AP Calculus AB Unit 4 Assessment

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1 Clss: Dte: 0-04 AP Clulus AB Unit 4 Assessment Multiple Choie Identify the hoie tht best ompletes the sttement or nswers the question. A lultor my NOT be used on this prt of the exm. (6 minutes). The slope field for ertin differentil eqution shown below. Whih of the following ould be solution to the differentil eqution with the initil ondition y(0) =? ) y = x d) y = os x b) y = x e) y = + x ) y = e x. (x x + ) dx = ) x x + C d) (x x + ) + C b) x x + x + C e) none of these ) x x + x + C

2 . Bsed on the funtion f shown in the grph below. On whih of the following intervls is f ontinuous? ) x 0 d) 0 < x < b) x e) none of these ) x dx 4. If the substitution u = x + is used, then is equivlent to 0 x x + du ) d) 0 (u )(u + ) du b) e) 0 u(u ) ) du u du u du u(u )

3 5. x x dx = ) ln d) b) ln e) 4 ) ln 6. The best liner pproximtion for f(x) = tnx ner x = π 4 is ) + (x π 4 ) d) + (x π 4 ) b) + (x π 4 ) e) + (x π 4 ) ) + (x π 4 ) 7. lim x 0 os x x is ) d) b) 0 e) ) none of these 8. If sinxy = x, then dy dx = se xy y ) x b) se xy e) ) se xy x d) se xy + se xy x

4 9. The funtion f is given by f(x) = x +. The figure below shows portion of the grph of f. Whih of the x + b following ould be the vlues of the onstnts nd b? ) =, b = d) =, b = 4 b) =, b = 4 e) =, b = ) =, b = 0. The funtion f is ontinuous on the losed intervl [0,6] nd hs the vlues given in the tble below. The trpezoidl pproximtion for f(x)dx found with subintervls of equl length is 5. Wht is the vlue of k? ) 0 d) b) 6 e) 7 ) x f(x) 4 k 8. Let f be the funtion given by f(x) = (x ) 5 (x + ). Whih of the following is n eqution for the line tngent to the grph of f t the point where x =? ) y = 0x + d) y = x 9 b) y = 0x 8 e) y = x + ) y = x 9. (t ) dt = 0 ) 0 d) 6 b) e) 4 ) 4 4

5 . If f is the funtion given by f(x) = t t dt, then f () = ) d) 0 b) 7 e) ) 4 x 4. The eqution of the urve whose slope t point (x,y) is x nd whih ontins the point (, ) is ) y = x d) y = x x 4 b) y = x 0 e) y = x 0 ) y = x x 5. The re of the shded region in the figure is equl extly to ln. If we pproximte ln using LRAM with n = nd RRAM with n =, whih inequlity follows? ) b) ) < x dx < d) < x dx < e) < x dx < 0 < x dx < 5 6 < x dx < 5

6 6. 4 t dt = ) b) (4 t) + C d) (4 t) / + C 4 (4 t) / + C e) (4 t) / + C ) 6 (4 t) + C 7. x lim x is x ) none of these d) b) 0 e) ) 8. The re of the lrgest retngle tht n be drwn with one side long the x-xis nd two verties on the urve of y = e x is ) d) e e b) e) e e ) e A grphing lultor is REQUIRED for some questions on this prt of the exm. (6 minutes) 9. If f(u) = sinu nd u = g(x) = x 9, then (f û g) () equls ) 6 d) none of these b) 9 e) 0 ) 0. The line y = x + k is tngent to the urve y = x when k is equl to ) 0 d) or b) or e) or ) 4 or 4 6

7 . os α dα = 0 π / 4 ) 0.5 d) 0.44 b).44 e).000 ).000 b. If f(x) is ontinuous on the intervl x b nd b, then f(x) dx is equl to ) f(x) dx f(x) dx d) f(x) dx + f(x) dx b b) f(x) dx f(x) dx e) f(x) dx + f(x) dx b ) f(x) dx f(x) dx b b b. The funtion f is differentible nd hs vlues s shown in the tble below. Both f nd f re stritly inresing on the intervl 0 x 5. Whih of the following ould be the vlue of f ()? x f(x) ) 0 d) 0 b) 7.5 e) 0.5 ) 9 Ï 4. If f(x) = Ô Ì ÓÔ x x x, x 0 k, x = 0 nd if f is ontinuous t x = 0, then k = ) d) b) e) 0 ) 7

8 4 5. The integrl 6 x dx gives the re of 4 ) qudrnt of irle of rdius 4 b) irle of rdius 4 ) semiirle of rdius 4 d) none of these e) n ellipse whose semi-mjor xis is 4 6. When 0 + x dx is estimted using n = 5 subintervls, whih is (re) true? Ê ˆ I. LRAM = Ë Á Ê ˆ II. MRAM = Ë Á III. Trpezoidl = 0. Ê ˆ Ë Á ) II nd III only d) III only b) I nd II only e) II only ) I nd III only 7. The figure below shows the grph of f, the derivtive of the funtion f, on the open intervl 7 < x < 7. If f hs four zeros on 7 < x < 7, how mny reltive mxim does f hve on 7 < x < 7? ) Three d) Two b) Five e) Four ) One 8

9 8. If f(x) is ontinuous on the intervl x b, if this intervl is prtitioned into n equl subintervls of length x, nd if x, is number in the kth subintervl, then lim ) f(b) f() b) F(x) + C, where df(x) dx ) f(x) dx b d) none of these e) F(b ), where df(x) = f(x) dx n = f(x) nd C is n rbitrry onstnt n f(x k ) x is equl to 9. If f(x) dx = 6 nd f(x) dx = 4, then Ê Ë Á + f(x) ˆ dx = 0 5 ) d) 50 b) 0 e) 0 ) Differentible funtion f nd g hve the vlues shown in the tble below. If D = g, then D () = ) b) 9 ) 9 d) e) 9

10 Free Response A grphing lultor is REQUIRED for some questions on this prt of the exm. (5 minutes) Ê. The funtion g is defined for x > 0 with g() =, g' (x) = sin x + ˆ Ë Á x, nd g (x) = Ê Ë Á x ˆ os Ê x + ˆ Ë Á x. () Find ll vlues of x in the intervl 0. x t whih the grph of g hs horizontl tngent line. (b) On wht subintervls of (0.,), if ny, is the grph of g onve down? Justify your nswer. () Write n eqution for the line tngent to the grph of g t x = 0.. (d) Does the line tngent to the grph of g t x = 0. lie bove or below the grph of g for 0. < x <? Why? A lultor my NOT be used on this prt of the exm. (0 minutes) È. The funtion g is defined nd differentible on the losed intervl ÎÍ 7,5 nd stisfies g(0) = 5. The grph of y = g (x), the derivtive of g, onsists of semiirle nd three line segments, s shown in the figure below. () Find g() nd g( ). (b) Find the x-oordinte of eh point of infletion of the grph y = g(x) on the intervl 7 < x < 5. Explin your resoning. () The funtion h is defined by h(x) = g(x) x. Find the x-oordinte of eh ritil point of h, where 7 < x < 5, nd lssify eh ritil point s the lotion of reltive minimum, reltive mximum, or neither minimum nor mximum. Explin your resoning. 0

11 . Solutions to the differentil eqution dy dx = xy lso stisfy d y dx solution to the differentil eqution dy dx = xy with f() =. () Write n eqution for the line tngent to the grph of y = f(x) t x =. = y ( + x y ). Let y = f(x) be prtiulr (b) Use the tngent line eqution from prt () to pproximte f(.). Given tht f(x) > 0 for < x <., is the pproximtion for f(.) greter thn or less thn f(.)? Explin your resoning. () Find the prtiulr solution y = f(x) with initil ondition f() =.

12 0-04 AP Clulus AB Unit 4 Assessment Answer Setion MULTIPLE CHOICE. ANS: E DIF: DOK. STA: C 7.0. ANS: B DIF: DOK. STA: C 5.0. ANS: D DIF: DOK. STA: C.0 4. ANS: D DIF: DOK. STA: C ANS: B DIF: DOK. STA: C ANS: A DIF: DOK. STA: C ANS: B DIF: DOK. STA: C. 8. ANS: A DIF: DOK. STA: C ANS: B DIF: DOK. STA: C ANS: A DIF: DOK. STA: C.0. ANS: D DIF: DOK. STA: C 4.. ANS: A DIF: DOK. STA: C 7.0. ANS: E DIF: DOK. STA: C ANS: D DIF: DOK. STA: C ANS: E DIF: DOK. STA: C.0 6. ANS: D DIF: DOK. STA: C ANS: B DIF: DOK. STA: C. 8. ANS: C DIF: DOK. STA: C.0 9. ANS: A DIF: DOK. STA: C ANS: D DIF: DOK. STA: C 4.. ANS: D DIF: DOK. STA: C 5.0. ANS: B DIF: DOK. STA: C 5.0. ANS: D DIF: DOK. STA: C.0 4. ANS: B DIF: DOK. STA: C.0 5. ANS: C DIF: DOK. STA: C ANS: A DIF: DOK. STA: C.0 7. ANS: C DIF: DOK. STA: C ANS: C DIF: DOK.4 STA: C ANS: C DIF: DOK. STA: C ANS: E DIF: DOK. STA: C 4.4

13 ESSAY. ANS: 00B # DIF: DOK.4 STA: C 4. / C 4.4

14 . ANS: 00 #5 DIF: DOK.4 STA: C 4. / C4. / C 6.0

15 . ANS: 00 #6 DIF: DOK.4 STA: C 4. / C4. / C 6.0 4

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