f(x)

Transcription

1 Calculus m y_0e1^y jkdudtdaw ZS[oifntCwxaCrJej ilhl[cq.k i qatlplm mrpiyg^hztbsz YrmePsqeWrNvxeEdG. Calculus Ch. 3 Review Given the graph of f '(x), sketch a possible graph of f (x). 1) f '(x) f(x) 8 8 Name Date Period x x For each problem, find the equation of the line tangent to the function at the given point. Your answer should be in slope-intercept form. ( ) y = -e -x at 1, - 1 e) 3) y = - 15 x + 5 at ( 15-3, - 1) For each problem, find the equation of the line normal to the function at the given point. If the normal line is a vertical line, indicate so. Otherwise, your answer should be in slope-intecept form. ) y = -sin (x) at ( p, -1 ) 5) y = sin (x) at ( - p, - ) For each problem, find the points where the tangent line to the function is horizontal. Indicate if no horizontal tangent line exists. 6) y = -cos (x); [-p, p] 7) y = -(x + ) 3 N J`0Y1cK _KauQtSae `SworfTtswsaRrjev vlxlmcr.f A HAZlQlS JrqiMgQhstBsz CrMeCs_ePrDvqeEd[.G ] KMEazeA owgijtdh^ BIGnsfIipnPiittew HCqa_lucXuSlMuBsL. -1-

2 Use the definition of the derivative to find the derivative of each function with respect to x. 8) y = -5x + 1 9) y = - 1 x - 1 For each problem, find the average rate of change of the function over the given interval and also find the instantaneous rate of change at the leftmost value of the given interval. 10) y = x + 1; [-, - 3 ] For each problem, find ( f -1) '(a) 11) f (x) = x 3 +, a = 1) f (x) = 3 x + 5, a = Differentiate each function with respect to x. 13) y = log x S pr0l1xm XKjuatwao gseomfutjwra]raeq ]LFLjCW.] K CAilFlU UrgiugihMtpsr ErIeusneGrDvXeddA.] K SMAaKd\eS GwPiVtchU pifnxf]iuntistpej GCXarl[cWuElhuysF. --

3 1) y = 3x - 1 log 5 x 5 15) y = 3x3 + 5x 16) y = cot 3x (x 5 + 5) 17) y = (-5x 3 + 1)sin x V `o0c1ep TKUuVtRaJ YSUobfCtQwMabrXen elylyct.d Z wa`lplt SrqiHgPhJtcs` grzensiekrevfeld[.m s GMEaodAe[ qwliptlhy XIJnkfiiWntigtIe_ KCVaFlGcVutlhuDs[. -3-

4 For each problem, use implicit differentiation to find in terms of x and y. dx 18) 5x 3 = (3y + ) 19) 5x = tan y Differentiate each function with respect to x. 0) y = cos -1 x 1) y = sin -1 x 5 For each problem, use implicit differentiation to find d y in terms of x and y. dx ) = x 3 - y For each problem, you are given a table containing some values of differentiable functions f (x), g(x) and their derivatives. Use the table data and the rules of differentiation to solve each problem. 3) x f (x) f '(x) g(x) g'(x) Part 1) Given h 1 (x) = f (x) g(x), find h 1 '(1) Part ) Given h (x) = f (x) g(x), find h '() Part 3) Given h 3 (x) = ( f (x)), find h 3 '(3) K Qb0`1rf FKcuetoac tseosfrtewbaorlev `LeLACc.k Y iaglslh yriidgphztes\ WraeJsceIrcvreudt.V o OMoa\daeT VwxiTthhk IIqnOfeiFnUiRtPeu MCyaal^cxuDlouDsJ. --

5 Calculus ` MC0L1Wi zk]uktya[ GSKopfLtzwHaWrveN olplicn.l K IAQlolk trfi[gchdtnsg ZrzessSeXr^vleJdR. Calculus Ch. 3 Review Given the graph of f '(x), sketch a possible graph of f (x). 1) f '(x) f(x) 8 8 Name Date Period x x For each problem, find the equation of the line tangent to the function at the given point. Your answer should be in slope-intercept form. ( ) y = -e -x at 1, - 1 e) 3) y = - 15 x + 5 at ( 15-3, - 1) y = 1 e x - e y = x For each problem, find the equation of the line normal to the function at the given point. If the normal line is a vertical line, indicate so. Otherwise, your answer should be in slope-intecept form. ) y = -sin (x) at ( p, -1 ) 5) y = sin (x) at ( - p, - ) Normal line is vertical line at x = p Normal line is vertical line at x = - p For each problem, find the points where the tangent line to the function is horizontal. Indicate if no horizontal tangent line exists. 6) y = -cos (x); [-p, p] ( (-p, -1), - p, 1 ), (0, -1), ( p, 1 ), (p, -1) 7) y = -(x + ) 3 No horizontal tangent line exits. l xq0i1ww \Kju[tPai QSeosfDtywzacrPek YLAL[Ch.M j YAdlLlb wrlibgwhvtbsp UrLeKsPecrwvue_da.i x FMEa_d`ea Aw]iWtDhl CIWn`fzinnViUtlex ccvatltcouzlquusj. -1-

6 Use the definition of the derivative to find the derivative of each function with respect to x. 8) y = -5x + 1 dx = - 5-5x + 1 9) y = - 1 x - 1 dx = 1 x - x + 1 For each problem, find the average rate of change of the function over the given interval and also find the instantaneous rate of change at the leftmost value of the given interval. 10) y = x + 1; [-, - 3 ] Average: - 7 Instant.: - For each problem, find ( f -1) '(a) 11) f (x) = x 3 +, a = ( f -1 )'(a) = 1 6 1) f (x) = 3 x + 5, a = ( f -1 )'(a) = 6 Differentiate each function with respect to x. 13) y = log x dx = 1 x ln x = x ln q mq0s1lq jkmurtbav hsboifntvwcawrpee hlcl_cd.b ^ IAhlXlL arsifg^hqtsso DrgeysaeAraveeMdt.s H imwa_deej qwbiitmhj YIwncf\innciStdex ZCNaHlIcGuGltubsy. --

7 1) y = 3x - 1 log 5 x 5 log 5 x5 6x - (3x - 1) dx = (log 5 x 5 ) = 6x log 5 x 5 ln 5-15x + 5 x ln 5 (log 5 x 5 ) 1 x 5 ln 5 10x 15) y = 3x3 + 5x dx = 9x - (3x 3 + ) 5x 5x ln 10x ( ) 5x = x(9x - 30x3 ln - 0 ln ) 5x 16) y = cot 3x (x 5 + 5) dx = cot 3x 5x + (x 5 + 5) -csc 3x 1x 3 = x 3 (5xcot 3x - 1x 5 csc 3x - 60 csc 3x ) 17) y = (-5x 3 + 1)sin x dx = (-5x3 + 1) cos x x 3 + sin x -15x = x (-0x cos x + xcos x - 15sin x ) W Wx0a1bZ EKeubt`aT `SvoYfmt`wHa[rGej ultlmch.d E faglrlj ir`ixgchqtfsm brmecsieurhvoe\da.l e nmlardke_ owfiotphb ]IQnifMiOnpi]tDeK _Cva[lkc]ulluuNsK. -3-

8 For each problem, use implicit differentiation to find in terms of x and y. dx 18) 5x 3 = (3y + ) dx = 5x 1y y 19) 5x = tan y dx = 15x ysec y sec y Differentiate each function with respect to x. 0) y = cos -1 x dx = (x ) x3 = - x x 8 1) y = sin -1 x 5 dx = (x 5 ) 0x 0x = 1-16x 10 For each problem, use implicit differentiation to find d y in terms of x and y. dx ) = x 3 - y d y dx = 6xy - 9x y 3 For each problem, you are given a table containing some values of differentiable functions f (x), g(x) and their derivatives. Use the table data and the rules of differentiation to solve each problem. 3) x f (x) f '(x) g(x) g'(x) h 1 '(1) = f (1) g'(1) + g(1) f '(1) = 5 h '() g() f '() - f () g'() = (g()) = 1 3 h 3 '(3) = f (3) f '(3) = - Part 1) Given h 1 (x) = f (x) g(x), find h 1 '(1) Part ) Given h (x) = f (x) g(x), find h '() Part 3) Given h 3 (x) = ( f (x)), find h 3 '(3) a XJ0J1Ak GK^uwtRaf MSDoRfttIwwabrneO RLVLbCf.y q fasltlu vrdixggh^ttsm HryeIsAesrCvUeOdu.A X um_aydrel Dw_iqtqhy `IrnjfDiLnliWtEeZ SCsa]lRcAutlZuFsO. --