Holy Cross College, Fall Semester, 2003 Math 131, Section 03, Midterm 2 Friday, October 31, High Noon

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  / 30
2  / 10
3  / 20
4  / 10
5  / 10
6  / 10
7  / 10
Total  /100

1. [5 points each] Compute the derivatives of the given functions. (a) f(t) = (t^{2} + 1)^{5}

(b) g(x) = x^{3} ln x

(c) z(u) = \frac{u}{e^{u} − 1}

(d) s(t) = te^{t^{2}}

(e) b(x) = ln(e^{x} − e^{−x})

(f) k(s) = \frac{s^{2} + ln s}{s}

2. [10 points] Find the tangent line to the curve xy^{2} = xy + 4 at the point (2, 2).

3. [10 points] Suppose \frac{dx}{dt} = \sqrt{1 + e^{2x}}. Find \frac{d^{2}x}{dt^{2}} as a function of x.

4. In both parts of this question, let f(x) = x^{2}.

(a) [10 points] Compute the derivative f′(x) from the definition.

(b) [10 points] Find the equation of the line obtained by zooming in on the graph at the point (1, 1), and sketch the line in the grid provided.

1  1

5. [5 points each] The position of a car (measured in feet) after t seconds is:

\begin{matrix}t & 0 & 1 & 1.5 \\ s(t) & 0 & 80 & 130 \end{matrix}

(a) Estimate the speed of the car when t = 0.5 and when t = 1.25. Give units.

Speed at t = 0.5 Speed at t = 1.25

(b) When t = 1, does the car appear to be accelerating or decelerating? Write a sentence or two in support of your answer.

Math 131-03, Midterm 6

6. [10 points] Let f(x) = e^{−x^{2}/2}. Find the interval(s) on which f is both decreasing and concave down.

Math 131-03, Midterm 7

7. [5 points each] (You must justify your answers for full credit.) Define

a function by

f(x) = \begin{cases}e^{x} & if x < 0, \\ 2 − cos x & if x ≥ 0. \end{cases}

(a) Is f continuous at x = 0? (Hint: One-sided limits!)

(b) Is f differentiable at x = 0?