Holy Cross College, Fall Semester, 2003 Math 131, Section 01, Final Exam Tuesday, December 16, 8:30 AM

Professor Hwang

Name:

Instructions Clearly mark your answers, and show work on the test itself. Use the back of the preceding page if you need more space for scratch work. You must show all work for full credit.

Please do not write in the space below

Problem Points/Poss
1  / 20
2  / 20
3  / 20
4  / 20
5  / 20
6  / 20
7  / 15
8  / 20
9  / 20
Total  /175

ℋ 𝒜𝒫𝒫𝒴 ℋ 𝒪ℒℐ𝒟𝒜𝒴𝒮 !

1. [10 points each] Let f(x) = e^{−x} cos x = \frac{cos x}{e^{x}}. (a) Use the product rule to compute f′(x). Simplify your answer.

(b) Use the quotient rule to compute f′(x). Simplify your answer.

2. [20 points] A small portion of a graph y = f(x) (with horizontal and vertical scales as shown) is given below. Determine (with justification) which of the following functions might be depicted, and circle your choice(s).

f_{1}(x) = − sin x,  f_{2}(x) = ln(1 − x),  f_{3}(x) = 1 − e^{x}

−10^{−6}  0  10^{−6} −10^{−6}0 10^{−6}

3. [10 points each] The graph of f(t) = 2 − e^{−t^{2}/2} on the interval [0, 2] is shown.

0  0.5  1  1.5  2 012

(a) On the graph above, sketch the left- and right-hand rectangles for 4 intervals of equal length, and shade the area representing the difference of the upper bound and the lower bound.

(b) Suppose left- and right-hand sums with n intervals are used to compute upper and lower bounds for ∫_{0}^{2} f(t) dt. Find the smallest number, n, of intervals so that

upper − lower ≤ 0.02 = \frac{2}{100}

Your answer to part (a) should help you recall the formula to use. Do not attempt to compute the corresponding sums!

4. [20 points] A rectangular poster will consist of a printed message, of area 150 square inches, surrounded by a border. The border is to be 1 inch wide along the sides and bottom, and 2 inches wide along the top. What dimensions of the poster minimize its total area?

Printed Area

Total width: Total height:

5. Let f(x) = \frac{1 − cos x}{x} for x ≠ 0. (a) [5 points] What value for f(0) makes f continuous at x = 0?

f(0) =

(b) [10 points] Compute f′(x) for x ≠ 0.

f′(x) =

(c) [5 points] Find lim_{x→0} f′(x).

lim_{x→0} f′(x) =

6. [10 points each] Each part of this question refers to the function pictured below. The areas enclosed are A = 25, B = 16, and C = 16.

0  1  2  3  4  5 0 20 40 60 A  B  C y = f(x)

(a) Find ∫_{0}^{3} f(x) dx (show all work for full credit)

∫_{0}^{3} f(x) dx =

(b) Find the average value of f on the interval [0, 5] Suggestion: Start by writing down the average value in terms of an integral.

Average value for 0 ≤ x ≤ 5:

7. The top of a 13-foot ladder rests against a vertical wall, and the foot of the ladder slides away from the wall at 3 ft/sec. Let x be the distance from the wall to the foot of the ladder in feet, and y the height of the top of the ladder in feet. Both parts of this question concern the instant at which the base of the ladder is 6 ft from the wall.

y  13 x 3 ft/sec

(a) [5 points] How high is the top of the ladder? Express your answer symbolically.

Height of ladder:

(b) [10 points] How rapidly is the top of the ladder dropping? For full credit, express your answer both symbolically and numerically, including units. No credit for answers off by more than a factor of 10.

Speed (symbolic): Speed (numerical):

8. [10 points each] Let a > 0 be constant, and define f(x) = (x − a)\sqrt[3]{x} for x real.

(a) Compute f′(x) and make a sign diagram. Find the critical point(s) of f, and classify as local maximum, local minimum, or neither. (Please see hints below.)

(b) Compute f′′(x) and make a sign diagram. Find the inflection point(s) of f, if any.

Hints: Much of this problem is algebra management. Unless you expand or factor when appropriate, you are likely to get bogged down. You will need to use properties of exponents, since your calculator may not be able to handle all the tasks you require.

For reference in checking your answers, the graph for a typical value of a is shown. Your answers should be deduced from calculation, not from the appearance of the graph.

9. [10 points each] A curve is defined implicitly by the equation x^{3} − 3xy^{2} = 2. (a) Find \frac{dy}{dx} in terms of x and y

\frac{dy}{dx} =

(b) Use part (a) to find the equation y = mx + b of the line tangent to the curve at the point (x_{0}, y_{0}) = (2, −1), and sketch this line. Hints: The horizontal and vertical scales in the figure are the same. You must determine the location of the origin and the size of the grid squares. For this, find the coordinates of the points on the curve where the tangent line is horizontal, then locate these points visually.

Tangent line at (2, −1):