SAMPLE MIDTERM 2, MAT 141, FALL 2000

The second midterm will be on Friday, November 17 at the usual class time (12:40pm) and place (P-118 in Physics building).

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) Find \frac{dy}{dx} at the point (3, 5) if y^{2} + y − 3 = x^{3}.

(a) 3 (b) 75/7 (c) 9/11 (d) 27/11 (e) 0 (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. 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.