Mathematics 133 – Intensive Calculus for Science Final Examination December 13, 2005

Directions

Do problems I - VIII in the blue exam booklet. You may use an approved calculator on any of the following. There are 200 possible points.

I. (20) One of the functions tabulated below is approximately linear and the other is approximately exponential. Say which is which and give possible formulas for each.

\begin{matrix}x & .3 & .6 & .9 & 1.2 \\ f(x) & 4.12 & 3.74 & 3.39 & 3.07 \\ g(x) & 4.31 & 4.07 & 3.83 & 3.59 \end{matrix}

II. A) (10) Give a possible formula for the function plotted here:

B) (10) How can you tell that the function from part A has an inverse function if it is restricted to the interval −1 ≤ x ≤ 1? What is the domain of the inverse function?

III. A) (10) Using the limit definition, compute f′(x) for f(x) = 5x^{2} − x + 3. Using appropriate derivative rules, compute the derivatives of the following functions: B) (10) g(x) = 3^{x} − ln(x^{3} + x) C) (10) h(x) = 3\sqrt[5]{x} + \frac{7}{\sqrt[3]{x}} + 3x^{8}

D) (10) i(x) = \frac{cos(x)}{sin^{2}(x)−1} E) (10) j(x) = arctan(3x + 2) F) (10) Find \frac{dy}{dx} if y^{2}\sqrt{x} − e^{y} = 3.

IV. The following graph shows the derivative y = f′(x) for some function f(x).

A) (10) Over which intervals is f increasing? decreasing? B) (10) Over which intervals is y = f(x) concave up? concave down? C) (10) Give a qualitative sketch of the graph y = f′′(x).

V. A frozen dinner is taken out of a grocery store freezer case at time t = 0 and placed in a shopping cart. Its temperature in degrees Fahrenheit t minutes later is given by H(t) = 70 − 50e^{−t/10}. A) (5) What is the temperature of the dinner at t = 0? B) (10) How fast is the temperature changing at t = 10? Give your answer including

correct units. C) (5) How long does it take for the temperature to reach 60^{∘}F?

VI. (15) Find the (global) maximum and minimum values of f(x) = e^{−x} sin(x) on [0, π].

VII. A town wants to build a water pumping station on a small island 2 miles from the straight shoreline of the lake that forms its water reservoir, as in this figure:

The town is 6 miles along the shore from the point nearest to the island. The town must be connected to the pumping station by a pipeline made up of straight line segments. It costs $3 million per mile to lay pipe under the water and $2 million per mile to lay pipe along the shoreline. A) (10) Give the cost of constructing the pipeline as a function of x. B) (10) Where along the shoreline should the pipeline hit land to minimize the costs of

construction?

VIII. (15) A block of dry ice (solid CO_{2}) is evaporating and losing volume at the rate of 10 cm^{3}/min. It has the shape of a cube at all times. How fast are the sides of cube shrinking when the block has volume 216 cm^{3}? Give the units of your answer.

Have a peaceful and joyous holiday season!