Spring 99, MTH 131 Brief Review for Exam 2

This is intended to supplement the homework and give you an idea of what type of problems to expect on Exam 2. Included is an outline of the main ideas in the sections.

The derivative at a point

The definition of the derivative

f¢(a) = 
lim
h ® ¥
  f(a+ h) - f(a)
h

    When the limit can be found exactly

1.) Let f(x) = 3x2 +1. Using the definition of the derivative, find the slope of the tanget to f(x)
where x = 2. Check your answer using the rule for derivatives of polynomials.

    When the limit must be estimated

2.) Let f(x) = sin(2x). Using the definition of the derivative, find f ¢((p)/ 8). To evaluate the limit, use the estimation technique. Check your answer, using the chain rule and formula for the derivative of sin x.

    When the limit doesn't exist

3.) Using the definition of the derivative show that the derivative of |2x + 1| does not exist for x = -1/2.

The derivative function

Using the definiton

    Finding the derivative of a function given graphically

4.) Sketch the graph of the derivative of the function shown below.


    Finding the derivative of a function given algebraically

5.) Let f(x) = 1/x. Using the definition of the derivative, find f ¢(x).

Interpreting the derivative as a rate of change

6.) A stone is dropped in a pool of water, creating circular ripples, the first one growing larger as time passes. Let r = f(t) be the radius (in inches) of the largest circle at time t seconds.
    a.) What does it mean that f(2) = 8? Write a complete sentence.
    b.) What are the units of f ¢(t)?
    c.) What does it mean that f ¢(2) = 3? Write a complete sentence.
    d.) What do you predict the value of f (3) is, given that f(2)=8  and f '(2)=3?

The second derivative

    Graphic implications of the second derivative - concavity

7.) Which of the points labeled by letters in the graph of f in the following figure have:
    a.) f positve? Do not include points where f = 0.
    b.) f ¢ = 0?
    c.) f ¢ positive? Do not include points where f ¢ = 0.
    d.) f ¢¢ positive? Do not include points where f ¢¢ = 0.
    e.) f ¢¢ negative? Do not include points where f ¢¢ = 0.

Some rules for differentiation

    Derivative of a constant

8.) What is the derivative of f(x) = p?

    Derivative of a linear function

9.) What is the derivative of f(x) = x? What is the derivative of f(x) = -2x + 4?

    Derivative of a constant times a function

    Derivative of Sums and Differences

    Power Rule

10.) What is the derivative of f(x) = x-2? What is the derivative of f(x) = x1/4?

    Derivative of a polynomial

11.) Find the equation of the tangent line to f(x) = x10 - 9x7 + 24 x4 - 3 x2 + 1/2    ,    at x = 1.

More Rules

    The derivative of sinx

    The derivative of cosx

    The Chain Rule

12.) Find the derivative of sin(2x). Find the derivative of cos(x2 + p/ 2).

    The derivative of lnx

13.) Find the derivative of sin( lnx).

    The derivative of ex

14.) Find the derivative of e1/x.

    The derivative of ax

15.) Find the derivative of 1/ 2x.


File translated from TEX by TTH, version 1.57.