Overview

Learning Outcomes

Functions and limits

·       Use real numbers, inequalities involving real numbers and their absolute values, the trigonometric functions, and the radian measure of angles.

·       Be able to move back and forth between the descriptions of a function by an equation, a table, a graph, and by words.

·       Be able to use exponential functions, sketch their graphs, and define the number e.

·       Define what it means for a function to be one-to-one and determine whether a function has an inverse or not and sketch its inverse if it does.

·       Be able to use logarithmic functions, sketch their graphs, and define the relationship between the natural exponential and natural logarithmic functions.

·       State in words what it means for a function to have a limit, be able to calculate limits, and be able to find the vertical and horizontal asymptotes of a function.

·       State in words what it means for a function to be continuous and be able to find limits for continuous functions.

Derivatives

·       Relate the notions of tangent to a curve, velocity, and rate of change, and illustrate them in a sketch.

·       State the definition of derivative as the limit of a difference quotient and explain how the derivative itself can be regarded as a function.

·       Be able to find derivatives of polynomials and exponential functions.

·       State the product and quotient rules for differentiation and be able to use them to differentiate functions.

·       Know the derivatives of sine and cosine and be able to use the quotient rule to determine the derivatives of the remaining four trigonometric functions.

·       State the chain rule and use it to differentiate functions obtained by composition.

·       Use the differentiation rules to differentiate implicitly, and to find higher order derivatives.

·       Be able to differentiate logarithmic functions, and functions involving them.

·       Define the hyperbolic functions and be able to differentiate them.

Applications of Differentiation

·       Be able to solve related rates problems. Understand them as an application of the chain rule.

·       Understand the connection between the derivative, the tangent line to the graph of a function, the linearization of a function, and the differential of a function.

·       Use the differential (or linearization) to solve “small change” and applied approximation problems.

·       Be able to state the Mean Value Theorem and give some of its consequences.

·       Describe how the signs of the first and second derivatives of a function affect the shape of its graph.

·       Define and recognize the various forms of indeterminate forms, and use L”Hospital’s Rule to determine their limits.

·       Be able to set up and solve optimization problems using calculus methods.

·       Be able to describe Newton’s method geometrically, and to use it to iteratively approximate the zeros of functions.

·       Define what the antiderivative of a function is and be able to find it for reasonable functions.

Integral Calculus

·       Describe the connection between the problems of finding areas and distances travelled, and how both problems lead to the same limit.

·       Know and be able to work with the properties of definite integrals.

·       State the Fundamental Theorem of Calculus in words, describe how it connects integral and differential calculus, and how it helps in finding antiderivatives and in evaluating definite integrals.

·       Define the indefinite integral of a function and state its relation to the antiderivative.

·       Be able to use the Substitution Rule to evaluate definite and indefinite integrals.

·       Be able to use integration by parts to evaluate appropriate integrals.