Slide 1

• Derivatives - Sign of Derivative
  • Introduction to signs of derivatives
  • Positive and negative derivatives
  • Importance in analyzing functions

Slide 2

• Calculating Derivatives
  • Using the limit definition
  • Rules of differentiation
  • Product rule, quotient rule, chain rule

Slide 3

• Derivative of Polynomial Functions
  • Derivatives of constant functions
  • Power rule for derivatives
  • Examples of differentiating polynomial functions

Slide 4

• Derivative of Trigonometric Functions
  • Derivatives of sin(x) and cos(x)
  • Derivatives of other trigonometric functions
  • Applications in physics and engineering

Slide 5

• Derivative of Exponential and Logarithmic Functions
  • Derivatives of e^x and ln(x)
  • Derivatives of other exponential and logarithmic functions
  • Applications in growth and decay problems

Slide 6

• Derivative of Composite Functions
  • Using the chain rule to differentiate composite functions
  • Examples of composite functions
  • Understanding nested functions

Slide 7

• Derivative of Implicit Functions
  • Implicit differentiation
  • Finding slopes of curves
  • Tangent and normal lines

Slide 8

• Higher Order Derivatives
  • Second and higher-order derivatives
  • Notation for higher-order derivatives
  • Understanding concavity and inflection points

Slide 9

• Applications of Derivatives
  • Optimization problems
  • Related rates problems
  • Graphical interpretation of derivatives

Slide 10

• Summary and Practice Problems
  • Recap of key concepts and formulas
  • Solving practice problems to reinforce understanding

Slide 11

• Higher Order Derivatives (continued)
  • Inflection points and concavity
  • Test for concavity using the second derivative test
  • Convex and concave upward functions
  • Examples of finding inflection points

Slide 12

• Optimization Problems
  • Finding maximum and minimum values
  • Using the first derivative test
  • Understanding critical points
  • Solving optimization problems using derivatives

Slide 13

• Related Rates Problems
  • Solving problems involving changing rates
  • Strategies for related rates problems
  • Setting up equations and differentiating
  • Solving for the desired rate of change

Slide 14

• Graphical Interpretation of Derivatives
  • Derivative as slope of tangent line
  • Relationship between derivative and graph
  • Identifying points of inflection
  • Analyzing increasing and decreasing intervals

Slide 15

• Summary and Practice Problems
  • Recap of key concepts and formulas
  • Solving practice problems to reinforce understanding
  • Reviewing common mistakes and misconceptions

Slide 16

• Integration - Introduction
  • Introduction to integration
  • Relationship between integration and differentiation
  • Importance in finding areas and calculating net change

Slide 17

• Definite Integrals
  • Definite integral as a limit of Riemann sums
  • Interpretation as area under a curve
  • Calculating definite integrals using antiderivatives
  • Applications in finding displacement and total distance

Slide 18

• Indefinite Integrals
  • Antiderivatives and indefinite integrals
  • General solution vs particular solution
  • Basic techniques for finding antiderivatives
  • Examples of finding indefinite integrals

Slide 19

• Integration by Substitution
  • Substitution rule for integrals
  • Selecting appropriate substitutions
  • Solving integrals using substitution
  • Examples and applications

Slide 20

• Integration Techniques
  • Integration by parts
  • Trigonometric substitutions
  • Partial fraction decomposition
  • Using tables and formulas for integrals

Slide 21

• Integration by Parts
  • Formula for integration by parts
  • Selecting which function to differentiate and which to integrate
  • Applying integration by parts to solve integrals
  • Examples of integration by parts

Slide 22

• Trigonometric Substitutions
  • When to use trigonometric substitutions
  • Common trigonometric identities
  • Applying trigonometric substitutions to solve integrals
  • Examples of trigonometric substitutions

Slide 23

• Partial Fraction Decomposition
  • Decomposing rational expressions into partial fractions
  • Common cases: linear factors, quadratic factors
  • Solving integrals using partial fraction decomposition
  • Examples of partial fraction decomposition

Slide 24

• Using Tables and Formulas for Integrals
  • Common integration formulas and tables
  • Using tables to quickly find antiderivatives
  • Applying theorems and properties to simplify integrals
  • Examples of using tables and formulas for integrals

Slide 25

• Applications of Integration
  • Area under a curve
  • Finding volumes of solids of revolution
  • Length of curves
  • Applications in physics and engineering

Slide 26

• Area Under a Curve
  • Relationship between definite integral and area
  • Graphical interpretation of definite integral
  • Finding areas of regions bounded by curves
  • Solving problems involving area under a curve

Slide 27

• Volumes of Solids of Revolution
  • Method of slicing
  • Washer and shell methods
  • Calculating volumes of solids of revolution
  • Examples of finding volumes using integrals

Slide 28

• Length of Curves
  • Arc length formula
  • Calculating length of curves
  • Parameterization and parametric equations
  • Examples of finding curve lengths

Slide 29

• Applications in Physics and Engineering
  • Work and energy problems
  • Center of mass and moments
  • Fluid pressure problems
  • Applications in electrical circuits

Slide 30

• Summary and Practice Problems
  • Recap of key concepts and techniques in integration
  • Solving practice problems to reinforce understanding
  • Reviewing common mistakes and misconceptions