**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 *e*^{x}

**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