Holy Cross College, Fall Semester, 2004

MATH 131, Section 01, Final Exam

Monday, December 13, 2:30 PM Professor Little

Your 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, but please place answers in the boxes provided where appropriate.

Please do not write in the space below

Problem Points/Poss
1  / 20
2  / 20
3  / 20
4  / 15
5  / 25
6  / 20
7  / 30
8  / 15
9  / 20
10  / 15
Total  /200

ℋ 𝒜𝒫𝒫𝒴 ℋ 𝒪ℒℐ𝒟𝒜𝒴𝒮 !

MATH 131-01, Final Exam 2 1. [5 points each] Circle the number of the graph showing each of the following functions.

• (a) f(x) = 3 − e^{−x} I II III IV V • (b) f(x) = x^{3} − 2x^{2} − x + 2 I II III IV V • (c) f(x) = 4 sin(πx) I II III IV V • (d) f(x) = 4 − 4 cos(πx) I II III IV V MATH 131-01, Final Exam 3 2. [20 points] One of the functions given in the following table is linear and the other is exponential. Find formulas of the appropriate type for each.

\begin{matrix}x & 1 & 2 & 3 & 4 & 5 \\ f(x) & 1.2 & 0.6 & 0.3 & 0.15 & 0.075 \\ g(x) & −2.3 & −0.6 & 1.1 & 2.8 & 4.5 \end{matrix}

f(x) = g(x) = 3.

• (a) [15 points] The depth of water in a tank oscillates sinusoidally once every 4 hours. The smallest depth is 2 feet and the maximum depth is 5 feet, which occurs at t = 0. Find a formula for the depth d(t) if t is the time in hours.

d(t) =

• (b) [5 points] How fast is the depth changing at t = 1.3 hours? Is it increasing or decreasing?

Rate of change = MATH 131-01, Final Exam 5 4. Compute the following limits [5 points each]. Any legal method is OK.

• (a) lim_{x→1} \frac{x^{2} + 1}{x − 3}

Limit =

• (b) lim_{x→2} \frac{x^{2} − 4x + 4}{cos(x − 2) − 1}

Limit =

• (c) lim_{x→∞} \frac{5x^{2} − x + 21}{3x^{2} + x + 1}

Limit = MATH 131-01, Final Exam 6 5.

• (a) [5 points] State the limit definition of the derivative:

f′(x) =

• (b) [10 points] Use the definition to compute f′(x) for f(x) = \sqrt{x}.

• (c) [10 points] Find the equation of the tangent line to the graph y = \sqrt{x} at the point [4, 2].

Tangent line = MATH 131-01, Final Exam 7 6. Compute the following derivatives using the derivative rules. You need not simplify. [5 points each]

• (a) f(t) = t^{3} − \frac{1}{\sqrt[4]{t}} + π^{t}.

f′(t) =

• (b) g(x) = \frac{x^{2} − 2}{cos(x) + 1}

g′(x) =

• (c) h(z) = ln(4z^{2} + 2e^{arctan(z)})

h′(z) =

• (d) Find \frac{dy}{dx} if x^{2}y − 2y^{3} = 3.

\frac{dy}{dx} =

7. Consider the family of curves defined by y = f(x) = x^{4} + 2ax^{2}, where a is any fixed real number.

• (a) [10 points] Find the critical points of f, construct a sign diagram for f′(x) in the case that a < 0. Which of your critical points are local maxima and which are local minima?

• (b) [10 points] Repeat part a, but assume now that a > 0.

• (c) [10 points] How many different inflection points does the graph y = f(x) have if a < 0? Explain.

8. [15 points] A cubical block of dry ice (solid CO_{2}) is evaporating and losing volume at the rate of 10 cm^{3}/min. How fast are the sides of cube shrinking when the block has volume 125 cm^{3}? Give the units of your answer.

Answer:

9. [20 points] Ship A travels along the path given by the parametric curve x = t, y = t^{2}. At the same time, ship B travels along the curve x = t, y = 4t − 5. At what time are the two ships closest to one another?

t =

MATH 131-01, Final Exam 11 10.

• (a) [7.5 points] Compute the left- and right-hand sums for the function f(t) = e^{−t^{2}} on the interval [0, 1], using n = 4 equal subdivisions.

LHS = RHS = • (b) [7.5 points] Consider the graph y = f(t) given below. Compute the average value

\overline{y} of f on the interval 0 ≤ t ≤ 5, given that the area marked A is 25 square units, and the areas marked B and C are both 16 square units.

\overline{y} =