Honors AP Calculus / Mr. Hansen

Name: _______________________________________

10/12/2006

Mr. Hansen’s use only (bonus point for spare batteries): _______

 

Test on Chapters 2 and 3

Please read: Calculator is OK throughout. Each multiple-choice problem is worth 4 points, with an additional point deducted for wrong guesses. (In other words, you lose 4 points if you omit a question, 5 points if you answer it incorrectly.) Free-response point values are shown in parentheses in the left margin.

 

MC

Multiple Choice Portion: Scoring as described above. Mark MC answers only on the bubble sheet.

1.

“Indefinite integral” means the same as . . .

 

(A) derivative
(B) second derivative
(C) antiderivative

(D) differential equation
(E) none of these

 

 

2.

If  and f (1) = 2, then f (3) equals . . .

 

(A) 2
(B) –2
(C) 2.5

(D) –2.5
(E) 0

 

 

3.



 

(A) 8x5/5
(B) 8x5/5 + C
(C) 8x4

(D) 0
(E) none of these

 

 

 

The following situation applies to problems 4, 5, 6, and 7. Dora throws a rock high into the air. Its height in meters above ground, h(t), is given by

h(t) = 30t − 4.9t2,

where t = time in seconds after the throw.

 

 

4.

Compute the average velocity from t = 5.0 seconds to t = 5.2 seconds.

 

(A) 20.18 m/sec
(B) −20.18 m/sec
(C) 19.98 m/sec

(D) −19.98 m/sec
(E) none of these

 

 

5.

Find an expression for average velocity from 5.0 seconds to some unspecified time t.

 

(A) −4.9t + 5.5
(B) −4.9 + 5.5t
(C) −4.9t + 4.32

(D) −9.8t + 30
(E) −9.8 + 30t

 

 

6.

What is the limit (as t  5) of the expression requested in question #5?

 

(A) h(5)
(B)
(C) average velocity from t = 5.0 to t = 5.0000001

(D) , i.e., a(5)
(E) none of these

 

 

7.

Compute the instantaneous velocity and acceleration when t = 5, using proper units.

 

(A) –19.000 m/sec, 9.800 m/sec2
(B) –19.000 m/sec, –9.800 m/sec2
(C) –19.450 m/sec, –10.000 m/sec2

(D) –19.450 m/sec, –9.800 m/sec2
(E) 19.450 m/sec, –10.000 m/sec2

 

 

 

8.

From the choices given below, find the largest possible 3-place decimal value of  such that
 for the function f (x) = 5 exp(x). If you have forgotten what the notation exp(x) means, you may raise your hand to purchase a hint for 1 point.

 

(A) .001
(B) .002
(C) .003

(D) .004
(E) .005

 

 

9.

Let us define the “Nick function” . As Nick taught us, N(4)  36. (Actually, the true value is 110/3, but his astonishing ability to estimate 36 in the blink of an eye is a feat that will live for years in HappyCal lore.) The function N(x) is an example of . . .

 

(A) a constant
(B) a definite integral
(C) an accumulator function

(D) a derivative
(E) an antiderivative of t2 + 3t

 

 

10.

Suppose that a lemma has been provided (with valid proof) stating that an accumulator function of any continuous function is continuous. Is there a value of x for which the Nick function (see question #9) satisfies N(x) = 36?

 

(A) Yes. By IVT,  such that N(x) = 36.
(B) Yes. By EVT,  such that N(x) = 36.
(C) Yes. By IVT,  such that N(x) = 36.
(D) Yes. By EVT,  such that N(x) = 36.
(E) No, not necessarily.

 

 

FR

Free Response: Point values are in parentheses.

11.
(8)

Let g be a function that is differentiable on its domain. Can the derivative of g at a point c  Dg be correctly defined as the limiting value of secant slopes for shorter and shorter secant segments having the point (c, g(c)) as one of their endpoints? If so, state two other ways of correctly thinking of g¢(c). If not, then state precisely where the error in the proposed definition lies.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

12.
(6)

Fill in the 3 blanks below to make a formal definition of a limit of a function f (x) as x increases without bound, a situation that is (somewhat incorrectly) denoted x. Although this definition differs from the definition for , where c is some constant, there are enough similarities that you should be able to figure out what to write.

 

 

 

We say  _____ for any _____ > 0, no matter how small, there exists a number M > 0 such that  _____________ .

 

 

13.
(3)

Why is the notation x invalid from a purist’s point of view? (Note: The notation is in widespread use and, because of the definition in question #12, is accepted as valid. However, there are grounds on which a purist might quibble.)

 

 

 

_____________________________________________________________________________

 

 

14.
(14)

Consider a “damping function”  for the questions that follow.

 

 

(a)

Make a rough sketch (very quickly—no accuracy is expected) of function D on the interval [0, 100].

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)

State . No work is expected. Answer: ________

(c)

Explain why taking M = 10,000 will always be sufficient to ensure that D(t) stays less than .1 unit away from its limiting value whenever t > M.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(d)

Wlog, let “.1” in part (c) be replaced by the symbol , where  > 0. Find an expression for M (in terms of ) so that the formal limit definition is satisfied. Because you may be able to do this in your head, work is optional. Answer: Let M = ____________ .

 

 

 

 

 

 

 

 

 

 

 

15.
(15)

Use the formal definition of the derivative function to prove that .

 

16.
(7)

Without simplifying your algebra, find  if .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

17.
(7)

A sinusoidal function S(t) has S(0) = 0, amplitude 2, period 70, and center line value –2. Write an equation for S(t). It is not necessary to show your work.

Hint: As a check on your answer, S ¢(t) is given to be –2 sin(2t/70) · 2/70. Such a hint would never have been provided in Precal.