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
Slide 1 Derivatives - Sign of Derivative Introduction to signs of derivatives Positive and negative derivatives Importance in analyzing functions