Slide 1

• Introduction to derivatives
• Definition of a derivative
• Notation for derivatives
• Importance of derivatives in calculus
• Applications of derivatives

Slide 2

• Basic rules of differentiation
• Constant rule
• Power rule
• Sum and difference rule
• Product rule
• Quotient rule

Slide 3

• Chain rule
• Derivatives of composite functions
• Application of chain rule
• Examples of chain rule
• Derivatives of trigonometric functions
• Derivatives of exponential and logarithmic functions

Slide 4

• Comparison between log(x) and ln(x)
• Definition of log(x)
• Definition of ln(x)
• Relationship between log(x) and ln(x)
• Differentiation of log(x)
• Differentiation of ln(x)

Slide 5

• Comparison between log(x) and ln(x) continued
• Example of differentiating log(x)
• Example of differentiating ln(x)
• Graphical representation of log(x) and ln(x) functions
• Properties of log(x) and ln(x) functions

Slide 6

• Common logarithms (log base 10)
• Natural logarithms (log base e)
• Change of base formula
• Applications of logarithms
• Logarithmic equations
• Logarithmic properties

Slide 7

• Review of exponential functions
• Definition of exponential functions
• Properties of exponential functions
• Differential equations involving exponential functions
• Examples of differentiating exponential functions
• Applications of exponential functions

Slide 8

• Growth and decay models
• Half-life and doubling time
• Compound interest
• Continuous compounding
• Applications of growth and decay models

Slide 9

• Implicit differentiation
• Definition of implicit differentiation
• Procedure for implicit differentiation
• Example of applying implicit differentiation
• Applications of implicit differentiation

Slide 10

• Higher order derivatives
• Definition of higher order derivatives
• Notation for higher order derivatives
• Relationship between higher order derivatives and concavity
• Examples of higher order derivatives
• Applications of higher order derivatives

Slide 11

• Limit definition of a derivative
• One-sided derivatives
• Relationship between limits and derivatives
• Example of finding the derivative using limit definition
• Applications of limit definition of derivatives

Slide 12

• Derivative as a slope
• Tangent lines and the derivative
• Finding the equation of a tangent line
• Example of finding the equation of a tangent line
• Applications of tangent lines and derivatives

Slide 13

• Derivatives of trigonometric functions
• Derivative of sin(x)
• Derivative of cos(x)
• Derivative of tan(x)
• Derivatives of other trigonometric functions

Slide 14

• Derivatives of logarithmic functions
• Derivative of log base a of x
• Derivative of ln(x)
• Example of finding the derivative of log base a of x
• Example of finding the derivative of ln(x)

Slide 15

• Antiderivatives and indefinite integrals
• Definition of antiderivatives
• Notation for antiderivatives
• Relationship between derivatives and antiderivatives
• Example of finding antiderivatives

Slide 16

• Basic rules of integration
• Constant rule for integration
• Power rule for integration
• Sum and difference rule for integration
• Examples of using basic rules for integration

Slide 17

• Integration by substitution
• Definition of integration by substitution
• Procedure for integration by substitution
• Example of using integration by substitution
• Applications of integration by substitution

Slide 18

• Techniques of integration
• Integration by parts
• Partial fraction decomposition
• Trigonometric substitutions
• Examples of using techniques of integration

Slide 19

• Definite integrals
• Definition of definite integrals
• Evaluation of definite integrals using antiderivatives
• Riemann sums and the definite integral
• Examples of evaluating definite integrals

Slide 20

• Applications of integration
• Area between curves
• Volume of revolution
• Average value of a function
• Differential equations and integration
• Applications in physics and engineering

Slide 21

• Techniques of integration continued
• Trigonometric substitutions
  • Substitutions using sin or cos
  • Substitutions using sec or csc
  • Substitutions using tan or cot
• Examples of using trigonometric substitutions
• Applications of trigonometric substitutions

Slide 22

• Definite integrals continued
• Fundamental Theorem of Calculus
  • Part 1: Evaluating definite integrals
  • Part 2: Finding antiderivatives
• Properties of definite integrals
• Examples of using the Fundamental Theorem of Calculus
• Applications of definite integrals

Slide 23

• Applications of integration continued
• Area between curves
  • Finding the area bounded by two curves
  • Finding the area between a curve and the x-axis
• Examples of finding the area between curves
• Volume of revolution
  • Finding the volume of a solid by rotating curves
• Examples of finding the volume of revolution

Slide 24

• Applications of integration continued
• Average value of a function
  • Definition of average value
  • Finding the average value using integration
• Examples of finding the average value of a function
• Differential equations and integration
  • Solving differential equations using integration
• Examples of solving differential equations using integration

Slide 25

• Applications of integration continued
• Applications in physics and engineering
  • Finding work and fluid force
  • Modeling growth and decay
  • Solving kinematic problems
• Examples of applications of integration in physics and engineering
• Summary of applications of integration

Slide 26

• Vectors in two and three dimensions
• Definition of vectors
• Components of vectors
• Addition and subtraction of vectors
• Scalar multiplication of vectors
• Examples of vector operations

Slide 27

• Dot product of vectors
• Definition of dot product
• Properties of dot product
• Finding the angle between vectors using dot product
• Applications of dot product
• Examples of dot product

Slide 28

• Cross product of vectors
• Definition of cross product
• Properties of cross product
• Finding the magnitude and direction of cross product
• Applications of cross product
• Examples of cross product

Slide 29

• Parametric equations and polar coordinates
• Definition of parametric equations
• Parametric equations in rectangular coordinates
• Converting between parametric equations and rectangular equations
• Definition of polar coordinates
• Converting between polar and rectangular coordinates

Slide 30

• Parametric equations and polar coordinates continued
• Graphing parametric equations
• Symmetry in parametric equations
• Graphing polar coordinates
• Converting polar equations to rectangular equations
• Examples of using parametric equations and polar coordinates