Fill in the Blnks for the Big Topis in Chpter 5: The Definite Integrl Estimting n integrl using Riemnn sum:. The Left rule uses the left endpoint of eh suintervl.. The Right rule uses the right endpoint of eh suintervl.. The Midpoint rule uses the midpoint of eh suintervl.. The Trpezoid rule uses the verge from the left nd right rules, i.e. f ( t) dt Formul for Trpezoid estimte using Left nd Right estimtes: Trp( n) If the grph of f is inresing on [, ], then f ( x ) If the grph of f is deresing on [, ], then f ( x ) If the grph of f is onve up on [, ], then f ( x ) If the grph of f is onve down on [, ], then f ( x ) = F '( t) dt = totl hnge of F (t) etween t = nd t = Averge vlue of f from to = If f is even, then f ( x) = If g is odd, then g ( x) = Given x, f ( x) = f x) ( + f ( x) = ( f ( x) g( x)) = f ( x) =
Ch 5 Review: Free Response Non-Clultor. Clulte the ext vlue of. It is known tht y f ( x) 8 nd x. (Use the geometri formul.) f ( x), find f ( x).. Show the following on the grph: ) f ( ) f ( ) f ( ) f ( ) ) line whose slope is ) n re F( ) F( ) where F ' f d) f ( )( ) e) verge vlue of f f ( x) Clultor. Set up n integrl nd lulte the verge veloity of vt ( ) 5(.) t in the first seonds if t represents seonds nd v represents ft/se. 5. Find the re under the urve --------------------- onsisting of semi-irle nd tringle 6. ) Grph f ( x) x x x. - ) Find 5 f ( x ). ) Set up integrls tht represent the totl re etween the urve nd the x-xis. d) Find the verge vlue of the funtion on the intervl [, ] (Alwys show your work nd give the nswer to the third deiml ple.) 7. If 5 5 f ( x) 5 nd f ( x), then f ( x)? 8. A prtile moves long line so tht its veloity t time t is v( t) t t 6 (mesured in meters per se). ) Find the displement of the prtile during the time period t. ) Find the distne trveled during this time period.. Use the Midpoint Rule with n 5 to pproximte x.
Non-Clultor Chpter 5 Review: Multiple Choie. x (A) (B) (C) ½ (D) (E) - For questions -5, x f ( x) f '( t) dt nd the grph of f is shown.. Whih of the following is/re true? I. f(-) = II. f() < f() III. f () < f () (A) I only (B) II only (C) III only (D) I nd II (E) I, II nd III. Whih of the following is/re true out the grph of f? I. f is inresing on (-, ) only II. f is inresing on (-, ) nd (, 5) III. f is deresing on (, ) (A) I only (B) II only (C) III only (D) I nd III (E) none. Whih of the following is/re true out the grph of f? I. f is onve up on (-, ) II. f is onve up on (, ) III. f is onve down on (, 5) (A) I only (B) II only (C) III only (D) II nd III (E) none 5. Whih of the following is/re true out the grph of f? I. f hs reltive minimum t x = II. f hs reltive minimum t x = III. f hs reltive mximum t x = (A) I only (B) II only (C) III only (D) I nd II (E) II nd III 6. The integrl 6 x gives the re of (A) irle of rdius (B) semiirle of rdius (C) qurter of irle of rdius (D) n ellipse whose minor xis is 7. If f(x) is ontinuous on the intervl [, ] nd < <, then f ( x) is equl to (A) f ( x) f ( x) (B) f ( x) ( ) (C) f ( x) f ( x) f x (D) f ( x) f ( x) (E) f ( x) ( ) f x
Clultor 8. Let f e ontinuous funtion on the losed intervl,. If f(x), then the gretest possile vlue of f (x) is (A) (B) (C) (D) 8 (E) 6. If f (x) nd f (x) 7, then f (x) (A) (B) (C) (D) (E). The grph of pieewise-liner funtion f, for x, is shown. Wht is the vlue of f (x)? (A) (B).5 (C) (D) 5.5 (E) 8. If F nd f re ontinuous funtions suh tht F( x) f ( x) for ll x, then f ( x) (A) F () F () (B) F () F () (D) F() F() (E) none of the ove (C) F() F(). The veloity of prtile moving on line t time t is v t 5t meters per seond. How mny meters did the prtile trvel from t to t? (A) (B) (C) 6 (D) 8 (E) 8 Prolems nd refer to the digrm ove. A ug egins to rwl up vertil wire t time t. The veloity v of the ug t time t, t 8, is given y the funtion whose grph is shown.. At wht vlue of t does the ug hnge diretion? (A) (B) (C) 6 (D) 7 (E) 8. Wht is the totl distne the ug trveled from t to t 8? (A) (B) (C) (D) 8 (E) 6 5. If the definite integrl e x is first pproximted y using two insried retngles of equl width nd then pproximted y using the trpezoidl rule with n, the differene etween the two pproximtions is (A) 5.6 (B).5 (C) 7.8 (D) 6.8 (E).78
6. A prtile with veloity t ny time t given y v(t) e t moves in stright line. How fr does the prtile move from t to t? e (A) e (B) e (C) e (D) e (E) 7. Find the verge vlue of y x on the intervl [-, -] (A).68 (B).75 (C). (D).8 (E).8 8. Clulte the pproximte re of the shded region in the figure y the trpezoidl rule, using divisions t x nd x 5. (A) 5 7 (B) 5 8 (C) 7 (D) 7 5 (E) 77 7 nd hs vlues. The funtion f is ontinuous on the losed intervl, 8 tht re given in the tle. Using the suintervls, 5, 5, 7, nd 7, 8 8, wht is the trpezoidl pproximtion of f (x)? x 5 7 8 f( x) (A) (B) (C) 6 (D) (E). During the worst -hour period of hurrine the wind veloity, in miles per hour, is given y v( t) 5t t, t. The verge wind veloity during this period (in mph) is (A) (B) (C) (D).667 (E) 8.667. If ftory ontinuously dumps pollutnts into river t the rte of 8 t tons per dy, then the mount dumped fter 7 weeks is pproximtely (A).7 ton (B). ton (C).55 tons (D). tons (E).7 tons Answers:. B. A. B. E 5. E 6. B 7. D 8. D. E. B. D. D. C. B 5. D 6. A 7. B 8. D. C. D. E 5