SAMPLE MIDTERM 2 MAT 141

The second midterm will be on Friday, November 19 at the usual class time (12:40pm). Section 1 will take the exam in room 201 of Heavy Enginnering (same as last time) and Section 2 will take in our usual lecture room, room 152 of Light Enginnering.

1. Place the letter corresponding to the correct answer in the box next to each question.

(i) What is the slope of the curve given by x^{3} + y^{3} − 9xy = 0 at the point (x, y) = (2, 4)? (a) 1 (b) \frac{24}{30} (c) \frac{3}{4} (d) \frac{9}{18} (e) \frac{6}{5} (f) none of these.

(ii) Suppose f(x) = |x^{2} − 2x|. The set of critical points of f is (a) {0} (b) {1} (c) {0, 1, 2} (d) {2} (e) {0, 2} (f) none of these.

(iii) Suppose g′(x) = sin^{1999}(x). The absolute maximum of g on [0, 2π] occurs at (a) 0 (b) π/4 (c) π/2 (d) π (e) 2π (f) none of these.

(iv) What is lim_{x→+∞} \frac{x^{2}+2}{x^{3}+x^{2}+x}?

(a) 0 (b) 1 (c) \frac{2}{3} (d) \frac{1}{2} (e) +∞ (f) none of these.

(v) Find the linearization of f(x) = x^{3} − x at x = 1.

(a) L(x) = 2x (b) L(x) = 2(x + 1) (c) L(x) = −2(x − 1) + 1 (d) L(x) = 2x + 1 (e) L(x) = 2(x − 1) (f) none of these.

(vi) Use differentials to estimate the change in the surface area of a cube S = 6x^{2} when the edge length goes from x_{0} to x_{0} + dx

(a) 6dx (b) 6x_{0}dx (c) 12x_{0}dx (d) 12dx (e) 18x_{0}dx (f) none of these.

(vii) The formula for finding sucessive approximations in Newton’s method is (a) x_{n+1} = x_{n} + f(x_{n})/f′(x_{n}) (b) x_{n+1} = x_{n} − f(x_{n})/f′(x_{n}) (c) x_{n+1} = x_{n} + f′(x_{n})/f(x_{n}) (d) x_{n+1} = x_{n} − f′(x_{n})/f(x_{n}) (e) x_{n+1} = x_{n} − f(x_{n})f′(x_{n}) (f) none of these.

(viii) The solution of the inital value problem \frac{dy}{dx} = x + 1, y(2) = 3 is (a) y = x + 1 (b) y = x^{2} − x (c) y = \frac{1}{2}x^{2} + 2 (d) y = x^{2} + x + 1 (e) y = \frac{1}{2}x^{2} + x (f) none of these.

(ix) Suppose f′(x) = x^{2} sin^{10}(x). Then on the interval [0, \frac{1}{2}π] the function f is (a) increasing and concave down (b) increasing and concave up (c) decreasing and concave down (d) decreasing and concave up (e) constant (f) none of these.

(x) The function f(x) = x^{3} − 3x^{2} + 1 has a point of inflection at x =? (a) -2 (b) -1 (c) 0 (d) 1 (e) 2 (f) none of these.

2. Find each of the following indefinite integrals

(i) ∫ x^{3} − x^{2} + 2dx,

(ii) ∫ sin(3x)dx,

(iii) ∫ cos(3x + 2)dx,

(iv) ∫ sin^{4}(t) cos(t)dt,

(v) ∫ t(t^{2} + 1)^{1/2}dt,

3. State the mean value theorem.

4. (5 pts) Suppose the second hand on a clock has length 20 cm. At what rate is the distance between the tip of second hand and the 12 o’clock mark changing when the second hand points to 3 o’clock?

5. Suppose it takes 2 hours to replace the drill bit while drilling for oil. A new drill bit digs quickly at first, but slows down with time. Suppose that in t hours it can drill though f(t) feet of rock.

(i) Suppose the drill bit is used for T hours before being replaced. What is the average speed of drilling (including the 2 hours to install the bit)?

(ii) Show that to maximize this average speed the bit should should be replaced after T hours of use where T satisfies f′(T) = f(T)/(T + 2).

(iii) If f(t) = 100t/(t + 5) find this time T.