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. Use the summation formulas to rewrite the n i i expression without the summation notation. i 1 n (n 1) ÎÍ (n 1) n È (n 1) ÎÍ n(n 1) 1 n (n 1) ÎÍ n(n 1) n È (n 1) ÎÍ n(n 1) n È (n 1) ÎÍ n(n 1) n. Find the indefinite integral 11 x 11 x 11 C 11 x 11 C x C 11 x C 11 x C dx.. Evaluate the following definite integral by the limit definition. 1 z dz 1 1. Use sigma notation to write the sum j 1 1 j 1 1 j 1 1 j 1 1 j 1 j 1 1 j 1 j 1 j 1 j
2 . Find the indefinite integral 1z ˆ 1z dz. 1z 1z C z 1z C z z z C 1z 1z z C z 1z z C. Use the properties of summation and Theorem. to evaluate the sum. i 1 i ˆ,,1,1,. Find the indefinite integral sint costdt. cost sint C cost sint C cos t sin t C cos t sin t C cost sint C. Find the indefinite integral t ˆ 1t dt.. Use left endpoints and rectangles to find the approximation of the area of the region between the graph of the function x x 1 and the x-axis over the interval ÎÍ,1. Round your answer to the nearest integer Use the limit process to find the area of the region between the graph of the function y x and x-axis over the interval ÎÍ,., 1,,,1 11. Find the sum given below. k k t t C t 1t C t t C 1t 1 C t 1t C
3 1. Find the indefinite integral ( 1z )dz. z z C 1z z C z z 1 none of the above 1. Evaluate the following definite integral by the limit definition. 1s ds 1. Write the limit lim c i c i ˆ x i, as a x i 1 definite integral on the interval ÎÍ, where c i is any point in the i th subinterval. c i c i c i ˆ x i x ( x) dx x( x) dx x ( x) dx n c i c i ˆ x i
4 1. Evaluate the integral. s ˆ ds given, x dx, 1. Evaluate the following definite integral by the limit definition. 1 u ˆ du x dx xdx 1,, 1. Write the following limit as a definite integral on the interval ÎÍ, where c i is any point in the i th subinterval. dx 1. lim x n i 1 c i c i x i 1 1 x dx 1x dx x x dx x x dx x x dx
5 1. Evaluate the definite integral u du. Use a 1 graphing utility to verify your results Use the summation formulas to rewrite the n 1k k 1 expression ( ) without the summation notation. k 1 n n n 1 n n n n n n n n. Evaluate the definite integral of the algebraic function. u du Use a graphing utility to verify your results n ˆ 1. Write the limit lim c x i, as a definite i 1 i integral on the interval ÎÍ,, where c i is any point in the i th subinterval. x dx x dx x dx x dx x dx. Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. s ˆ ds..
6 . Find the average value of the function fx ( ) 1x over the interval s Evaluate the integral. ( 1z ) dz given, x dx x dx xdx dx 1., 1, 1,. Evaluate the definite integral of the algebraic function. ( t )dt Use a graphing utility to verify your results Find the area of the region bounded by the graphs of the equations y x x, x, y. Round your answer to the nearest whole number. 1 1
7 . Write the following limit as a definite integral on the interval [, ], where c i is any point in the ith subinterval.. Determine the area of the given region. y x ( 1 x) lim x n i 1 ( x )dx ci ˆ x i ˆ x x dx ( x )dx ( x )dx x ˆ x dx. Evaluate the definite integral of a function x Use a graphing utility to verify your results.,,,,,1 1 dx. 1. Find F (x) given x Fx ( ) s ds. x F(x) x F(x) F (x) 1x F(x) x F(x)
8 1. Evaluate the definite integral of the algebraic function. x 1dx Use a graphing utility to verify your results.. Determine all values of x in the interval ÎÍ 1, for x ˆ which the function fx ( ) equals its x average value. x x x x x. Find F (x) given x Fx ( ) ( 1s )ds. x F (x) 1s F (x) 1x F (x) x F (x) F (x) s
9 M1c Answer Section MULTIPLE CHOICE 1. ANS: B PTS: 1 DIF: Medium REF: Section. OBJ: Rewrite a sum without summation notation. ANS: B PTS: 1 DIF: Easy REF: Section.1 OBJ: Evaluate the indefinite integral of a function. ANS: B PTS: 1 DIF: Easy REF: Section. OBJ: Evaluate a definite integral by the limit definition. ANS: E PTS: 1 DIF: Easy REF: Section. OBJ: Write a sum in sigma notation. ANS: C PTS: 1 DIF: Easy REF: Section.1 OBJ: Evaluate the indefinite integral of a function. ANS: B PTS: 1 DIF: Medium REF: Section. OBJ: Evaluate a sum using summation properties. ANS: B PTS: 1 DIF: Easy REF: Section.1 OBJ: Evaluate the indefinite integral of a function. ANS: C PTS: 1 DIF: Easy REF: Section.1 OBJ: Evaluate the indefinite integral of a function. ANS: A PTS: 1 DIF: Medium REF: Section. OBJ: Approximate the area bounded by a function using rectangles 1. ANS: B PTS: 1 DIF: Medium REF: Section. OBJ: Calculate the area bounded by a function using the limiting process MSC: Application 11. ANS: B PTS: 1 DIF: Easy REF: Section. OBJ: Calculate a sum given in sigma notation 1. ANS: A PTS: 1 DIF: Easy REF: Section.1 OBJ: Evaluate the indefinite integral of a function 1. ANS: A PTS: 1 DIF: Easy REF: Section. OBJ: Evaluate a definite integral by the limit definition 1. ANS: D 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. 1. ANS: D PTS: 1 DIF: Easy REF: Section. OBJ: Evaluate a definite integral by the limit definition 1. ANS: E PTS: 1 DIF: Easy REF: Section. OBJ: Write a limit as a definite integral on an interval 1. ANS: D PTS: 1 DIF: Easy REF: Section. 1. ANS: E PTS: 1 DIF: Medium REF: Section. OBJ: Rewrite a sum without summation notation. ANS: B PTS: 1 DIF: Difficult REF: Section. 1
10 1. ANS: D PTS: 1 DIF: Easy REF: Section. OBJ: Write a limit as a definite integral on an interval. ANS: B PTS: 1 DIF: Easy REF: Section. OBJ: Evaluate a definite integral geometrically. ANS: B PTS: 1 DIF: Easy REF: Section. OBJ: Calculate the average value of a function over a given interval. ANS: C PTS: 1 DIF: Easy REF: Section.. ANS: C PTS: 1 DIF: Medium REF: Section.. ANS: D PTS: 1 DIF: Medium REF: Section. OBJ: Calculate the area bounded by a function MSC: Application. ANS: A PTS: 1 DIF: Easy REF: Section. OBJ: Evaluate a definite integral by the limit definition. ANS: C PTS: 1 DIF: Medium REF: Section.. ANS: D PTS: 1 DIF: Medium REF: Section. OBJ: Calculate the area bounded by a function MSC: Application. ANS: B PTS: 1 DIF: Medium REF: Section. OBJ: Calculate the derivative of an integral using the Second Fundamental Theorem of Calculus 1. ANS: B PTS: 1 DIF: Medium REF: Section.. ANS: C PTS: 1 DIF: Easy REF: Section. OBJ: Identify the points where a function equals its average value over a given interval. ANS: D PTS: 1 DIF: Medium REF: Section. OBJ: Calculate the derivative of an integral using the Second Fundamental Theorem of Calculus
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