AP Calculus AB Unit 5 (Ch. 6): The Definite Integral: Day 12 Chapter 6 Review


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1 AP Clculus AB Unit 5 (Ch. 6): The Definite Integrl: Dy Nme o Are Approximtions Riemnn Sums: LRAM, MRAM, RRAM Chpter 6 Review Trpezoidl Rule: T = h ( y + y + y +!+ y + y 0 n n) **Know how to find rectngle nd trpezoidl pproximtions when the intervls re not evenly spced (different se widths)** Simpson s Rule: S = h ( y + y + y + y +!+ y + y + y 0 n n n) o Integrl s re clcultion o Rules for Definite Integrls. Order of Integrtion: =. Zero: = 0. Constnt Multiple: k f ( x) = k f ( x). Sum nd Difference ( f x) ± g( x) ) = f ( x) g( x) ± ( 5. Additivity: If < < c, then + = 6. MxMin Inequlity If f min nd f mx re the minimum nd mximum vlues of f on [, ], c c then f min ( ) f (x) f mx ( ). Domintion If f (x) g(x) on [, ] then f (x) g(x) 8. Averge (Men) Vlue vg( f ) = o Men Vlue Theorem for Integrls ( ) If f is continuous on [, ], then t some point c in [, ], f (c) = o Averge Vlue of functions from integrls: vg( f ) = f (x) (only if f integrle on [, ]) f (x) ( ) f (x) o Fundmentl Theorem of Clculus, prt : F "(x) = d f (t)dt = f (x) o Fundmentl Theorem of Clculus, prt : f (x) = F() F() x
2 Let R e the re in the first qudrnt enclosed y the xxis nd the grph of the function y = x x.. Sketch four equl suregions to find the LRAM pproximtion of R.. Sketch four equl suregions to find the MRAM pproximtion of R. c. Sketch four equl suregions to find the RRAM pproximtion of R. d. Sketch four equl suregions to find the trpezoidl pproximtion of R. e. Find the exct vlue of R y using the Fundmentl Theorem of Clculus (prt ).
3 . Show tht y = x cos(x) + e t dt stisfies oth of the following conditions: i. y!! = + sin x e cos x cos x e cos x ii. y! = π nd y = π when x = π. Which of the following is the grph of the function whose derivtive is dy? Explin your nswer. = x nd whose vlues t x = is Multiple Choice Questions. A prticle moves long line with velocity v(t) = t t. During the intervl from t = 0 to t =, wht is the net chnge in the prticle s position (i.e., the displcement)? (A) /6 (B) / (C) / (D) / (E) none of these 5. The verge vlue of the function f ( x) = 9 + x on the intervl x 0 is (A) 0.0 (B) 0.08 (C) 0. (D) 0.6 (E) none of these 6. If = 5 nd g ( x) =, then ll of the following must e true except (A) ( f ( x) g( x)) = (B) ( g ( x) f ( x)) = 6 (C) f ( x + ) = 5 (D) ( f ( x)) = 0 (E) g( x) = 5. Suppose f(x) nd F(x) re continuous functions for ll rel x, nd F(x) is n ntiderivtive of f(x). Which of the following equls + h lim f ( x)? h 0 h (A) f ʹ() (C) f() (E) none of these (B) f(0) (D) F()
4 8. The grph of the function f, consisting of three line segments, is shown elow. Let g( x) = x f t) dt. Find g( ) nd g ( ) (.. Find the instntneous rte of chnge of g, with respect to x, t x =. c. Find the solute minimum vlue of g on the closed intervl [, ]. Justify your nswer. d. The second derivtive of g is not defined t x = nd x =. How mny of these vlues re x coordintes of points of inflection of the grph of g? Justify your nswer. 9. Do this prolem without clcultor until you get to question d. Show complete work done y hnd for prts nd c. Let A represent the re underneth the function f (x) = cos x ( ) etween = 0 x nd π x =.. Tell how A could e found if you hd prior knowledge of the vlue of c cos( x) for some nd c of your choice.. Estimte A using the midpoint rectngle pproximtion method (MRAM) with four rectngles of equl width. Go s fr s possile without using clcultor. c. Find A exctly using the Fundmentl Theorem, without using your clcultor. d. Now use your clcultor to find deciml vlues for your nswers to questions nd c. Briefly comment on the ccurcy of the Trpezoidl pproximtion in this prolem. 0. Including strtup costs, it costs printer $50 to print 5 copies of newsletter, fter which the mrginl cost of x copies is dc = dollrs per copy. Find the totl cost of printing 500 newsletters. x
5 Answers:...5,..5, c..5, d..5, e... y' = x + d cos(x) e t dt = x e cos x sin x y FTC (Prt ), y'' = # $ e cos x cos x + sin x e cos x ( sin x) % & = + sin x e cos x cos x e cos x cos(π ). When x = π, y = (π ) + e t dt = π + e t dt = π, nd y' = (π ) sin(π )e cosπ = π 0 e = π. y = x +, so the middle grph.. D 5. B 6. E. C 8. See AP scoring guide on ck π / 9.. A = cos( ) = 0 π / x cos( x), ecuse ( x) 0 cos is horizontl shrink y of ( x) π π π 5π π π. MRAM = (cos( ) + cos( ) + cos( ) + cos( )) = ( = π ) cos. c. A = π / 0 cos( ) π x = sin( ) sin( 0) ( ) = 0 = 8. d. Prolem : MRAM 0., prolem c: A 0., so MRAM is out 0.0 off (or out 5% off) from the ctul re. 0. C(x) = x + 0, C(500)=$0 5
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