Calculus | mathuniversity

CALCULUS INFO

/ LIMITS & CONTINUITY

UNIT LESSON OUTLINE

Intermediate Value Thm

01

This first topic in calculus deals with an in-depth review of the representations, types, and properties of functions as discussed in previous courses.

With this strong foundation established, we will explore the concept of limits and how to estimate and evaluate limits using the various representations of functions. Also, students will use limits to investigate continuity as a property of functions and the different types of discontinuities.

02

Derivatives represent an instantaneous rate of change, or the slope of the tangent line. This topic builds upon the last by defining a derivative as the limit of the average rate of change. We will discuss basic rules to determine derivatives of algebraic, trigonometric, and rational functions at a single point and as a function. We will also find derivatives of more complex functions and use derivatives to write equations of tangent lines. Finally we will discuss differentiability and how it relates to theorems like the Mean Value Theorem.

/ DERIVATIVES

Rates of Change

Definition & Power Rule

Converting Limit Definition

Product & Quotient Rule

Chain Rule

Implicit Differentiation

Tangent Lines

Higher Order Derivatives

Inverse Functions & Tables

Differentiability

Mean Value Theorem

UNIT LESSON OUTLINE

Position, Velocity, Acceleration [10/27]

Related Rates [10/31]

Absolute Extrema [11/4]

Function Behavior [11/9]

Function Concavity [11/14]

Curve Sketching [11/16]

Optimization [12/1]

Approximation [12/5]

L'Hopital's Rule [12/6]

/ APPLICATIONS OF DERIVATIVES

Now that students have discovered the meaning of a derivative and how to calculate them for various types of functions, they will use derivatives to discover properties of functions and solve problems in real-world contexts. We will solve word problems involving position, velocity, and acceleration; related rates; and optimization. We will also make connections between the graphs of functions and their derivatives. Finally, students will explore slope fields and how to approximate values using differentials and tangent lines.

03

UNIT LESSON OUTLINE

Logarithm & Exponential [1/19]

Log & Ex Derivatives [1/24]

Indefinite Integrals [1/26]

Initial Conditions [1/30]

Substitution [2/3]

Slope Fields [2/7]

Approximating Area [2/15]

Definite Integrals [3/8]

Fundamental Theorem [3/14]

Other Integral Theorems [3/16]

Definite Substitution [3/20]

Log & Ex Integrals [3/22]

/ INTEGRALS

An indefinite integral is also called an antiderivative because it is simple the opposite process of differentiation. Instead of being given the function and finding the derivative, we are given the derivative and must find the original function. Integration can be used to find a general solution to a differential equation, which can be graphed as a slope field, or given an initial condition we can find a particular solution representing a single curve.

A definite integral represents the area between two curves, and can be defined as the limit of Riemann sums which are used to approximate the area beneath a curve.