Holy Cross College, Fall Semester, 2004 MATH 131, Section 01, Final Exam Monday, December 13, 2:30 PM Professor Little

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

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}

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.

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

4. Compute the following limits [5 points each]. Any legal method is OK.

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

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

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

5.

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

• (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].

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}.

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

• (c) h(z) = ln(4z^{2} + 2e^{tan^{−1}(z)})

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

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.

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?

10. A car moves along a straight line road with acceleration at time t (seconds) given by a(t) = cos(t) − \frac{t^{2}}{100} (meters per second per second).

• (1) [7.5 points] Find the velocity v of the car as a function of time if v(0) = −2 meters per second.

• (2) [7.5 points] Find the position x of the car as a function of time if x(0) = 0 meters.