Honors AP Calculus / Mr. Hansen

Name: _______________________________

11/10/2009

Spare battery bonus (Mr. Hansen’s use only): __________

 

Test #4 (100 points): Calculator Required

 

 

Part I: Standard Proofs (9 pts. each)
Clarity and quality of presentation matter. Perfection is not expected, but a significant point penalty will be applied for sloppiness.

 

 

1.

Using the chain rule, the product rule, and the power rule for derivatives as givens, prove the quotient rule.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.

Prove that the derivative of any even function is odd. On this problem, you must state the “given” and “prove” statements explicitly for full credit.

 

 

 

Given: __________________________________________

 

 

 

Prove: __________________________________________

 

 

 

Proof:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.

Use a reference triangle to prove the “chain rule ready” formula for .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4.

Prove by mathematical  induction that , where n is any integer greater than or equal to 7.

 

 

Part II: Definitions (6 pts. each)

 

 

5.

Define precisely what it means for a function f to be continuous on . A multi-part definition is required for full credit.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6.

Define what it means for a function g to be “one-to-one” on its domain. Zero (0) points will be awarded for a precal-type description based on any sort of “line test.”

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7.

Give a synonym for each of the following (a one-word answer is expected in each case):

 

 

(a)

one-to-one: ______________________

 

 

(b)

locally linear with finite slope: ______________________

 

Part III: Problems (12 pts. each)

 

Show adequate justification. Circle or box your answer for full credit.

 

 

8.

Find .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9.

Let y2 = 2 cos(xy + y).

 

 

(a)

Compute .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)

State a point where  = −1. (Work is optional.)

 

 

 

 

 

 

 

 

(c)

For the point described in part (b), find the derivative of x with respect to y.

10.

Let f (x) be defined as x3 + 1 if , and let f (x) = a(x − 2)2 + b for all other real values of x. Find the values of constants a and b such that  exists. Also compute . Place answers in the blanks, supported by work beneath.

a = _____________                  b = _____________                   = _____________

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11.

Sketch the function defined parametrically by x = 5 cos t, y =  sin t. Indicate your domain for t in some clear fashion, and compute the slope of the tangent line at the point (2.5, −1.5).