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 .
(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
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 x-axis 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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