7.2 Trapezoidal Approximation

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b f(x)dx a Trapezoidal Approximation for interval [1,3] with n

1 7. Trapezoidal Approximation NOTES Write your questions here! Riemann Sum f(x) = x + 1 [1,3] f(x) = x + 1 The Definite Integral b f(x)dx a Trapezoidal Approximation for interval [1,3] with n subintervals How close is it the true value? f(x) = x + 1 n = n = 4 n = 6

2 Given the definite integral (1 x )dx 1 (a) Use the Trapezoid Rule with three equal subintervals to approximate its value. (b) Is your answer from part (a) an overestimate or underestimate? (c) Calculate the exact value on your calculator. Compare to your answer in part (a). The rate at which customers are being served at Starbrusts is given by the continuous R(t). A table of selected values of R(t), for the time interval < t < 1 hours, is given below. At t = there had been customers served. Time (hours) R(t) (people/hour) Use a trapezoidal sum with 4 subintervals to approximate 1 R(t)dt Approximately how many customers had been served after 1 hours? SUMMARY: Now, summarize your notes here!

3 7. Trapezoidal Approximation PRACTICE You can use a calculator on 1-13 Sketch the trapezoidal approximations. Find the width of each subinterval. Write a definite integral. 1. n = 6 subintervals on [1,4]. n = 4 subintervals on [-3,] 3. n = 1 subintervals on [-1,4] Width of each subinterval = Width of each subinterval = Width of each subinterval = v(t) f(x) g(x) Definite Integral = Definite Integral = Definite Integral = Approximate the definite integral use trapezoidal approximation (9 + x 4 x )dx 5. ( x + 5 ) dx Trapezoidal approximation with 6 subintervals Trapezoidal approximation with 4 subintervals 6. (3 x + )dx 1 Trapezoidal approximation with 3 subintervals Use the calculator to find the exact value of the definite integral ( 1 5 x3 x + 1) dx 8. (e x + 4)dx 3π 9. (sin x)dx e 1. (4 9 x ) dx 1 π

4 Use the information provided to answer the following. 11. Let y(t) represent the rate of change of the population of a town over a -year period, where y is a differentiable function of t. The table shows the population change in people per year recorded at selected times. The population at t = was 5,5 people. Time (years) y(t) (people per year) a. Use a trapezoidal approximation with four subintervals to approximate y(t)dt b. What is the approximate population after years? 1. The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by the twicedifferentiable and strictly increasing function R of time t. A table of selected values of R(t) for the time interval t 9 minutes is shown below. At t = the plane had already consumed 84 gallons of fuel. Time (minutes) R(t) (gallons per min) a. Use data from the table to find an approximation for R (45). Show the computations that led to your answer. Indicate units of measure. 9 b. Approximate the value of R(t)dt data in the table. using trapezoidal approximation with five subintervals indicated by the c. Approximately how much fuel has the plane consumed after 9 minutes?

5 7. Trapezoidal Approximation TEST PREP MULTIPLE CHOICE You can use a calculator on If three equal subdivisions of [,4] are used, what is the trapezoidal approximation of dx? (A) e 4 + e + e + e (B) e 4 + e + e + e (C) 1 (e4 + e + e + e ) (D) 1 (e4 + e + e + e ) (E) 1 4 (e4 + e + e + e ) 4 ex. Suppose that f (x) > on (, 1), f (x) < on ( 1,1) and f (x) > on (1, ). Which of the following functions could be f? (A) f(x) = x x (B) f(x) = x 3 3x (C) f(x) = x + x (D) f(x) = x 4 4x (E) f(x) = x 3 + 3x 3. A continuous function f(x) has the values shown in the table. What is the value of a trapezoidal approximation 3 of f(x) using six equal subintervals? (A) 9 (B) 14 (C) 18 (D) 8 (E) 56 x f(x) Determine the slope of the normal line to the curve x 3 + xy = 1y at the point (,1). (A) (B) (C) 7 3 (D) 6 13 (E) 1

6 CALCULATOR ACTIVE FREE RESPONSE Your score: out of 6 A test plane flies in a straight line with positive velocity v(t), in miles per minute at time t minutes where v is a differentiable function of t. Selected values of v(t) for t 4 are shown in the table below. t (min) v(t) (mpm) (a) Use a midpoint Riemann sum with four subintervals of equal length and values from the table to approximate 4 4 v(t) dt. Show the computations that lead to your answer. Using correct units, explain the meaning of v(t) dt in terms of the plane s flight. (b) Based on the values in the table, what is the smallest number of instances at which the acceleration of the plane could equal zero on the open interval < t < 4? Justify your answer. (c) The function f, defined by f(t) = 6 + cos ( t ) + 3 sin (7t), is used to model the velocity of the plane, in 1 miles per minute, for t 4. According to this model, what is the acceleration of the plane at t = 3? Indicate units of measure. 4

A.P. Calculus BC First Semester Exam Calculators Allowed Two Hours Number of Questions 10

A.P. Calculus BC First Semester Exam Calculators Allowed Two Hours Number of Questions 10 Each of the ten questions is worth 10 points. The problem whose solution you write counted again, so that the maximum