Math 110: Exam 2

Here is some information about Exam 2. There are no cheat sheets allowed. Graphing Calculators are allowed. The question topics are as follows (all questions besides BONUS have equal weight):

1. Differentiation Rules.
2. True/False: Differentials, Related Rates, Chain Rule, Differentiability, Continuity
3. Visualization of the Derivative
4. Definition of the Derivative, Limits.
5. Chain Rule, Tangent Lines, Implicit Functions, Logarithmic Differentiation.

MATH 110

Ron Buckmire

Fall 2007

MATH 110

Ron Buckmire

Fall 2007

Study Guide for Exam II

The exam will cover material that appears on homeworks 13 through 24, Classes 13 through 24 and Quiz 4 through 7. I try to make the study guides complete, but there is no guarantee. You should review homeworks, homework solutions, quizzes, handouts, labs, and class notes.

This might change, but my current plan is not to allow calculators.

Topics

· Derivatives

o know all the different notations for derivative

o compute a derivative using the limit definition(s)

o use the definition of derivative to prove differentiation formulas (e.g.,). I will assume that you know the values of the limits and . In fact I might give you information about certain limits like , and have you use these to prove the derivative of an elementary function

o all the derivatives of elementary functions on the Derivatives To Remember handout

o compute using all the derivative formulas on the Rules of Differentiation handout

o Ability to evaluate or estimate numbers such as

o From graph of , sketch the graph of , given graph of sketch graph of

o Multiple meanings of Chain Rule: related rates, implicit differentiation and logarithmic diferentiation

o estimate derivatives with left estimates, right estimates, symmetric difference quotients, and tables of values (i.e., successive approximations).

o Equation of the line tangent to a function at a particular point

o Local linear approximation to estimate the value of a function near a value you already know such as sin(3) (i.e. near π) or (near =2).

o compute higher-order derivatives such as

o the units on a derivative are always

o Physical interpretation of derivative as the rate of change of the output relative to the input

o Know that when the function is increasing(decreasing), the derivative function is positive(negative)

o Differentiability implies continuity, a function not continuous at a point implies the derivative does not exist at that point

o Difference between average rate of change and instantaneous rate of change and their graphical interpretation (secant line versus tangent line)

o Use differentials to estimate errors and distinguish dx, dy from Δx and Δy

Review Exercises

(1) Let . Use the definition of derivative to compute .

(2) Use the definition of derivative to prove that .

(3) If , find .

(4) Determine whether is the solution to the differential equation .

(5) Find . [Hint: recognize the limit as a derivative, and use differentiation rules.]

(6) In this problem, you will prove the quotient rule in two different ways. [The two purposes of this problem are (1) to see that the quotient rule is true, and (2) to get differentiation practice.]

(a) Use the product and chain rules to continue the computation

. After computing the derivative, use algebra to make your answer look like the quotient rule.

(b) Let , so that . Take derivatives of both sides and solve for . While simplifying your answer, you will use again that .

(7) Several times, we’ve used the fact that differentiable implies continuous. (For example, when sketching from the graph of , you draw open circles in the graph of wherever the graph of is discontinuous.) Your task in this problem is to prove this fact. You’ll need the “limits are nice” theorem, the definition of continuous, the version of the definition of derivative that you developed in problem (13) on homework 6, and two applications of the “introduce a canceling pair” algebra trick! Assume that is differentiable at , and fill in the blanks to show that is continuous at .