Problem Set Four Integration AP Calculus AB

Size: px

Start display at page:

Download "Problem Set Four Integration AP Calculus AB"

Godfrey Henry
5 years ago
Views:

1 Multiple Choice: No Calculator unless otherwise indicated. ) sec 2 x 2dx tan x + C tan x 2x +C tan x x + C d) 2tanxsec 2 x x + C x 2 2) If F(x)= t + dt, what is F (x) x x 2 + 2x x 2 + d) 2(x2 +) ) Let A = cosxdx. We estimate A using the LRAM, RRAM and Trapezoidal approximations with n = subintervals. Which is true? L < A < T < R L < T < A < R R < A < T < L d) R < T < A < L e) The order cannot be determined. ) The graph of f(x) consists of line segments and quarter circles as sown in the graph above. What is the value of 5π +5π 2+5π d) 2 5π f(x)dx? 5) Let R be the region between the functionf(x) = x + 6x 2 + x +, the x-axis and the lines x = and x =. Using the Trapezoidal Rule, compute the area when there are equal subdivisions. (Calculator) d) 96 6) What is f(x) if f (x) = 2x and f(2) = (Calculator) x 2 f(x) = ln x 2 f(x) = ln x 2 ln f(x) = ln x 2 + ln d) f(x) = 2lnx x 2 e) f(x) = 2lnx x 2 2ln2 + 7) What value of c on the closed interval [, ] satisfies the Mean Value Theorem for Integrals for f(x) = 2lnx? (calculator) d).296 s 8) Find the value of x at which the function y = x 2 reaches its average value on the interval [, ] (calculator) d) 5.77 e) ) If f(x)dx = k and f(x)dx =, what is the 7 value of x + f(x)dx? k + k 6 k d) -6 + k e) -6 + k

2 ) If a particle is moving in a straight line with a velocity of v(t) = 2t ft/sec and its position at t = 2 sec is -ft, find its position at t = 5 sec. -22 ft 2 ft ft d) 2 ft e) 22 ft x ) Let F(x) = t dt. Which of the following +e statements is/are true? (calculator) I. F ()=5 II. F(2)<F(6) III. F is concave upward I only II only III only d) I and II only e) I and III only Questions 2 &. The graph below consists of a quarter circle and two line segments and represents the velocity of an object during a 6-second interval. ) The object s acceleration (units/sec 2 ) at t =.5 is - -2 d) -/ e) π / ) Using midpoint approximation with three subintervals, what is the approximate area of the function y = 6x x 2, on the interval [, 6] d) 8 e) 5 5) Using a Trapezoid approximation, what is the approximate area of the function in Problem using 6 subintervals? d) 6 e) 6 2) The object s average speed in (units/se during the 6-second interval is π+ 6 π 6 - d) -/ e)

3 Free Response I (Non-Calculator) X f(x) 7 7 f (x) -5-5 Let f(x) be a twice-differentiable function on the closed interval [-, ]. The table above gives the values of f and its derivative f for selected points along the interval. The second derivative of f has the property f > for all x in the open interval (-, ). Compute ( 2 f '( x) f "( x)) dx. Show the work that leads to your answer. Write an equation for the line tangent to the graph f at the point (-2, ). Use this line to approximate f(-2.). Is this approximation greater or less than the actual value of f(-2.)? Justify your answer. Find a positive number r such that there must be a point on c on the open interval (, ) with f ( = r. Give a reason for your answer. d) Is it possible that f (x) is increasing for all x on the open interval (-, )? Justify your answer. Free Response II (Calculator) Water is draining out of a tank at a variable rate as given by the chart. t R(t) gal/min R(t) is a differentiable function Approximate the volume of water that has leaked from the tank t 5 using a Riemann sum with a righthand endpoint for the five unequal intervals indicated by the data in the chart. Interpret the meaning of R ( t) dt and find its value with appropriate units using the data from part a. 2 Use the data from the table to find R (25). Show the computations that lead to your answer. d) If the rate of the leak is modeled by Q ( t) 6.78sin(.5x.25). 6, at what time is the rate of the leak increasing the fastest?

4 Answer Sheet ) 2) ) ) 5) 6) 7) 8) 9) ) ) 2) ) ) 5) Free Response I (No-Calculator) d)

5 Free Response II (Calculator) d)

CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS. Second Fundamental Theorem of Calculus (Chain Rule Version): f t dt

CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS d d d d t dt 6 cos t dt Second Fundamental Theorem of Calculus: d f tdt d a d d 4 t dt d d a f t dt d d 6 cos t dt Second Fundamental