Mathematics 133 – Intensive Calculus for Science 1 Sample Exam II – October 28, 2005

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

A) (15) Using the information here, construct a “qualitative” plot of y = f′′(x). B) (10) Over which intervals is f increasing? C) (5) Is f′ continuous at x = 1? Why or why not? What happens on the graph y = f(x)

at x = 1?

II. A) (10) The function H(t) gives the number of hours of daylight t days after the start of

the year in Worcester. At t = 304 days (October 31 in a non-leap year), H′(304) = −0.083. Give the meaning of this equation as a sentence, using appropriate units.

B) (10) The table below gives the position s (in miles) of a freight train moving along a straight line track as a function of time t (in hours).

\begin{matrix}t & .5 & 1 & 1.5 & 2 & 2.5 \\ s & 10 & 25 & 42 & 50 & 55 \end{matrix}

Estimate the train’s instantaneous velocity at t = 1.5 hours as closely as you can from this information.

III. (15) Using the limit definition, find f′(x) for f(x) = 1/x.

IV. Find derivatives of each of the following functions by applying the appropriate “shortcut” derivative rules:

A) (10) f(x) = 5x^{7} − \frac{3}{\sqrt{x}} − 4^{2x}

B) (10) g(x) = (x^{2} + 1)^{12}2^{x}

C) (10) h(x) = \frac{x^{2}}{e^{x}−1}

V. (5) Say whether the following statement is true or false, and explain your reasoning: If the time interval is short enough, then we expect the average velocity of a car over the interval will be close to its instantaneous velocity at any time in the interval.