# 10 Multiple Choice Calc

1.

Find the particular solution to y ‘ = sin(x) given the general solution is y = C – cos(x) and the initial condition . (5 points)

 -cos(x) 2 – cos(x) -1 – cos(x) 1 – cos(x)

2.

The slope of the tangent to a curve at any point (x, y) on the curve is . Find the equation of the curve if the point (2, -2) is on the curve. (5 points)

 x + y = 0 x2 – y2 = -2 x2 + y2 = 16 x2 + y2 = 8

3.

The rate of decay in the mass, M, of a radioactive substance is given by the differential equation , where k is a positive constant. If the initial mass was 100g, then find the expression for the mass, M, at any time t. (5 points)

 M = e-kt M = 100 e-kt M = 100 ekt M = 100ln(kt)

4.

The temperature of a pot of coffee varies according to Newton’s Law of Cooling: , where T is the temperature of the coffee, A is the room temperature, and k is a positive constant. If the water cools from 90°C to 85°C in 1 minute at a room temperature of 30°C, find the temperature, to the nearest degree Celsius of the coffee after 4 minutes. (5 points)

 72 42 68 81

5.

The differential equation (5 points)

I. produces a slope field with horizontal tangents at y = 2
II. produces a slope field with vertical tangents at y = -1
III. produces a slope field with columns of parallel segments

 I only II only I and II III only

6.

Which of the following differential equations is consistent with the following slope field? (5 points)    7.

The general solution of the differential equation dy – 0.2x dx = 0 is a family of curves. These curves are all (5 points)

 lines hyperbolas parabolas ellipses

8.

Estimate the value of by using the Trapezoidal Rule with n = 4. (5 points)

 8 19 22 36

9.

The table below gives selected values for the function f(x). With 5 rectangles, using the midpoint of each rectangle to evaluate the height of each rectangle, estimate the value of . (5 points)

x 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
f(x) 1 0.909 0.833 0.769 0.714 0.667 0.625 0.588 0.556 0.526 0.500

 0.7456 0.6456 0.6919 0.6932

10.

Given f(x) > 0 with f ′(x) < 0, and f ′′(x) < 0 for all x in the interval [0, 1] with f(0) = 1 and f(1) = 0.3, the left, right, trapezoidal, and midpoint rule approximations were used to estimate . The estimates were 0.7915, 0.8405, 0.8410, 0.8421 and 0.8895, and the same number of subintervals were used in each case. Match the rule to its estimate. (5 points)

 abcde 1 trapezoidal abcde 2 right endpoint abcde 3 actual area abcde 4 midpoint abcde 5 left endpoint
 a. 0.8895 b. 0.7915 c. 0.8421 d. 0.8405 e. 0.841