dollars for a week of sales t weeks after January 1. What is the total revenue (to the nearest hundred dollars) earned from t = 10 to t = 16?

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1 MATH 7 RIOHONDO SPRING 7 TEST (TAKE HOME) DUE 5//7 NAME: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. ) A department store has revenue from the sale of electrical kitchen appliances that is given approximately by R(t) =. +. cos t for t 5, where R(t) is revenue in hundreds of dollars for a week of sales t weeks after January. What is the total revenue (to the nearest hundred dollars) earned from t = to t = 6? ) Provide an appropriate response. / ) Find the exact value of cos x sin x dx. ) ) Find the indefinite integral - 6x sin x dx. ) ) Integrate: sin x cos x dx ) Find the derivative. 5) y = x cos (8x ) 5) 6) d dx cos(x - ) 6) 7) y = tan 5 x 7) Evaluate using integration by parts. 8) x dx 8) x + 9) x ln 9x dx 9) ) x e x dx ) ) 6x ln x dx )

2 Compute the area of the shaded region. ) ) y = x + y = -x + - Use the midpoint rule with the given number of subintervals to approximate the integral. Round to three decimal places, when necessary. ) e x dx ; n = ) - Use a Riemann sum to approximate the area under the graph of f(x) on the given interval, with selected points as specified. ) f(x) = e -x + from x = - to x = 6, n = ; use right endpoints round the answer to two decimal places ) 5) The rate at which an assembly line worker's efficiency E (expressed as a percent) changes with respect to time t is given by E'(t) = 8-8t, where t is the number of hours since the worker's shift began. Assuming that E() = 96, find E(t). 5) Evaluate the improper integral whenever it is convergent. If it is divergent, state this. 6) dx x(ln x) 6) 7) This is a sketch of the region between the two curves f(x) = x, g(x) = 8 - x. Compute this area. 7) Enter just a reduced fraction of form a b.

3 8) In a certain city, the cost of a taxi ride is computed as follows: There is a fixed charge of $.95 as soon as you get in the taxi, to which a charge of $.6 per mile is added. Find a linear equation that can be used to determine the cost, C, of an x-mile taxi ride. 8) Determine the value of k that makes the function f(x) continuous at x =. 9) f(x) = k(x + 9) for x + x for x < 9) Find f'(x) using limits. ) f(x) = x x - 8 ) ) Suppose that the cost C of removing p% of the pollutants from a chemical dumping site is given by C(p) = $, - p. Can a company afford to remove % of the pollutants? Explain. ) Provide an appropriate response. ) Find the equation(s) of the tangent line(s) to the graph of y - xy - = at x =. ) Round your answer, if appropriate. ) Water is being drained from a container which has the shape of an inverted right circular cone. The container has a radius of. inches at the top and a height of 5. inches. At the instant when the water in the container is. inches deep, the surface level is falling at a rate of in./sec. Find the rate at which water is being drained from the container. ) ) A rectangular garden is to be fenced in and divided into three parallel sections. The fencing for the boundary costs $ per foot whereas the fencing for the dividing fences costs $ per foot. Consider the problem of finding the dimensions of the largest garden possible if the gardener can spend for the fencing. Find the values of l and w for which the total area of the garden is as large as possible. ) Provide an appropriate response. 5) A man 6. ft tall walks at a rate of 6 ft/sec away from a lamppost that is 5 ft high. At what rate is the length of his shadow changing when he is 67 ft away from the lamppost? 5) 6) Graph of F(x) = x State the intercepts, asymptotes, inflection points, relative x + extrema, domain, and intervals where the funstion in increasing, decreasing, concave upward and downward. 6)

4 7) A company has found that its rate of expenditure (in hundreds of dollars) on a certain type of job is given by E'(x) = x +, where x is the number of days since the start of the job. Find the total expenditure if the job takes days. 7) 8) Find the area of the region bounded by y = 6x - x and y = x - x. Enter a reduced fraction of form a b. 8) Evaluate the improper integral whenever it is convergent. If it is divergent, state this. 9) 8 dx 9) (x + ) ) Suppose the annual rate of income from an investment at any time t is K(t) = - + 5t. What is the formula for the present value of the income over the next 5 years at a 6% interest rate compounded continuously? ) / ) Consider f(x)dx, where f(x) = e -x. Find a number A such that f (x) A for all x satisfying x. Use this A to obtain a bound on the error of using Simpson's rule with ) n = 5 to approximate the definite integral. Determine the integral by making an appropriate substitution. ) e x + xe x + xe x dx ) Enter your answer with any coefficients in front as integers or reduced fractions of form a b. Approximate the integral by the trapezoidal rule. 5 ) (ln x) 5 dx; n = Express your answer to five decimal places. ) Evaluate the integral. / ) x cos x dx ) Use Simpson's rule to approximate the integral using the indicated value of n (so there are n subintervals). 5) (x + ) dx ; n =, Write answer as whole number or reduced fraction. 5)

5 6) x dx; n =, Round to two decimal places. 6) + Determine the integral by making an appropriate substitution. 7) sec x dx 7) ( - tan x) / 8) Find an upper bound for ET in estimating (6x + 6) dx with n = 5 steps. 8) Use the Trapezoidal Rule to approximate the integral using the indicated value of n. 9) 5x dx ; n =, Write answer as a whole number or reduced fraction. 9) ) x + 9 dx ; n =, Round to three decimal places. ) 5

6 Answer Key Testname: MATH7TESTTAKEHOME ) $ ) - ) cos x + C ) -8(cos x) / + C 5) y' = - 6x sin (8x ) + x cos (8x ) 6) - x sin(x -)[cos (x - )] -/ 7) tan x sec x 8).9 9) 5 x5 ln 9x - 5 x5 + C ) x ex - x ex + ex + C ). ) 9 ).65 ). 5) E(t) = 8t - t + 6) ln 7) 6 8) C =.6x ) k = 9 ) f'(x) = -8 (x - 8) ) No, the cost of removing p% of the pollutants increases without bound as p approaches. ) y = x + and y = - x - ). in./s ) l = 5 ft, w = ft 5) 7 ft/sec 6

7 Answer Key Testname: MATH7TESTTAKEHOME 6) 7) $ 8) 6 9) 8 5 ) (5t - )e -.6t dt ) ) ln + xe x + C ).59 ) - 5) 7 6).9 7) - ( - tan x)/ + C 8) 5 9) 55 ) 7.8 7