MATH 2413 TEST ON CHAPTER 4 ANSWER ALL QUESTIONS. TIME 1.5 HRS
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1 MATH 1 TEST ON CHAPTER ANSWER ALL QUESTIONS. TIME 1. HRS M1c Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the general solution of the differential equation below and check the result by differentiation. dy du 9 u Yu ( ) 9 1 u Yu ( ) u 9 Yu ( ) 1 1 u Yu ( ) 9 9 u e. Yu ( ) u 9. Find the indefinite integral ( 8t )dt. 8t t t t t t 8 e. none of the above. Find the indefinite integral 1u ˆ u du. u u 1u u 1u u u u e. u. Find the indefinite integral 9t ˆ 1t dt. 9t 1t t t 1t t t t t 18t 1t e. 9t 1t. Find the indefinite integral 1s ˆ 1s ds. 9s 1s s s s s s s s s s e. s s. Find the indefinite integral 1 x 8 e. 8 1 x x x x x 1 1 dx.
2 . Find the indefinite integral 11tan x 1 dx. 11tanx x 11 tan x 1x 11 tan x 1x 11tanx x e. 11tanx x 8. Use the properties of summation and Theorem. to evaluate the sum. i 1 ( i ) 111 e Use the limit process to find the area of the region between the graph of the function y 1 x and È x-axis over the interval ÎÍ,. e Evaluate the following definite integral by the limit definition. 1 s ds 9. Use the properties of summation and Theorem. to evaluate the sum. 9 i 1 i ˆ 9,,1,81 e., Use the properties of summation and Theorem. to evaluate the sum. 1 e. 1 i 1 i i ˆ 8,1, 8,0 8,00 e.,0
3 1. Evaluate the following definite integral by the limit definition. 10 1u du e. 1. Evaluate the following definite integral by the limit definition. 9s ˆ ds Write the following limit as a definite integral on the interval [, ], where c i is any point in the ith subinterval. lim x 0 n i 1 ci ˆx i x ˆ x dx ˆ x x dx ( x )dx ( x )dx e. ( x )dx 19 e. 1
4 1. Write the following limit as a definite integral on È the interval ÎÍ, where c i is any point in the i th subinterval. 18. Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. lim x 0 n i 1 c i c i x i t dt x dx x x dx 9x 1x dx e. 81 x x dx e. x dx 1. Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. 0 ( t 1)dt e. 10
5 19. Evaluate the integral. xdx given, x dx x dx xdx dx ,01 e.,0 110, 1, 1, 0. Find the area of the region bounded by the graphs of the equations y x x, x, y 0. Round your answer to the nearest whole number e Find the average value of the function fx ( ) 8 1x over the interval s e.. Find the average value of the function over the given interval and all values t in the interval for which the function equals its average value. ft () t,1t t Use a graphing utility to verify your results. The average is 9 and the point at which the function is equal to its mean value is. The average is 9 and the point at which the function is equal to its mean value is and. The average is 9 and the point at which the 0 function is equal to its mean value is. The average is 9 and the point at which the 0 function is equal to its mean value is. e. The average is 9 and the point at which the 0 function is equal to its mean value is and.
6 È. Determine all values of x in the interval ÎÍ 1, for x 1 ˆ which the function fx ( ) equals its x average value 1. x x x 1 x e. x. Find F (x) given x Fx ( ) t dt. x F(x) 9x F(x) 0 F (x) 18x F(x) 8x e. F(x). Solve the differential equation. df du 1u 8u ˆ u 1 fu ( ) 8u ˆ u 1 1 fu ( ) 8u ˆ u 1 8u fu ( ) 8u ˆ u 1 8 fu ( ) 8u ˆ u 1 1 e. fu ( ) 8u ˆ u 1. Find the indefinite integral of the following function. cosudu cosu sinu sinu sinu e. sinu. Find the indefinite integral of the following function. sinu cos u du e. ( cos u) ( sinu) ( cos u) ( cosu) ( sinu) 8. The rate of depreciation dv / dt of a machine is inversely proportional to the square of t,where V is the value of the machine t years after it was purchase The initial value of the machine was $00,000, and its value decreased $100,000 in the first year. Estimate its value after years. Round your answer to the nearest integer. $00,000 $1,000 $,000 $0,000 e. $0,000
7 9. The sales S (in thousands of units) of a seasonal product are given by the model S..sin t where t is the time in months, with t 1 corresponding to January. Find the average sales for the first quarter ( 0 t ). Round your answer to three decimal places thousand units 9. thousand units.89 thousand units 10.9 thousand units e. 9.9 thousand units 0. Find the smallest n such that the error estimate from the error formula in the approximation of the definite integral x dx is less than using the Trapezoidal Rule. 9 0 e. 1 0
8 M1c Answer Section MULTIPLE CHOICE 1. ANS: B PTS: 1 DIF: Medium REF: Section.1 OBJ: Calculate the general solution of a differential equation. ANS: C PTS: 1 DIF: Easy REF: Section.1. ANS: A PTS: 1 DIF: Easy REF: Section.1. ANS: C PTS: 1 DIF: Easy REF: Section.1. ANS: C PTS: 1 DIF: Easy REF: Section.1. ANS: E PTS: 1 DIF: Easy REF: Section.1. ANS: A PTS: 1 DIF: Medium REF: Section.1 8. ANS: A PTS: 1 DIF: Medium REF: Section. OBJ: Evaluate a sum using summation properties 9. ANS: B PTS: 1 DIF: Medium REF: Section. OBJ: Evaluate a sum using summation properties 10. ANS: E PTS: 1 DIF: Medium REF: Section. OBJ: Evaluate a sum using summation properties 11. ANS: B PTS: 1 DIF: Medium REF: Section. OBJ: Calculate the area bounded by a function using the limiting process MSC: Application 1. ANS: B PTS: 1 DIF: Easy REF: Section. OBJ: Evaluate a definite integral by the limit definition 1. ANS: A PTS: 1 DIF: Easy REF: Section. OBJ: Evaluate a definite integral by the limit definition 1. ANS: B PTS: 1 DIF: Easy REF: Section. OBJ: Evaluate a definite integral by the limit definition 1. ANS: C PTS: 1 DIF: Easy REF: Section. OBJ: Evaluate a definite integral by the limit definition 1. ANS: B PTS: 1 DIF: Easy REF: Section. OBJ: Write a limit as a definite integral on an interval 1. ANS: C PTS: 1 DIF: Easy REF: Section. OBJ: Evaluate a definite integral geometrically 18. ANS: A PTS: 1 DIF: Easy REF: Section. OBJ: Evaluate a definite integral geometrically 19. ANS: B PTS: 1 DIF: Easy REF: Section. OBJ: Evaluate the definite integral of a function 0. ANS: D PTS: 1 DIF: Medium REF: Section. OBJ: Calculate the area bounded by a function MSC: Application 1
9 1. ANS: B PTS: 1 DIF: Easy REF: Section. OBJ: Calculate the average value of a function over a given interval. ANS: A PTS: 1 DIF: Medium REF: Section. OBJ: Calculate the average value of a function over a given interval and identify the point at which it occurs. ANS: E PTS: 1 DIF: Easy REF: Section. OBJ: Identify the points where a function equals its average value over a given interval. ANS: B PTS: 1 DIF: Medium REF: Section. OBJ: Calculate the derivative of an integral using the Second Fundamental Theorem of Calculus. ANS: E PTS: 1 DIF: Medium REF: Section. OBJ: Solve a differential equation. ANS: C PTS: 1 DIF: Easy REF: Section. using substitution. ANS: A PTS: 1 DIF: Medium REF: Section. using substitution 8. ANS: C PTS: 1 DIF: Medium REF: Section. OBJ: Solve a differential equation in applications MSC: Application 9. ANS: D PTS: 1 DIF: Medium REF: Section. OBJ: Evaluate the definite integral of a function in applications MSC: Application 0. ANS: D PTS: 1 DIF: Medium REF: Section. OBJ: Identify the smallest value of n needed to approximate a definite integral to within a desired degree of accuracy
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