MAC Find the x-value that maximizes the area of the shaded rectangle inscribed in a right triangle below.

Transcription

1 MAC 23. Find the x-value that maximizes the area of the shaded rectangle inscribed in a right triangle below. (x, y) y = 3 x + 4 a. x = 6 b. x = 4 c. x = 2 d. x = 5 e. x = 3 2. Consider the area of the region between the curve f(x) = x and the x-axis on the interval [, 7]. Find the right-endpoint approximation, R 3, to estimate the area of the region. f(x) = x a b c d e A particle moving along a line has acceleration function a(t) = 6t 2 ft/sec 2. The initial velocity is 2 ft/sec and the initial displacement is 4 ft. Find its position after t = 2 seconds. a. 2 ft b. 6 ft c. 23 ft d. 48 ft e. 9 ft

2 4. Find the general antiderivative of e 2x + x. a. 2e 2x x 2 + C b. 2e2x + x 2 + C c. 2 e2x x 2 + C d. 2 e2x + ln x + C e. 2e 2x + ln x + C 5. Consider the area of the region between the curve f(x) = x and the x-axis on the interval [, ]. Find the right-endpoint approximation, R 3, to estimate the area of the region. f(x) = x a b c d e A particle moving along a line has acceleration function a(t) = 6t + 2 ft/sec 2. The initial velocity is 3 ft/sec and the initial displacement is 5 ft. Find its position after t = 2 seconds. a. 2 ft b. 6 ft c. 23 ft d. 48 ft e. 9 ft

3 7. Find the x-value that maximizes the area of the shaded rectangle inscribed in a right triangle below. (x, y) y = 4 x + 2 a. x = b. x = 2 c. x = 3 d. x = 4 e. x = 5 8. Find the general antiderivative of x + e3x. a. x 2 + 3e3x + C b. x 2 + 3e3x + C c. x e3x + C d. ln x + 3e 3x + C e. ln x + 3 e3x + C 9. Find the x-value that maximizes the area of the shaded rectangle that has its base on the positive x-axis and a vertex lying on the parabola as below. (x, y) y = 6 x 2 a. x = 2 b. x = 4 c. x = 2 d. x = 3 e. x = 3

4 . If the slope of the tangent line to y = f(x) at any point is sec 2 x ( tan x π ) and the point 4, is on the curve, find f(). a. 2 b. 3 2 c. d. 3 e Find the absolute extreme values of f(x) = x on [, 2] first and then find the lower and upper bounds of the integral 2 x2 + 2 dx using the comparison property for integrals. a. 2 c. 3 2 e x2 + 2 dx 2 6 b. x2 + 2 dx 3 3 x2 + 2 dx 3 6 d x2 + 2 dx x2 + 2 dx 3 2. The population of Gainesville is changing at the rate of (2t + ) people 2 per year, where t is the number of years since the year 2. Find the net change of population in Gainesville from 2 to 22. a. 4 more people b. 8 more people c. 6 more people d. 4 fewer people e. 8 fewer people

5 3. ( Find d ) x sec( + t 2 ) dt. dx a. sec( + x) tan( + x) 2 x b. d. sec( + x 2 ) e. sec( + x) 2 x sec( + x) x c. sec( + x) tan( + x) 4. Evaluate the integral: cos(π/x) x 2 dx a. 2π b. 3 2π 3 c. π d. 3 2π e. 3 π 5. The integral following using substitution. x(2x 2 + ) /3 dx can be converted to which of the a. u /3 du b. 4 u/3 du c. 4 u/3 du d. 4u /3 du e. 4u /3 du

6 6. If f is a continuous even function on (, ) with 3 f(x) dx = 4 and 4 f(x) dx = 3, find 4 f(x) dx. a. b. 2 c. d. 2 e. 7. Find the x-value that maximizes the area of the shaded rectangle that has its base on the positive x-axis and a vertex lying on the parabola as below. (x, y) y = 9 x 2 a. x = 2 b. x = 3 c. x = 2 d. x = 4 e. x = 3

7 8. Find the absolute extreme values of f(x) = x on [ 2, ] first and then find the lower and upper bounds of the integral x2 + 4 dx 2 using the comparison property for integrals. a. 2 c. 5 e x2 + 4 dx 2 2 b. x2 + 4 dx 5 x2 + 4 dx 2 2 d. 6 x2 + 4 dx x2 + 4 dx ( Find d ) x tan( + t 2 ) dt. dx a. sec2 ( + x) 2 x d. tan( + x) 2 x b. tan( + x) c. tan( + x 2 ) e. tan( + x) x 2. Evaluate the integral: 2 sin(π/x) x 2 dx a. π b. π c. 2π d. 3 2π 3 e. π

8 2. The population of Gainesville is changing at the rate of (3t + ) people 2 per year, where t is the number of years since the year 2. Find the net change of population in Gainesville from 2 to 23. a. 3 fewer people b. 9 fewer people c. 6 fewer people d. 3 more people e. 9 more people 22. If the slope of the tangent line to y = f(x) at any point is 2 sec2 x tan x ( π ) and the point 4, is on the curve, find f(). a. 4 3 b. 3 4 c. 7 4 d. 3 e.

9 23. If f is a continuous odd function on (, ) with 3 f(x) dx = 5 and 4 f(x) dx =, find 4 f(x) dx. a. b. 2 c. 3 d. 4 e. 24. The integral following using substitution. x(3x 2 + ) /5 dx can be converted to which of the a. d. 4 6 u/5 du b. u /5 du e. 4 6u /5 du c. 6u /5 du 6 u/5 du

10 a. A cylindrical tube i s to be constructed of heavy cardboard with two plastic ends having a volume of 48π cubic feet. What radius and height will minimize the cost of the tube if cardboard is $2 per square foot and plastic is $6 per square foot? Primary function: Constraint: h r r = h = ft ft Use either the First Derivative Test or the Second Derivative Test to confirm your result.

11 b. A cylindrical tube with open top i s to be constructed of heavy cardboard with plastic bottom having a volume of 28π cubic feet. What radius and height will minimize the cost of the tube if cardboard is $2 per square foot and plastic is $4 per square foot? Primary function: Constraint: h r r = h = ft ft Use either the First Derivative Test or the Second Derivative Test to confirm your result.

12 2a. Consider the area of the region between the curve f(x) = x and the x-axis on the interval [, 2]. Find the Riemann sum R n (right endpoint approximation with n subintervals of equal width). x = ; x i = ; R n = n i= 2b. Consider the area of the region between the curve f(x) = x 2 + and the x-axis on the interval [, 3]. Find the Riemann sum R n (right endpoint approximation with n subintervals of equal width). x = ; x i = ; R n = n i=