AP Calculus AB / Mr. Hansen
2/9/2004


Check one (1):

           Name: ____________________________
c Regular time (42 minutes, all problems)
c Extra time (51 minutes, omit “*” problems)

Test on Chapters 6 and 7

Instructions and Scoring. Please read carefully before you begin. Check (þ) each item as you read it (1 pt. each).

¨     Do not discuss this test with anyone until I inform you officially that everyone has taken it. This is an honor issue. Even seemingly minor infractions—such as discussing the shape of a slope field or asking “What did you get for the third multiple choice question?”—should not be tolerated by anyone concerned about honor. Unbeknownst to you, another student could overhear your conversation and compromise the security of the test, even if you yourself did not seek to commit a cheating offense.

¨     There will be a 1-point deduction for each unnecessary disturbance during the test: speaking out of turn, asking a question that has already been answered, asking a question that is clarified later in the problem, etc.

¨     If you have a legitimate question, there is no point penalty. Please raise your hand and keep working until I spot you. This may take a minute. Please do not call out my name, since that creates unnecessary noise.

¨     If you see a typographical error, please mark it clearly for extra credit. The test contains at least one intentional typo. There are probably a few unintentional ones, too!

¨     No calculator is allowed for Part I (multiple choice). Mark answers on bubble sheet today.

¨     A graphing calculator is required for Part II (free response).

¨     If you have extended time, your time limit is 51 minutes, and you should omit the problems marked with an asterisk (*). In Part I, you will answer 4 problems in 12 minutes, 6 points each. In Part II, you will answer 57 points’ worth of problems in 39 minutes, which will be rescaled to 66 points for comparability with the regular-time students’ scores. (In other words, point values in Part II should be multiplied by about 1.158.)
If you have regular time, your time limit is 42 minutes. In Part I, you will answer 6 problems in 12 minutes, 4 points each. In Part II, you will answer 66 points’ worth of problems in 30 minutes. Point values are in parentheses in the left margin.

¨     Please remember that there is an assignment posted on the Web for tomorrow.

¨     If you finish early, please work on the bonus question or the opinion question until time is called. Then leave quietly so that the extra-time students may finish without disturbance. That means no talking (see second checkbox above).

 

 

 

 

Part I: Multiple Choice. No calculator, no partial credit. Mark answers on bubble sheet.

 

 

1.
(4 pts.)

Compute

 

 

 

(A) ln |x|
(B) ln |x| + C
(C) ln |t| + C

(D) 1/t
(E) 1/x

 

 

2.
(4 pts.)

The weight (in pounds) of a leaking bucket, including the water that remains in it, is given by
W(h) = 20 – 0.04h, where h denotes height above ground. The work required to lift this bucket from a height of 2 feet to a height of 6 feet equals . . .

 

 

 

(A) 79.18 ft. lbs.
(B) 79.36 ft. lbs.
(C) 79.84 ft. lbs.

(D) 119.28 ft. lbs.
(E) 120 ft. lbs.

 

 

3.
(4 pts.)

Compute y ¢ if y = ln[(4x – 7)(x + 10)].

 

(A)
(B)
(C)

(D)
(E)

 

 

*4.
(4 pts.)

For any real number c,  equals . . .

 

(A)  by function-interchange prop.
(B)  by continuity of sine & cosine
(C)  by continuity and limit properties

(D) does not exist (infinite)
(E) does not exist (by oscillation)

 

 

5.
(4 pts.)

Let x and y be the two variables of interest. Give examples, in the order specified, of

  • a second-order separable differential equation
  • a first-order differential equation that is not separable.

 

 

 

(A) (y ¢)2 = 2x, y ¢¢ = 2y
(B) y ¢¢ = 2y, (y¢)2 = 2x
(C) y ¢¢ = 2x, y ¢¢ = 2x + y

(D) y ¢¢ = 2x, y ¢ = 2y
(E) y ¢¢ = 2x, (y ¢)2 = 2x + y

 

 

*6.
(4 pts.)

 equals . . .

 

 

 

(A) 0
(B) 0.01
(C) 0.10

(D) 10
(E) does not exist

 

 

 

Part II: Free Response. Calculator is required.

 

 

1.

In this problem, all units are in hundreds of meters. A commando is attempting to parachute from a stationary helicopter at an altitude of 5 (i.e., 500 m) onto an offshore oil drilling platform located on the interval [–1, 1]. Because of winds in the vicinity, she will not parachute straight down. We will neglect motion in 3 dimensions and will analyze only the slope field affecting her fall in 2 dimensions (x and y). Because her parachute opens .5 unit (50 m) after her bail-out point and then keeps her at a constant downward velocity, gravity is not an issue in this problem. The only factor affecting her descent is the slope field defined by

y ¢ = 0.05(3y – 25)(.5x + 1)2/3

 

 

*(a)
(4 pts.)

What is the width of the oil drilling platform in meters?

 

 

(b)
(16 pts.)

Sketch the slope field.

 

 

(c)
(4 pts.)

Show graphically an example of a starting point from altitude 5 that does not result in a successful landing on the platform. (The commando would drown in such a situation.)

 

 

(d)
(4 pts.)

Show graphically an example of a starting point from altitude 5 that does result in a successful landing.

 

 

BONUS

Explain why bailing out from a lower altitude does not improve the odds of a successful landing. Hint: There is some uncertainty in the determination of the altitude and the precise bail-out x value.

 

2.

Any teacher will confirm that, in general, larger classes are harder to teach than smaller classes. There are exceptions, of course; small classes can be difficult, and “large lecture” sections (n > 40 students) can be quite easy, especially if the teacher is well-rehearsed, enjoys public speaking, and has grading assistants to deal with the mountain of paperwork.

(a)
(1+3+1+1)

Nevertheless, the general observation that larger classes are harder to teach than smaller classes remains true. For Mr. Hansen, the key variable is I(n), the mean number of interruptions per minute as a function of n, where n = __________________ . (Fill in the blank.)

Several teachers have suggested that I(n) is an increasing exponential function of n. In other words, they believe that

[1]          ________________ (fill in the blank),

where k is a ________________  ________________ (fill in the blanks).

 

 

(2 pts.)

However, Mr. Hansen doubts that equation [1], which is a ________________  _______________ (fill in the blanks), is valid. Observations suggest that dI/dn varies jointly with I and the square of the number of students. In other words,

[2]          

for some constant H > 0. The “Hansen constant” H is known to be approximately 0.001.

 

 

(b)
(2 pts.)

In Algebra II, you learned what joint variation means. If you have forgotten, you may be able to use context clues to refresh your memory. Joint variation means that a quantity is

_________________________________________________________________ .

 

 

(c)
(12 pts.)

Solve equation [2] subject to the initial condition that Mr. Hansen is happiest when there are about 3 interruptions every 4 minutes, and this happy state occurs in a class of 14 students. If there are any decimal approximations in your solution, give them to 5 decimal places.

           

(d)
(3 pts.)

Because H = 0.001 is at best an educated guess, the solution to equation [2] that you found or tried to find in (c) implies a degree of precision that is not really present. For the remainder of the problem, we will use

[3]          

as the _________________________ solution of equation [2].

 

(Write “general” or “particular” in the blank.)

 

 

(e)
(4 pts.)

Mr. Hansen needs, on average, at least 1 interruption every 3 minutes to avoid boredom. Based on equation [3], what is the smallest class that he can teach in the normal fashion? (Note: Classes smaller than this would need to be taught using a small-group seminar or independent study format.) Justify your answer numerically for full credit or some other way for partial credit.

 

 

*(f)
(5 pts.)

Stress is directly proportional to I(n), and Mr. Hansen can handle approximately 2 interruptions per minute without experiencing an uncomfortable level of stress. Based on equation [3], what is the largest class that he can comfortably teach? Justify your answer graphically for full credit or some other way for partial credit.

 

 

(g)
(4 pts.)

Mr. Hansen once taught an AP Statistics class of 19 students, and for the first semester Mr. Constantine audited the class as an additional student. Based on equation [3] and part (f), by what factor was Mr. Hansen’s comfort threshold exceeded in the fall of that academic year? in the spring?

 

 

OPINION
(0 points)

Classes larger than 20 students are exceedingly rare at St. Albans. What would you suggest Mr. Hansen do to be a more effective and helpful teacher for a class of 21 calculus students?