Central Bucks School District



UNIT 3: DERIVATIVES – STUDY GUIDESection 1:Limit Definition (Derivative as the Slope of the Tangent Line)Section 2:Calculating Rates of Change (Average vs Instantaneous)AVERAGE VELOCITYINSTANTANEOUS VELOCITYvavg=st2-st1t2-t1s't=vtAVERAGE ACCELERATIONINSTANTANEOUS ACCELERATIONaavg=vt2-vt1t2-t1s''t=v't=atSection 3:Power Rule and Exponentials/LogsPOWER RULEy=xn→y'=n?xn-1fx=exfx=ax fx=lnx fx=logaxf'x=exf'x=axln?(a)f'x=1xf'x=1xln(a)Section 4:Product & Quotient RulesPRODUCT RULEQUOTIENT RULEy=fx?gx→y'=fx?g'x+f'x?gxy=fxgx→y'=gx?f'x-fx?g'xgx2Section 5:Chain RuleCHAIN RULEy=fgx→y'=f'gx?g'xSection 6:Complex DerivativesSection 7:Trigonometric Derivative RulesTRIGONOMETRIC DERIVATIVESy=sinx→y'=cosxy=cosx→y'=-sinxy=tanx→y'=sec2xy=cscx→y'=-cscxcotxy=secx→y'=secxtanxy=cotx→y'=-csc2xSection 8:Trigonometric Derivatives using Trig IdentitiesSection 9:Trigonometric Derivatives using Chain RuleUnusual Angle2.) Raised to a Power3.) CombinationSection 10:Higher Order Derivatives (Projectile motion applications)Section 11:Implicit Derivativesf' NOTATIONy' NOTATIOND NOTATIONLEIBNIZ NOTATIONf'xy'DxydydxUNIT 3: DERIVATIVES – REVIEWDirections: Find f'x.1.)fx=12x4-2x3+x+62.)fx=4x+5x23.)fx=4x+63x24.) fx=4ex+6x7 5.) fx=log3(x2-8)Directions: Find y'.6.)y=x2+3xx3-17.)y=2x5cotx8.)y=3x5x2-19.)y=x2-2x+5x2+2x-3Directions: Find dydx.10.)y=3x4-5611.)y=3x2+6x312.) y=ln?(4x3+6x-1)13.) y=4x2+3xDirections: Use the appropriate rule when finding the derivative.14.)If f2=-2, f'2=1 and g2=13, g'2=4a.)f+g'2=b.)g-f'2=c.)g?f'2=d.)fg'2=Directions: Find Dxy.15.)y=csc4x16.)y=tan5-4x217.)y=cos43xDirections: Find f'x.18.)fx=secxcosx+sin2xcscx19.)fx=tanxcosxcotxDirections: Find f4x and f4-1 for the following function.20.)fx=3x6-5x4+6x3-7xDirections: Find the implicit derivative.21.)8x2-2y2=422.)cosy+6xy=3x2Directions: Find the slope and equation of the tangent line on the graph of the following.23.)fx=2x3-5x at the point 2, 624.)x2y2=9 at the point 1, 3Directions: Answer the following questions.25.)Find the point(s) on the graph of fx=x3-6x2-10x where the slope is 5.26.)The position of an object is given by st=t3+4t.What is the average velocity of the object in the time interval 1, 4?27.)An object travels so that its position is given by st=2t4+5t3-t+2.Find the instantaneous velocity at t=4.28.)A particle has a position function of st=8t5-5t4.Find the instantaneous acceleration at t=3.29.)A particle has a position function of st=3t2-6t+18.At what time will the velocity equal 6 units/sec?30.)Phillies star, Shane Victorino “The Flyin’ Hawaiian” jumps off a cliff into the water below and his position (feet) is defined after t seconds by the following: st=-16t2+32t+48. Use the Limit Definition of the Derivative to solve these problems.a.)What is his initial height? What is his initial velocity?b.)When does he reach his maximum height? What is his maximum height?c.)What other moment in time is his height the same as his initial height? What is his velocity?d.)When does “The Flyin’ Hawaiin” reach the water? What is his impact velocity?31.)Show the setup for finding the derivative using the limit definition.a.)fx=x2-2x+5b.)gx=2x+3c.)hx=cosxSOLUTIONS1.)f'x=2x3-6x2+12.)f'x=-4x2-10x33.)f'x=2x+43x4.) f'x=4ex+42x6 5.) f'x=2x(x2-8)ln36.)y'=5x4+12x3-2x-37.)y'=-2x5csc2x+10x4cotx8.)y'=-15x2-35x2-129.)y'=4x2-16x-4x2+2x-3210.)dydx=72x33x4-5511.)dydx=-18x-54x2+6x412.) dydx=12x2+64x3+6x-113.) dydx=2x+3ln44x2+3x14.)a.) 5b.)3c.)5d.)2116915.)Dxy=-4csc4xcotx16.)Dxy=-8xsec25-4x217.)Dxy=-12sin3xcos33x18.)f'x=cosx19.)f'x=-sinx20.)f4x=1080x2-120f4-1=96021.)y'=4xy22.)y'=6x-6y-siny+6x23.)mtan=19 y=19x-3224.)mtan=-3 y=-3x+625.)-1, 3 & 5, -7526.)vavg=25 units/sec27.)v4=751 units/sec28.)a3=3780 units/sec229.)t=2 sec30.)a.)s0=48ft, v0=32ft/secb.)t=1 sec, s0=64ftc.)t=2 sec, v2=-32ft/secd.)t=3sec, v3=-64ft/sec31.)a.)b.)c.) ................
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