Honors AP Calculus / Mr. Hansen

Name (2 pts.): _______________________________________

4/3/2007

 

 

Bonus points (for Mr. Hansen’s use only): __________

 

Test on §§10-3, 10-6, 10-7, 11-2, 11-3, 11-6

 

Instructions

 

 

For problems 1 through 3, consider a particle whose 2-dimensional position function is given by s(t) = e3t i – (cos 2t) j.

 

 

1.

Compute the velocity vector at time t = 0.5.

 

(A) 4.482 i + .540 j
(B) 4.482 i – .540 j
(C) 13.445 i + 1.683 j
(D) 13.445 i – 1.683 j
(E) –13.445 i – 1.683 j

 

 

2.

Compute the tangential component of the acceleration vector at time t = 0.5.

 

(A) 39.979 i + 5.004 j
(B) 39.979 i – 5.004 j
(C) 119.938 i + 6.426 j
(D) 119.938 i – 6.426 j
(E) 541.719 i + 67.808 j

 

 

3.

Is the particle speeding up or slowing down at time t = 0.5, and how can we tell?

 

(A) Speeding up, since the norm of the acceleration vector is positive.
(B) Speeding up, since the dot product of acceleration and velocity vectors is positive.
(C) Speeding up, since the dot product of acceleration and velocity vectors is negative.
(D) Slowing down, since the dot product of acceleration and velocity vectors is positive.
(E) Slowing down, since the dot product of acceleration and velocity vectors is negative.

 

 

4.

Lake Flatland (in the 2-dimensional kingdom of Flatland) consists of the water bounded by the x-axis and the curve y = –x3/2 +2.5x2 + 4x – 24. The x-axis represents the surface of the lake on a calm day. Compute the average depth of Lake Flatland.

 

(A) 13.818 units
(B) 14.000 units
(C) 14.084 units
(D) 14.188 units
(E) 14.292 units

 

 

 

5.

Let t denote time in seconds. A leaky bucket having weight W(t) = 40 + 0.5t2 – 2t1.5 lbs. is hoisted from ground level to a height of 3 feet over a period of 5 seconds by a purely linear motion. In other words, height at time 0 equals 0, height at time 5 equals 3, and in between, height is a linear function of time. Compute the work done during the interval 0 £ t £ 5.

 

(A) 67.218 ft. lbs.
(B) 105.667 ft. lbs.
(C) 112.029 ft. lbs.
(D) 176.112 ft. lbs.
(E) 293.520 ft. lbs.

 

 

6.

A crude-oil tanker (see sketch on board) is docked in a port. The cargo hold is 480 feet long and is squared off at both ends as shown. The hull has semicircular cross sections of radius 60 ft. and is filled with oil to a depth of 30 ft. The density of crude oil in this particular tanker load is 55.686 lbs./ft.3. Compute the work done in pumping all the oil to the height of the tanker wall (60 ft.).

 

(A) 25 million ft. lbs.
(B) 250 million ft. lbs.
(C) 2.5 billion ft. lbs.
(D) 25 billion ft. lbs.
(E) 250 billion ft. lbs.

 

 

7.

Heating degree days are a measure of how much energy a home furnace must consume in order to bring the temperature indoors up to a comfortable level (say, 68° F.). If the outside temperature is constant at 50° F. for one full day, then we say that 18 heating degree days have been incurred, because the difference has been 18 degrees for one day. Of course, outside temperatures are seldom constant, and they seldom follow “nice” functions. Therefore, the calculation of heating degree days usually requires what aspect of the calculus?

 

(A) antiderivative
(B) rate of change
(C) rate of change of rate of change (i.e., second derivative)
(D) linear approximation
(E) variable-factor product

 

 

8.

Using the target inside temperature of 68° F., as in problem 7, compute the number of heating degree days incurred in a single day if temperature can be approximated by the sinusoidal relation

T(t) = 30 + 15 cos(pt/12) + 6 sin(pt/6),

where T is in degrees Fahrenheit and t is in hours. Note that this function is periodic with period 24 hours.

 

(A) 30
(B) 32
(C) 34
(D) 36
(E) 38

 

 

 

Instructions for today’s last problem (free response):

·         Justify your steps and show adequate work.

·         Circle or box your final answers.

·         Final answers, if approximated by decimals, must be correct to at least 3 places after the decimal point. Units must be included if appropriate.

·         As on the actual AP exam, the subparts are often separable. If you cannot get one part, keep right on going, because you can usually do the other parts if you understand the material well.

·         Write “OVER” if you need more room. Use additional sheets if necessary.

 

 

9.

All dimensions for this problem are in centimeters (cm).

The graph of the function y = 17 – x3 in the first quadrant is rotated about the y-axis to produce a solid of revolution called a cubic paraboloid. (See sketch on board.) A right circular cone that has the y-axis as the cone’s axis and the origin as the cone’s vertex is inscribed in the cubic paraboloid.

Very important note: The domain for the radius r of the cone is restricted to be 1 £ r £ 2.5.

 

 

(a)
(10 pts.)

Prove rigorously that the minimum height h of the cone is 11/8 cm. (A vague “hand-waving” proof may earn a few points, but for full credit, you must use the calculus.)

 

 

(b)
(20 pts.)

Compute the maximum and minimum volume of the cone subject to the given domain restriction. Clearly indicate the values of r and h that give rise to these maximum and minimum volumes, and justify your reasoning.

 

 

(c)
(20 pts.)

For the inscribed cone of minimum height (i.e., 11/8 cm as given in part (a)), find the cone’s mass if the cone’s density varies linearly as a function of y, with the density being 2 g/cm3 at the origin and 8 g/cm3 when y = 11/8 cm.

 

 

Bonus
(up to 7 pts.)

Prove that the bottom of Lake Flatland, given in problem 4, has roots at x = –3 and x = 4 only. This is straightforward by using precalculus (2 pts.) but is also worth doing by using the calculus (4 pts.). Finally (1 pt.), what type of root is the point (4, 0)? _________________