Formulas

Complex Numbers

• Imaginary Unit:
• Standard Form of a Complex Number:
• Modulus of a Complex Number:
• Argument of a Complex Number:
• Equation of a Circle in the Complex Plane:
• Trigonometric Form:
• Euler’s Formula:
• Exponential (Polar) Form:
• System for Square Roots of :
• De Moivre’s Theorem (Powers):
• De Moivre’s Theorem (Roots): for .

Functions

• Even Part of a Function:
• Odd Part of a Function:
• Function Composition:
• Finding an Inverse Function:

Trigonometry

• Pythagorean Identity:
• Inverse Trigonometric Identity:
• Inverse Sine as an Odd Function:
• Inverse Reciprocal Identity:

Hyperbolic Functions

• Hyperbolic Sine Definition:
• Hyperbolic Cosine Definition:
• Hyperbolic Identity:
• Hyperbolic-Inverse Trig Identities:
• Logarithmic Form of arsinh:
• Logarithmic Form of arcosh:
• Logarithmic Form of artanh:
• Logarithmic Form of arcsch:
• Logarithmic Form of arsech:
• Logarithmic Form of arcoth:

Algebra and Inequalities

• Binomial Theorem:
• Sum of Cubes:
• Triangle Inequality:
• Bernoulli’s Inequality: For and integer , .

Sequences and Limits

• Arithmetic Sequence (n-th term):
• Arithmetic Sequence (Sum):
• Geometric Sequence (n-th term):
• Geometric Sequence (Sum):
• Limit Definition:
• Divergence to Infinity Definition:
• Squeeze Theorem: If for and , then .
• Sum Rule for Limits:
• Constant Multiple Rule for Limits:
• Product Rule for Limits:
• Quotient Rule for Limits:
• The Number e:
• Limit Definition of e:
• Ratio Test for Sequences: Given , if then .
• Cauchy Sequence Definition: For every , there exists an such that for any , we have .
• Recursive Limit Evaluation: If and where is continuous, then .

Series

• Series Notation:
• nth Partial Sum:
• Sum of a Convergent Series:
• General Telescoping Series Sum: If , then .
• Mengoli Series:
• Generalized Telescoping Sum:
• Geometric Series Sum: For , .
• Harmonic Series Divergence:
• -Series Convergence: converges if and only if .
• Linear Property of Series:
• Necessary Condition for Convergence: If converges, then .
• nth-Term Test for Divergence: If or the limit does not exist, then diverges.
• Direct Comparison Test: If , then: (1) converges converges; (2) diverges diverges.
• Limit Comparison Test: If where , then and have the same behavior.
• Ratio Test: Let . If , the series converges absolutely. If , the series diverges. If , inconclusive.
• Root Test: Let . If , the series converges absolutely. If , the series diverges. If , inconclusive.
• Leibniz Test: If is non-increasing and , then converges.
• Absolute Convergence Test: If converges, then converges.
• Alternating Harmonic Series Sum:
• Cauchy Product: where

Power Series

• Power Series Form:
• Radius of Convergence (Ratio Test):
• Radius of Convergence (Root Test):

Function Limits and Continuity

• Limit Definition (-):
• Right-Hand Limit:
• Left-Hand Limit:
• Limit Existence:
• Limit at Infinity:
• Infinite Limit:
• Sum Rule for Function Limits:
• Product Rule for Function Limits:
• Quotient Rule for Function Limits: (if denominator )
• Power Rule for Function Limits:
• Root Rule for Function Limits: (with domain restrictions)
• Polynomial Limit: If is a polynomial, then
• Rational Function Limit: if
• Sandwich (Squeeze) Theorem: If near and , then
• Important Trigonometric Limit: ( in radians)
• Related Trigonometric Limits: ,
• Tangent Limit:
• Exponential Limit (Form 1):
• Exponential Limit (Form 2):
• General Exponential Limit:
• Exponential-Logarithmic Limit: (for )
• Indeterminate Form : If and , then
• Continuity at a Point: is continuous at if
• Composition of Continuous Functions: If is continuous at and is continuous at , then is continuous at
• Limit of Continuous Composition: If and is continuous at , then
• Intermediate Value Theorem: If is continuous on and is between and , then such that

Differentiation

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 (at ):
  • 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