Topics
Pre Calculus Review
Prerequisites for Calculus: Precalculus Review
Average rate of change
Elementary functions and graphs
Function arithmetic and composition
Limits
Concept of limit
Finding limit graphically and numerically
Properties of limits
Intermediate theorem
Squeeze theorem
Introduction to Derivative
Concepts and types of continuity and discontinuity
Average and instantaneous rate of change
Definition of derivative
Computing derivative function from the definition
Local linearity
Derivative at a given point
Secant and tangent line at a given point
Derivative as a rate of change
Estimating the average and instantaneous rate of change from data and it graphs
Motion, position, velocity and acceleration functions
Differentiability and continuity
Power rule
Product rule
Quotient rule
Chain rule
Special Functions
Trigonometric function, exponential functions, and logarithmic functions
Graphs of trigonometric function, exponential functions, and logarithmic functions
Derivatives of trigonometric Functions, exponential functions, and logarithmic functions
Derivative of ln x and ex
Implicit Differentiation and Calculus of Inverse Function
Finding derivative implicitly
Differentiation notation
Inverse functions: exponential and logarithmic functions
Calculus of Inverse functions
Inverse trigonometric functions
Calculus of Inverse trigonometric functions
Calculus of Hyperbolic functions
Comparing graphs of f and f '
Practical Application of the Derivative
Position, velocity, and acceleration
Critical points
Finding extrema and increasing and decreasing behavior of a curve
Local and absolute extreme values
Higher derivative and linear approximation
Newton’s method
Related rates
Optimization
Curve Sketching
Finding critical points, local minimum and maximum on a curve
Comparing the graphs of f , f ' and f ''
The first and second derivative tests
Concavity and inflection points
Estimating derivative from data and graphs
Asymptotes and infinite limits
Intermediate value theorem
Mean value theorem
Rolle’s theorem
The Basics of Integration
Antidifferentiation
Antiderivatives of Powers of x
Antiderivatives of Trigonometric and Exponential
Functions
Undoing the Chain Rule
Integrating Polynomials by Substitution
Integrating Composite Trigonometric Functions by
Substitution
Integrating Composite Exponential and Rational
Functions by Substitution
More Integrating Trigonometric Functions by
Substitution
Choosing Effective Function Decompositions
Approximating Areas of Plane Regions
Areas, Riemann Sums, and Definite Integrals
The Fundamental Theorem of Calculus, Part II
Illustrating the Fundamental Theorem of Calculus
Evaluating Definite Integrals
An Introduction to the Integral Table
Deriving the Trapezoidal Rule
An Example of the Trapezoidal Rule
Applications of Integration
Motion of a particle: net and total distance traveled
Gravity and vertical motion
Area bounded by curves
Average value of a function
Volume using cross-section
Disk method
Washer method
Shell method
Arch length
Work and Hooke’s Law
Moments and center of mass
Separable differential equations
Direction fields
Growth and decay problems
Logistic growth
Exponential growth and radioactive decay