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Unformatted text preview: A.P. Calculus Copy original problem. Convince me that you understand the concept!
4 Name _______________________________________ Per ____________ Date _______________ Chapters 7 & 8 I II III
Use the definition of definite integral to show that ∫ 6 dx = 12 .
2 Exam
(10 pts) (10 pts) (hint: The definition has nothing to do with a graph.) Differentiate: g( x ) = x ∫ 2 2 cos t dt + ∫ cos t dt
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2 2 3 (20 pts total) Given the three equations: x + y = 4, y = x + 2, and y = x. A) Sketch the three equations on the same axis. Shade the region above the xaxis bounded by these curves. B) Set up, but Do Not Evaluate the definite integral(s) which describe the shaded region in part A. Give a very explicit explanation of your methods. The value IV 1 b−a
av ∫ f ( x ) dx a b is called the mean (or average) value of f on the interval [ a, b ] and is usually
2 denoted f . Let f ( x ) = sin x for 0 ≤ x ≤ a) b) c) d) e) f) π (20 pts total) Describe the window setting necessary to yield a clear view of the function in the given interval on your calculator. Sketch your graph. Predict the mean value, f , of f on 0, π . Call your prediction A. av Use the program RiemannC with n = 100 to find the approximation of the integral. Compute f using av this value and call this value B. Use fnInt (Y , X , 0, π ) / π (found in the MATH menu). Call this value C. 1 Compare the values A, B, and C. Briefly explain. [ V The velocity v (t ) , in ft/sec, of a car traveling on a straight road, for 0 ≤ t ≤ 50 , is shown in a table of values for v (t ) , at 5 second intervals of time.
50 Approximate the definite integral, ∫ v (t )dt with a riemann sum, using the mid0 VI points of five subintervals of equal length. Using correct units, explain the meaning of this integral. (20 pts) 1 Given f ( x ) = x and g( x ) = . Draw the graph showing the area computed x and compute the exact area described. (tot 20) a) t v(t ) (seconds) (feet per second) 0 0 5 12 10 20 15 30 20 55 25 70 30 78 35 40 45 50 81 75 60 72 ∫ f ( x ) dx − ∫ g( x ) dx
1 1 2 2 b) 5 pts ∫ f ( x ) dx + ∫ g( x ) dx
0 1 1 2 Extra Credit Lemma 3, used by our text in the proof of the Fundamental Theorem of Calculus states: f ( X )(b − a) = ∫ f ( x ) dx . a b Using the function in Section IV. Compute the “cap X” value. ...
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This note was uploaded on 03/12/2010 for the course CALCULUS 1561234586 taught by Professor Zalla during the Spring '10 term at Air Force Institute of Technology, Ohio.
 Spring '10
 Zalla
 Calculus

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