evaluate:

Answer:

4.30. Limits Without L’Hôpital - Problem 2 (Preparing for Final, Limits 1)

Evaluate (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Factor numerator. Since is a root, factor out .

  1. Factor numerator: Using polynomial division or synthetic division:
  2. Simplify:

Answer:

4.31. Limits Without L’Hôpital - Problem 3 (Preparing for Final, Limits 1)

Evaluate (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Rationalize the numerator.

  1. Rationalize:
  2. Evaluate:

Answer:

4.32. Limits Without L’Hôpital - Problem 4 (Preparing for Final, Limits 1)

Evaluate (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Use the identity .

  1. Rationalize using cube root identity: Let and . Then:

    Therefore:

  2. Evaluate:

Answer:

4.33. Derivative by Definition - Problem 1 (Preparing for Final, Differentiation 1)

By the definition, find the derivative of:

Click to see the solution

Key Concept: Use the definition .

  1. Apply the definition:
  2. Combine fractions:
  3. Expand and simplify: Numerator:
  4. Evaluate:

Answer:

4.34. Derivative by Definition - Problem 2 (Preparing for Final, Differentiation 1)

By the definition, find the derivative of:

Click to see the solution

Key Concept: Use the definition of the derivative.

  1. Apply the definition:
  2. Expand:

Answer:

4.35. Find the Sum of Series (Preparing for Final, Series 2)

Find the sum:

Click to see the solution

Key Concept: Use partial fractions and telescoping series.

  1. Use partial fractions:

    Solving: ,

  2. Write partial sum and take limit: The series telescopes, and after taking the limit:

Answer:

4.36. Limits Without L’Hôpital - Problem 5 (Preparing for Final, Limits 2)

Find the limit (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Consider left and right limits separately due to absolute value.

  1. Right limit (): For , we have , so:
  2. Left limit (): For , we have , so:
  3. Conclusion: Since the left and right limits differ, the limit does not exist.

Answer: Limit does not exist

4.37. Limits Without L’Hôpital - Problem 6 (Preparing for Final, Limits 2)

Find the limit (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Rationalize the numerator.

  1. Rationalize:
  2. Evaluate:

Answer:

4.38. Limits Without L’Hôpital - Problem 7 (Preparing for Final, Limits 2)

Find the limit (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Use the limit and .

  1. Rewrite:

  2. Evaluate:

    Therefore:

Answer:

4.39. Limits Without L’Hôpital - Problem 8 (Preparing for Final, Limits 2)

Find the limit (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Use substitution and known limits.

  1. Substitute : As , we have . Also, .
  2. Rewrite:

Answer:

4.40. Limits Without L’Hôpital - Problem 9 (Preparing for Final, Limits 2)

Find the limit (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Use substitution and known limits.

  1. Substitute : As , we have .
  2. Rewrite:

Answer:

4.41. Limits Without L’Hôpital - Problem 10 (Preparing for Final, Limits 2)

Find the limit (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Rewrite in terms of sine and cosine, then use known limits.

  1. Rewrite:

  2. Use known limits:

    Therefore:

Answer:

4.42. Limits Without L’Hôpital - Problem 11 (Preparing for Final, Limits 2)

Find the limit (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Factor and use known limits.

  1. Factor:
  2. Use known limits:
    • , so
    Therefore:

Answer:

4.43. Limits Without L’Hôpital - Problem 12 (Preparing for Final, Limits 2)

Find the limit (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Rewrite in terms of sine and cosine, then use known limits.

  1. Rewrite:

  2. Use known limits:

    Therefore:

Answer:

4.44. Logarithmic Differentiation (Preparing for Final, Differentiation 2)

Find the derivative of the function for all .

Click to see the solution

Key Concept: Use logarithmic differentiation.

  1. Take natural logarithm:
  2. Differentiate both sides:
  3. Solve for :

Answer:

4.45. Find the Sum of Series (Preparing for Final, Series 3)

Find the sum:

Click to see the solution

Key Concept: This is a telescoping series.

  1. Use partial fractions:
  2. Write partial sum:
  3. Take limit:

Answer:

4.46. Limits at Infinity - Problem 1 (Preparing for Final, Limits 3)

Find (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Divide by the dominant exponential term.

  1. Divide numerator and denominator by :

  2. Evaluate: As : and

    Therefore:

Answer:

4.47. Limits at Infinity - Problem 2 (Preparing for Final, Limits 3)

Find (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Divide by the dominant exponential term. As , and approach slowly.

  1. Divide numerator and denominator by :

  2. Evaluate: As : and

    Therefore:

Answer:

4.48. Limits at Infinity - Problem 3 (Preparing for Final, Limits 3)

Find (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Substitute and use limit properties.

  1. Substitute : As , we have .
  2. Rewrite:
  3. Evaluate: Since grows much faster than as :

Answer:

4.49. Limits at Infinity - Problem 4 (Preparing for Final, Limits 3)

Find (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Divide by the highest power of in the denominator.

  1. Divide numerator and denominator by :

  2. Simplify exponents:

  3. Evaluate: As :

    • All other terms approach

    Therefore the limit is .

Answer:

4.50. Limits at Infinity - Problem 5 (Preparing for Final, Limits 3)

Find (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Rationalize the expression.

  1. Rationalize:
  2. Evaluate: As , both square roots approach , so:

Answer:

4.51. Limits at Infinity - Problem 6 (Preparing for Final, Limits 3)

Find (without using L’Hôpital’s Rule):

Click to see the solution

Key Concept: Rationalize the expression.

  1. Rationalize:

  2. Evaluate: As :

    Therefore:

Answer:

4.52. Taylor Polynomials for at (Preparing for Final, Differentiation 3)

Find the Taylor Polynomials of orders 1, 2, and 3 generated by at .

Click to see the solution

Key Concept: Compute derivatives and use Taylor polynomial formula.

  1. Find derivatives:
    • , so
    • , so
    • , so
    • , so
  2. Construct Taylor polynomials:

Answer:

4.53. Taylor Polynomials for at (Preparing for Final, Differentiation 3)

Find the Taylor Polynomials of orders 1, 2, and 3 generated by at .

Click to see the solution

Key Concept: Compute derivatives at .

  1. Find derivatives at :
    • , so
    • , so
    • , so
  2. Construct Taylor polynomials:

Answer:

4.54. Taylor Polynomials for at (Preparing for Final, Differentiation 3)

Find the Taylor Polynomials of orders 1, 2, and 3 generated by at .

Click to see the solution

Key Concept: Compute derivatives at .

  1. Find derivatives:
    • , so
    • , so
    • , so
    • , so
  2. Construct Taylor polynomials:

Answer:

4.55. Find the Sum of Series (Preparing for Final, Series 4)

Find the sum:

Click to see the solution

Key Concept: This is a telescoping series.

  1. Write partial sum:

  2. Take limit: Since and , we have:

Answer: (for all )

4.56. Limits with Indeterminate Powers - Problem 1 (Preparing for Final, Limits 4)

Find the limit:

Click to see the solution

Key Concept: This is type . Use the identity .

  1. Rewrite:

  2. Find limit of exponent:

    (Using as )

  3. Compute the limit:

Answer:

4.57. Limits with Indeterminate Powers - Problem 2 (Preparing for Final, Limits 4)

Find the limit:

Click to see the solution

Key Concept: Type . Use exponential form.

  1. Rewrite:

  2. Find limit of exponent:

    Using for small :

    Therefore:

Answer:

4.58. Limits with Indeterminate Powers - Problem 3 (Preparing for Final, Limits 4)

Find the limit:

Click to see the solution

Key Concept: Type . Use exponential form.

  1. Rewrite:

  2. Find limit of exponent:

    Using as :

  3. Compute the limit:

Answer:

4.59. Limits with Indeterminate Powers - Problem 4 (Preparing for Final, Limits 4)

Find the limit:

Click to see the solution

Key Concept: Type . Use exponential form and substitution.

  1. Substitute : As , we have . Also, .

  2. Rewrite:

  3. Use exponential form:

  4. Find limit of exponent: As : , , and

    Therefore:

  5. Compute the limit:

Answer:

4.60. Taylor Series for at (Preparing for Final, Differentiation 4)

Find the Taylor series of near .

Click to see the solution

Key Concept: Use geometric series or find derivatives.

  1. Method 1: Geometric series:

    Valid for .

  2. Method 2: Derivatives: , so

    Taylor series:

Answer: for

4.61. Taylor Series for at (Preparing for Final, Differentiation 4)

Find the Taylor series of near .

Click to see the solution

Key Concept: Use geometric series with substitution.

  1. Rewrite:
  2. Use geometric series: For , i.e., :

Answer: for

4.62. Taylor Series for at (Preparing for Final, Differentiation 4)

Find the Taylor series of near .

Click to see the solution

Key Concept: Use geometric series with substitution.

  1. Rewrite:
  2. Use geometric series: For , i.e., :

Answer: for

4.63. Taylor Series for at (Preparing for Final, Differentiation 4)

Find the Taylor series of near .

Click to see the solution

Key Concept: Use known series for and multiply by .

  1. Use known series:

  2. Multiply by :

    Or reindexing:

Answer:

4.64. Taylor Series for at (Preparing for Final, Differentiation 4)

Find the Taylor series of near .

Click to see the solution

Key Concept: Use substitution and known series.

  1. Substitute : Then and:

  2. Use series for :

  3. Multiply:

    Substituting back : (with convention that )

Answer: (where )

4.65. Taylor Series for at (Preparing for Final, Differentiation 4)

Find the Taylor series of near .

Click to see the solution

Key Concept: Use known series for and multiply.

  1. Use known series:
  2. Multiply by :

Answer:

4.66. Taylor Series for at (Preparing for Final, Differentiation 4)

Find the Taylor series of near .

Click to see the solution

Key Concept: Use substitution and known series.

  1. Substitute : Then and:

  2. Use series for and :

  3. Multiply and combine:

    Substituting back :

Answer:

4.67. Find the Sum of Series (Preparing for Final, Series 5)

Find the sum:

Click to see the solution

Key Concept: This is a telescoping series.

  1. Write partial sum:

  2. Take limit:

    Therefore:

Answer:

4.68. Limit Using Taylor Series - Problem 1 (Preparing for Final, Differentiation 5)

Find the limit:

Click to see the solution

Key Concept: Use Taylor series expansions. (This problem was solved in Example 4.13, answer is .)

  1. Expand each function:
  2. Compute numerator:
  3. Compute the limit:

Answer:

4.69. Limit Using Taylor Series - Problem 2 (Preparing for Final, Differentiation 5)

Find the limit:

Click to see the solution

Key Concept: Use Taylor series expansions. (This problem was solved in Lab 15 Problem 7, but answer given is .)

  1. Expand each function:
    • , so
  2. Compute numerator:
  3. Compute denominator:
  4. Compute the limit:

Answer:

4.70. Find the Sum of Series (Preparing for Final, Series 6)

Find the sum:

Click to see the solution

Key Concept: This is a geometric series.

  1. Rewrite:

  2. Use geometric series formula: For :

    Here , and , so:

Answer:

4.71. Sketch Graph - Problem 1 (Preparing for Final, Differentiation 6)

Sketch the graph of:

Click to see the solution

Key Concept: This problem was solved in detail in Example 4.17. See that solution for complete analysis.

Answer: See Example 4.17 for complete solution. Key features: horizontal asymptote , local min at , local max at , inflection points at .

4.72. Sketch Graph - Problem 2 (Preparing for Final, Differentiation 6)

Sketch the graph of:

Click to see the solution

Key Concept: This problem was solved in detail in Example 4.18. See that solution for complete analysis.

Answer: See Example 4.18 for complete solution. Key features: vertical asymptote at (approaching from right, from left), horizontal asymptote on both sides, always decreasing, inflection point at .

4.73. Sketch Graph - Problem 3 (Preparing for Final, Differentiation 6)

Sketch the graph of:

Click to see the solution

Key Concept: This problem was solved in detail in Example 4.19. See that solution for complete analysis.

Answer: See Example 4.19 for complete solution. Key features: cusp at origin , local max at , crosses -axis at and .

4.74. Sketch Graph - Problem 4 (Preparing for Final, Differentiation 6)

Sketch the graph of:

Click to see the solution

Key Concept: Analyze domain, asymptotes, derivatives, and behavior.

  1. Domain:

  2. Asymptotes:

    • Vertical:
    • Oblique: Perform division: , so is oblique asymptote
    • No horizontal asymptote

  3. Derivatives and critical points:

    Critical points:

  4. Behavior:

    • is increasing on , decreasing on , increasing on , decreasing on , increasing on , decreasing on
    • Local max at and , local min at

Answer: See sketch with vertical asymptotes at , oblique asymptote , local maxima at , local minimum at .

4.75. Sketch Graph - Problem 5 (Preparing for Final, Differentiation 6)

Sketch the graph of:

Click to see the solution

Key Concept: Analyze domain, asymptotes, derivatives, and behavior.

  1. Domain: (due to )

  2. Asymptotes:

    • Vertical: (as , )
    • As : (since dominates)

  3. Derivatives:

    The numerator has discriminant , so for all .

    Therefore is increasing on .

  4. Second derivative:

    Find inflection points where .

Answer: See sketch with vertical asymptote at , always increasing, going to as .

4.76. Sketch Graph - Problem 6 (Preparing for Final, Differentiation 6)

Sketch the graph of:

Click to see the solution

Key Concept: This problem was solved in detail in Lab 15 Problem 8 (Example 4.27). See that solution for complete analysis.

Answer: See Example 4.27 for complete solution. Key features: even function, defined on , decreasing on , increasing on , horizontal asymptote at both ends, maximum value at .

4.77. Find the Sum of Series (Preparing for Final, Series 7)

Find the sum:

Click to see the solution

Key Concept: Split into two geometric series.

  1. Split the series:

  2. Use geometric series formula: For :

Answer:

4.78. Find the Sum of Series (Preparing for Final, Series 8)

Find the sum:

Click to see the solution

Key Concept: This is a geometric series.

  1. Rewrite:
  2. Use geometric series formula: Since :

Answer:

4.79. Find the Sum of Series (Preparing for Final, Series 9)

Find the sum:

Click to see the solution

Key Concept: This is a geometric series.

  1. Rewrite:
  2. Use geometric series formula: Since :

Answer:

4.80. Find the Sum of Series (Preparing for Final, Series 10)

Find the sum:

Click to see the solution

Key Concept: This is a geometric series.

  1. Rewrite:

  2. Use geometric series formula: For :

    Since , we have , so:

Answer: (for all )

4.81. Find the Sum of Series (Preparing for Final, Series 11)

Find the sum:

Click to see the solution

Key Concept: Split the sum and use the fact that .

  1. Split the sum:

  2. Evaluate each sum:

    • For :
    • For :

    Therefore:

    And:

  3. Compute the sum:

Answer:

4.82. Find the Sum of Series (Preparing for Final, Series 12)

Find the sum:

Click to see the solution

Key Concept: Split the sum and use properties of the exponential series.

  1. Split the sum:

  2. Evaluate : Write :

    • For :
    • For :

    Therefore:

    And from the previous problem:

    So:

  3. Compute the sum:

Answer:

4.83. Find the Sum of Series (Preparing for Final, Series 13)

Find the sum:

Click to see the solution

Key Concept: Rewrite the summand and use properties of the exponential series.

  1. Change index: Let , so :

  2. Split the sum:

  3. Use known results: From previous problems:

  4. Compute the sum:

Answer:

4.84. Study Convergence - Part (a) - Problem 1 (Preparing for Final, Convergence a)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Rationalize and check if the terms approach zero.

  1. Rationalize:
  2. Compare with harmonic series: Since as , and diverges (p-series with ), the series diverges.

Answer: Diverges

4.85. Study Convergence - Part (a) - Problem 2 (Preparing for Final, Convergence a)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Rationalize and check convergence.

  1. Rationalize:
  2. Check if terms approach zero: As , , so the terms do not approach zero.

Answer: Diverges (terms do not approach zero)

4.86. Study Convergence - Part (a) - Problem 3 (Preparing for Final, Convergence a)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Use limit comparison test with a known series.

  1. Find limit of terms: As , , so .
  2. Since terms do not approach zero: The series diverges by the divergence test.

Answer: Diverges (terms approach 1, not zero)

4.87. Study Convergence - Part (a) - Problem 4 (Preparing for Final, Convergence a)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Find the general term and use comparison test.

  1. Identify the pattern: Numerators: Denominators:

    The denominator appears to be (check: , , etc.)

  2. General term:

  3. Check limit:

  4. Conclusion: Since terms do not approach zero, the series diverges.

Answer: Diverges

4.88. Study Convergence - Part (b) - Problem 1 (Preparing for Final, Convergence b)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Use the integral test or comparison with harmonic series.

  1. Use integral test: Consider for .

    (Substitution: , )

  2. Conclusion: Since the integral diverges, the series diverges.

Answer: Diverges

4.89. Study Convergence - Part (b) - Problem 2 (Preparing for Final, Convergence b)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Use ratio test or comparison with geometric series.

  1. Rewrite:
  2. Use ratio test:
  3. Conclusion: The series converges by the ratio test.

Answer: Converges

4.90. Study Convergence - Part (b) - Problem 3 (Preparing for Final, Convergence b)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Use comparison test with geometric series.

  1. Compare with geometric series: For large , dominates, so:

  2. Use ratio test on :

    So converges.

  3. Conclusion: By comparison test, the original series converges.

Answer: Converges

4.91. Study Convergence - Part (b) - Problem 4 (Preparing for Final, Convergence b)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Use comparison test with geometric series.

  1. Compare: For large , dominates , so:
  2. Since converges (geometric with ): The original series converges by limit comparison test.

Answer: Converges

4.92. Study Convergence - Part (b) - Problem 5 (Preparing for Final, Convergence b)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Use comparison test with p-series.

  1. Compare with p-series: For , we have (for sufficiently large ), so:
  2. Since converges (p-series with ): The original series converges by comparison test.

Answer: Converges

4.93. Study Convergence - Part (b) - Problem 6 (Preparing for Final, Convergence b)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Use comparison test with geometric series.

  1. Compare: For all :
  2. Since converges (geometric with ): The original series converges by comparison test.

Answer: Converges

4.94. Study Convergence - Part (b) - Problem 7 (Preparing for Final, Convergence b)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Simplify and use comparison test.

  1. Simplify: For large , , so:
  2. Compare:
  3. Since diverges (harmonic series): The original series diverges by limit comparison test.

Answer: Diverges

4.95. Study Convergence - Part (c) - Problem 1 (Preparing for Final, Convergence c)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Use limit comparison test with harmonic series.

  1. Compare with harmonic series:
  2. Since diverges and the limit is positive: The original series diverges by limit comparison test.

Answer: Diverges

4.96. Study Convergence - Part (c) - Problem 2 (Preparing for Final, Convergence c)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Use comparison test with p-series.

  1. For large :
  2. Since converges (p-series with ): The original series converges by limit comparison test.

Answer: Converges

4.97. Study Convergence - Part (c) - Problem 3 (Preparing for Final, Convergence c)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Simplify and use p-series test.

  1. Simplify:
  2. Since converges (p-series with ): The original series converges by limit comparison test.

Answer: Converges

4.98. Study Convergence - Part (c) - Problem 4 (Preparing for Final, Convergence c)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Simplify and use p-series test.

  1. For large :
  2. Since diverges (harmonic series): The original series diverges by limit comparison test.

Answer: Diverges

4.99. Study Convergence - Part (c) - Problem 5 (Preparing for Final, Convergence c)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Rationalize and check convergence.

  1. Rationalize:
  2. For large :
  3. Since diverges: The original series diverges by limit comparison test.

Answer: Diverges

4.100. Study Convergence - Part (d) - Problem 1 (Preparing for Final, Convergence d)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Use ratio test.

  1. Apply ratio test:
  2. Conclusion: The series converges by the ratio test.

Answer: Converges

4.101. Study Convergence - Part (d) - Problem 2 (Preparing for Final, Convergence d)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Use ratio test.

  1. Apply ratio test:
  2. Conclusion: The series converges by the ratio test.

Answer: Converges

4.102. Study Convergence - Part (d) - Problem 3 (Preparing for Final, Convergence d)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Use ratio test. Note that .

  1. Apply ratio test:
  2. Conclusion: The series converges by the ratio test.

Answer: Converges

4.103. Study Convergence - Part (d) - Problem 4 (Preparing for Final, Convergence d)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Use root test.

  1. Apply root test:

    (Since grows much faster than )

  2. Conclusion: The series converges by the root test.

Answer: Converges

4.104. Study Convergence - Part (d) - Problem 5 (Preparing for Final, Convergence d)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Simplify and use ratio test.

  1. Simplify:
  2. Use ratio test:
  3. Conclusion: The series converges by the ratio test.

Answer: Converges

4.105. Study Convergence - Part (d) - Problem 6 (Preparing for Final, Convergence d)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Check for absolute convergence first, then conditional convergence.

  1. Check absolute convergence:

    This diverges (from Problem 4.52).

  2. Check conditional convergence using alternating series test: Let .

    • for all
    • Need to check if is decreasing: for sufficiently large

    Since for large , the series converges conditionally.

Answer: Conditionally convergent

4.106. Study Convergence - Part (d) - Problem 7 (Preparing for Final, Convergence d)

Study the convergence or divergence of:

Click to see the solution

Key Concept: Use alternating series test and integral test for absolute convergence.

  1. Check absolute convergence:

    Using integral test with :

    So the series does not converge absolutely.

  2. Check conditional convergence using alternating series test: Let .

    • for all
    • is decreasing (since is decreasing for )

    The series converges conditionally.

Answer: Conditionally convergent

4.107. Study Convergence - Part (d) - Problem 8 (Preparing for Final, Convergence d)

Study the convergence or divergence of:

Click to see the solution

Key Concept: This is an alternating harmonic series.

  1. Check absolute convergence:

    This diverges (harmonic series).

  2. Check conditional convergence: The series is times the alternating harmonic series, which converges by the alternating series test.

Answer: Conditionally convergent

4.108. Study Convergence - Part (e) - Problem 1 (Preparing for Final, Convergence e)

Study the absolute and conditional convergence of:

Click to see the solution

Key Concept: Check absolute convergence using ratio test.

  1. Check absolute convergence:
  2. Use ratio test:
  3. Conclusion: The series converges absolutely.

Answer: Absolutely convergent

4.109. Study Convergence - Part (e) - Problem 2 (Preparing for Final, Convergence e)

Study the absolute and conditional convergence of:

Click to see the solution

Key Concept: Use root test for absolute convergence.

  1. Check absolute convergence:
  2. Use root test:
  3. Conclusion: The series converges absolutely.

Answer: Absolutely convergent

4.110. Study Convergence - Part (e) - Problem 3 (Preparing for Final, Convergence e)

Study the absolute and conditional convergence of:

Click to see the solution

Key Concept: Check absolute convergence first.

  1. Check absolute convergence:

    For large :

    Since diverges, the series does not converge absolutely.

  2. Check conditional convergence: The terms do not approach zero monotonically, so the alternating series test may not apply directly. However, we can check:

    But the terms are not decreasing, so we need a different approach. The series diverges because the absolute value series diverges and the terms don’t alternate in a way that would allow cancellation to produce convergence.

Answer: Diverges

4.111. Analyze Convergence - Part (f) (Preparing for Final, Convergence f)

Find, using the Cauchy product, the general term of the series , prove that this series is convergent and find its sum.

Click to see the solution

Key Concept: Use Cauchy product formula for multiplying series.

  1. Rewrite the series:

  2. Find the Cauchy product: Let .

    The Cauchy product is:

  3. The squared series:

  4. Prove convergence: Use ratio test:

    The series converges.

  5. Find the sum: Since , we have:

Answer: General term: ; The series converges and its sum is .

4.112. Power Series - Radius and Interval of Convergence 1 (Preparing for Final, Power Series)

For the power series, find the radius of convergence and the interval of convergence:

Click to see the solution

Key Concept: Use ratio test to find radius of convergence, then check endpoints.

  1. Apply ratio test:

  2. Find radius: For convergence, we need , so , which means .

    Radius of convergence: .

  3. Check endpoints:

    • When :
    • When :

    Check convergence at these points using appropriate tests.

Answer: Radius: ; Interval: Check endpoints and .

4.113. Power Series - Radius and Interval of Convergence 2 (Preparing for Final, Power Series)

For the power series, find the radius of convergence and the interval of convergence:

Click to see the solution

Key Concept: Use ratio test, then check endpoints.

  1. Apply ratio test:

  2. Find radius: For convergence: , so .

    Radius: .

  3. Check endpoints:

    • When : → series becomes (diverges)
    • When : → series becomes (converges conditionally)

Answer: Radius: ; Interval:

4.114. Power Series - Radius and Interval of Convergence 3 (Preparing for Final, Power Series)

For the power series, find the radius of convergence and the interval of convergence:

Click to see the solution

Key Concept: Use ratio test.

  1. Apply ratio test:

  2. Find radius: For convergence: .

    Radius: .

  3. Check endpoints:

    • : diverges (terms → 1)
    • : diverges (terms don’t approach zero)

Answer: Radius: ; Interval:

4.115. Power Series - Radius and Interval of Convergence 4 (Preparing for Final, Power Series)

For the power series, find the radius of convergence and the interval of convergence:

Click to see the solution

Key Concept: Use ratio test.

  1. Apply ratio test:

  2. Find radius: For convergence: .

    Radius: .

  3. Check endpoints:

    • : converges (p-series, )
    • : converges absolutely

Answer: Radius: ; Interval:

4.116. Power Series - Radius and Interval of Convergence 5 (Preparing for Final, Power Series)

For the power series, find the radius of convergence and the interval of convergence:

Click to see the solution

Key Concept: This is a power series in . Use substitution.

  1. Substitute : The series becomes .

  2. Apply ratio test:

  3. Find radius: For convergence: , so , which means , so .

    Radius: .

  4. Check endpoints:

    • : , series becomes (diverges)

Answer: Radius: ; Interval:

4.117. Power Series - Radius and Interval of Convergence 6 (Preparing for Final, Power Series)

For the power series, find the radius of convergence and the interval of convergence:

Click to see the solution

Key Concept: Use ratio test.

  1. Apply ratio test:

  2. Find radius: For convergence: .

    Radius: .

  3. Check endpoints:

    • : diverges (limit comparison with harmonic series)
    • : converges conditionally

Answer: Radius: ; Interval:

4.118. Power Series - Radius and Interval of Convergence 7 (Preparing for Final, Power Series)

For the power series, find the radius of convergence and the interval of convergence:

Click to see the solution

Key Concept: Use ratio test.

  1. Simplify:

  2. Apply ratio test:

  3. Find radius: For convergence: , so .

    Radius: .

  4. Check endpoints: Check .

Answer: Radius: ; Interval: Check endpoints .

4.119. Power Series - Radius and Interval of Convergence 8 (Preparing for Final, Power Series)

For the power series, find the radius of convergence and the interval of convergence:

Click to see the solution

Key Concept: Use root test, noting that .

  1. Apply root test:

  2. Find radius: For convergence: , so .

    Radius: .

  3. Check endpoints:

    • : Terms approach , so diverges
    • : Terms don’t approach zero, so diverges

Answer: Radius: ; Interval:

4.120. Find Asymptotes - Problem 1 (Preparing for Final, Asymptotes)

Find the equations of the asymptotes (horizontal, vertical, oblique) of the graph of:

Click to see the solution

Key Concept: Check horizontal, vertical, and oblique asymptotes.

  1. Horizontal asymptotes:

    Horizontal asymptotes: (as ) and (as ).

  2. Vertical asymptotes: The denominator is never zero, so no vertical asymptotes.

  3. Oblique asymptotes: Since horizontal asymptotes exist, there are no oblique asymptotes.

Answer: Horizontal asymptotes: (as ), (as ); No vertical or oblique asymptotes.

4.121. Find Asymptotes - Problem 2 (Preparing for Final, Asymptotes)

Find the equations of the asymptotes (horizontal, vertical, oblique) of the graph of:

Click to see the solution

Key Concept: Check all types of asymptotes.

  1. Vertical asymptote: when . Check:

    Vertical asymptote: .

  2. Horizontal asymptote:

    No horizontal asymptote.

  3. Oblique asymptote: Perform polynomial division:

    As , the remainder , so:

    Oblique asymptote: .

Answer: Vertical asymptote: ; Oblique asymptote: ; No horizontal asymptote.

4.122. Find Asymptotes - Problem 3 (Preparing for Final, Asymptotes)

Find the equations of the asymptotes (horizontal, vertical, oblique) of the graph of:

Click to see the solution

Key Concept: Check all types of asymptotes.

  1. Vertical asymptotes: Factor denominator:

    Check each:

    • : (vertical asymptote)
    • : (removable discontinuity, not asymptote)
    • : Similar check shows vertical asymptote

    Vertical asymptotes: and .

  2. Horizontal asymptote:

    Horizontal asymptote: .

  3. Oblique asymptote: Since horizontal asymptote exists, no oblique asymptote.

Answer: Vertical asymptotes: , ; Horizontal asymptote: ; No oblique asymptote.