ࡱ>  `b_MTTTg jbjbjVV .r<r<}&%NN8 t"$:fպo,պQλ888888rWTW>C08WWD8պպPN n: Section 2.2: Derivatives of Polynomials and Exponential Functions Practice HW from Larson Textbook (not to hand in) p. 99 # 1-49 odd, 75-81 odd Alternative Notations for the Derivative If  EMBED Equation.3 , then  EMBED Equation.3  is known as the derivative of y with respect to x. For example, if we have the function  EMBED Equation.3 , then we can write  EMBED Equation.3 . For doing intermediate computations, we have the following notation:  EMBED Equation.3  Thus, we can say  EMBED Equation.3  Also, to evaluate a derivative at a point, say x = a, we write  EMBED Equation.3  Hence, if  EMBED Equation.3 , then   EMBED Equation.3 . Basic Derivative Formulas  1.  EMBED Equation.3 , where k is a constant (for our purposes, a real number).  2.  EMBED Equation.3  3.  EMBED Equation.3  4.  EMBED Equation.3  5.  EMBED Equation.3  Example 1: Differentiate EMBED Equation.3 .  Solution: % Example 2: Differentiate EMBED Equation.3 .  Solution:    % Example 3: Differentiate  EMBED Equation.3 .  Solution:       %  Example 4: Differentiate EMBED Equation.3 .  Solution:            %   Properties of Differentiation 1.  EMBED Equation.3  (k is a constant) 2.  EMBED Equation.3  Example 5: Differentiate EMBED Equation.3 .  Solution:          % Example 6: Differentiate EMBED Equation.3 . Solution:         % Example 7: Differentiate EMBED Equation.3 . Solution:            %  Example 8: Differentiate EMBED Equation.3 . Solution: Using the following property of fractions that  EMBED Equation.3 , we first rewrite the function as  EMBED Equation.3  Differentiating  EMBED Equation.3 , we obtain  EMBED Equation.3  % Example 9: Find the equation of the line tangent to the graph of  EMBED Equation.3  at the point  EMBED Equation.3 .  Solution: To find the equation of any line, including a tangent line, we need a point (this is given to be  EMBED Equation.3 ) and the slope. Recall that the derivative at a point gives the slope of the tangent line at that point. To find a formula for calculating the slope, we calculate the derivative of the function which is given by  EMBED Equation.3  Then,  EMBED Equation.3  Then, using the slope intercept equation of a line (in terms of t) given by  EMBED Equation.3  we use the slope of the tangent line we just found m = 1 to find the equation of the tangent line as follows:  EMBED Equation.3  Hence, substituting the slope m = 1 and  EMBED Equation.3  into  EMBED Equation.3  gives the tangent line equation:   EMBED Equation.3  (Continued on next page) The graph of the function and its tangent line can be seen as follows using the Maple command > plot([sin(t)+2*t, t + Pi], t = -Pi..2*Pi, y = -10..15, color = [red, blue],thickness = 2, title = "Graph f(t) = sin(t)+t (red), tangent line y = t+Pi (blue) at point (Pi, 2Pi)");  % Example 10: Find the point)s on the graph of  EMBED Equation.3  that has a horizontal tangent line.  Solution: On this problem, a horizontal tangent line means that the slope of the tangent line is 0. Since the derivative gives a formula for the slope of the tangent line, we can find the point that gives a tangent line slope of 0 by taking the derivative of the function, setting it equal to 0, and solving for x. The result of this calculation is as follows:   EMBED Equation.3   To complete the problem, we must find the y-coordinate of the point by substituting  EMBED Equation.3  back into the original function  EMBED Equation.3 . Keeping in mind the inverse property  EMBED Equation.3 , this is done as follows:   EMBED Equation.3  Thus, the coordinates of the point that have a horizontal tangent line is   EMBED Equation.3  % Average Velocity Suppose the position of a moving object starting from rest is given by the position function  EMBED Equation.3  feet where t is the time given in seconds.  t EMBED Equation.3 0 1 2 3 4         We define the average velocity on the time interval from t = a to t = b as follows: Formula For Average Velocity  EMBED Equation.3 . Example 11: Find the average velocity for the time intervals [1, 3] and [3, 4] for an object if the position starting from rest is given by  EMBED Equation.3  .  Solution:    % Suppose we now desire to find the velocity of the object precisely when t = 1 second for the position function  EMBED Equation.3 . A method for approximating would involve finding average velocities on an interval that is close to t = 1. Example 12: Find the average velocity for the time intervals [1, 1.01] and [1, 1.001] for an object if the position starting from rest is given by  EMBED Equation.3 . Solution: To assist in the calculations, we find the position function  EMBED Equation.3  at the following times.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Then, using the average velocity formula  EMBED Equation.3  we obtain  EMBED Equation.3   EMBED Equation.3      % In general, for an object moving from time  EMBED Equation.3  to time  EMBED Equation.3 ,    Average Velocity on the time interval =  [t, t + h]  To get the instantaneous velocity at  EMBED Equation.3 , we let h!0 which gives the following definition.  Instantaneous Velocity and Instantaneous Rate of Change Given a position function  EMBED Equation.3 , the instantaneous velocity  EMBED Equation.3  is given by the derivative of the position function. That is,   EMBED Equation.3  In general, if we are given a function  EMBED Equation.3 ,  EMBED Equation.3   EMBED Equation.3 . Example 13: Find the instantaneous velocity for the position function  EMBED Equation.3  at t = 1.  Solution:     %  Example 14: The position function representing the height of a freely falling object is given by  EMBED Equation.3 , where  EMBED Equation.3  is the initial velocity of the object and  EMBED Equation.3  is the initial height of the ball at time t = 0. Here the height s is in feet and the time t is in seconds. Suppose someone throws a baseball from 6 feet off the ground with a initial velocity of 100 ft/s. a. Determine the position and velocity functions for the ball. b. Find the average velocity for the time intervals [4, 4.1], [4, 4.01], and [4, 4.0001]. c. Find the instantaneous velocity when t = 4 and t = 5 seconds. d. Find the time required for the ball to reach ground level.   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Find the velocity of the coin at impact. Solution part a: Since the ball starts 6 ft off the ground, the initial height is  EMBED Equation.3 . The initial velocity is  EMBED Equation.3  Substituting into the equation  EMBED Equation.3  gives the position equation   EMBED Equation.3  To get the velocity equation, we take the derivative of the position equation  EMBED Equation.3 . This gives   EMBED Equation.3   Solution part b: Recall that given a time interval  EMBED Equation.3 , if s represents the height of the ball (the position) after time  EMBED Equation.3 , then  EMBED Equation.3 .  To find the average velocities for the given intervals, we will use the following calculations:   EMBED Equation.3 .  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Hence, Continued on the next page  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Note that the negative average velocities indicate that the ball is falling down instead of going up. Solution part c: The average velocities found in part b indicate the instantaneous velocity at the specific time  EMBED Equation.3  should be close to -28 ft/sec. From part a, we found the equation for the instantaneous velocity at a particular time t to be  EMBED Equation.3   Thus, at  EMBED Equation.3  we have  EMBED Equation.3  We can easily use this same equation to find the velocity at t = 5 seconds.   EMBED Equation.3  Solution part d: When the ball reaches ground level, its height s = 0. Thus, the find the time when the ball reaches ground level, we set the position equation  EMBED Equation.3  and solve for t. Since this quadratic equation is not easily factorable, we use the quadratic formula to find its approximate solution. Continued on the next page Recall that the quadratic formula says that the solution to the quadratic equation is given by  EMBED Equation.3  is given by  EMBED Equation.3  For  EMBED Equation.3 , setting a = -16, b = 100, and c = 6 we obtain  EMBED Equation.3   Thus, the ball hits the ground in approximately 6.3 seconds. Solution part e: From part d, we found out the ball hits the ground after t = 6.3 seconds. To find the velocity when the ball impacts the ground, we substitute t = 6.3 into the velocity equation we found in part a  EMBED Equation.3 . 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KA?H1Z4깡jx|K2B* Rvfz `[YB2sSRs:V~. _D\Yt9 %;HgE%ļ2cd>`,aZ 0C@J {H^F[XAF[ٳL {+ss&p8|", su4C̶ [\{W I)$5dP"CD|b@#3XT{?Dd b ) c $A'? ?3"`?(2TrY4+?35>e~D`!]TrY4+?35>Z "@{5$+xڝOhW߼7v֝N֬`4XzAZ؍tddEDD<-KQ, S=`RzB%y>yo4`N?&0hAM- nll+6ޗ~&\.qt ,˸uƇX(cj4=FR: $oXc qí4Rsxry^Ш?Ve#ޢ{Q̓??wUSdQXϒr KDxo`p^5CompObj fObjInfoEquation Native A_1261038309F C Cy% x=ln(4) FMicrosoft Equation 3.0 DS Equation Equation.39qy+ N e lnu =uOle CompObjfObjInfoEquation Native G_1261038376F C COle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39qy¯ y=8ln(4)"2e ln(4) =8ln(4)"(2)(4)=8ln(4)"8Equation Native _1261038483&F C COle  CompObj!f FMicrosoft Equation 3.0 DS Equation Equation.39qyMH.& (ln(4), 8ln(4)"8) FMicrosoft Equation 3.0 DS EqObjInfo#Equation Native $i_1188467868F C COle &CompObj'fObjInfo)Equation Native *N_1188468872F C Cuation Equation.39q2[  s(t)=5t 2 FMicrosoft Equation 3.0 DS Equation Equation.39qOle ,CompObj-fObjInfo/Equation Native 0N2G s(t)=5t 2 FMicrosoft Equation 3.0 DS Equation Equation.39qI Average Velocityon t_1188469424F C COle 2CompObj3fObjInfo5Equation Native 6_1188469621w@F C COle >CompObj?fhe time interval[a,b]t=a  to  t=b   =   Change in DistanceChange in Time=s(b)"s(a)b"a FMicrosoft Equation 3.0 DS Equation Equation.39q2; s(t)=5t 2 FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoAEquation Native BN_1188469765F C COle D     < "!#$%&'()+*,.-/1024357689:=;>?@ABDCEFHGIJKLOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~CompObjEfObjInfoGEquation Native HN_1259667497F C C28]  s(t)=5t 2 FMicrosoft Equation 3.0 DS Equation Equation.39qPWP 0 s(1)=5(1) 2 =5(1)=5Ole JCompObjKfObjInfoMEquation Native Ns_1259667530F C COle PCompObjQfObjInfoS FMicrosoft Equation 3.0 DS Equation Equation.39qP—\ s(1.01)=5(1.01) 2 =5(1.0201)=5.1005Equation Native T_1259667576F C COle WCompObjXf FMicrosoft Equation 3.0 DS Equation Equation.39qP¯ s(1.001)=5(1.001) 2 =5(1.002001)=5.010005ObjInfoZEquation Native [_1159698316FCCOle _CompObj`fObjInfobEquation Native c_1259667730F C C FMicrosoft Equation 3.0 DS Equation Equation.39q, Average velocity on thetime interval [a,b]t=a  to  t=b=Change in distance (height)Change in Time=s(b)"s(a)b"a FMicrosoft Equation 3.0 DS Equation Equation.39qP  AveraOle kCompObjlfObjInfonEquation Native oge velocity on thetime interval [1, 1.01]t=1 to  t=1.01=s(1.01)"s(1)1.01"1=5.1005"50.01=0.10050.01=10.05 ft/sec_1259667841N F C COle xCompObj  yfObjInfo { FMicrosoft Equation 3.0 DS Equation Equation.39qP/*l Average velocity on thetime interval [1, 1.001]t=1 tEquation Native |K_1188469983F C COle CompObjfo  t=1.001=s(1.001)"s(1)1.001"1=5.010005"50.001=0.0100050.001=10.005 ft/sec FMicrosoft Equation 3.0 DS Equation Equation.39q# t 1 =t FMicrosoft Equation 3.0 DS Equation Equation.39q+ lL t 2 =t+ObjInfoEquation Native ?_1188469999F C COle CompObjfObjInfoEquation Native G_1188470201 !F C Ch FMicrosoft Equation 3.0 DS Equation Equation.39q# t 1 =t FMicrosoft Equation 3.0 DS EqOle CompObjfObjInfoEquation Native ?_1188470382F C COle CompObjfObjInfo uation Equation.39q <  s(t) FMicrosoft Equation 3.0 DS Equation Equation.39qXGB v(t)Equation Native 5_11884704080#F C COle CompObj"$fObjInfo%Equation Native 5_1281534791(F C COle  FMicrosoft Equation 3.0 DS Equation Equation.39q’:Lg v(t)=2s(t)='lim h!0 s(t+h)"s(t)hCompObj')fObjInfo*Equation Native _1188470535-F C COle CompObj,.fObjInfo/Equation Native = FMicrosoft Equation 3.0 DS Equation Equation.39q!` y=f(x) FMicrosoft Equation 3.0 DS Equation Equation.39q_1188470555+52F C COle CompObj13fObjInfo4Equation Native _1188470642:7F C COle CompObj68f0L' Average Rateof change on[a,a+h]=f(a+h)"f(a)h FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo9Equation Native 1_1188470779<F C COle `)` Instantaneous Rateof change att=a=2f(a)='lim h!0 f(a+h)"f(a)h FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj;=fObjInfo>Equation Native N_1259669493XDAFCC2xM s(t)=5t 2 FMicrosoft Equation 3.0 DS Equation Equation.39qPj 0 s(t)="1Ole CompObj@BfObjInfoCEquation Native 6t 2 +v 0 t+s 0 FMicrosoft Equation 3.0 DS Equation Equation.39qP`t v 0_1259669519IFF C COle CompObjEGfObjInfoHEquation Native 6_1259669569KF C COle CompObjJLf FMicrosoft Equation 3.0 DS Equation Equation.39qPR s 0 FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoMEquation Native 6_1259670001?PF CCOle CompObjOQfObjInfoREquation Native ?_1259670103UFCCP#  s 0 =6 FMicrosoft Equation 3.0 DS Equation Equation.39qP/\ v 0 =100.Ole CompObjTVfObjInfoWEquation Native K_1259669415ZFCCOle CompObjY[fObjInfo\ FMicrosoft Equation 3.0 DS Equation Equation.39qPW 0 s(t)="16t 2 +100t+6 FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native s_1259670155Sb_FCCOle CompObj^`fObjInfoaEquation Native 5_1259670175dFCCOle P  s(t) FMicrosoft Equation 3.0 DS Equation Equation.39qPUjĠ v(t)=2s(t)="32t+100CompObjcefObjInfofEquation Native q_1159628360iFCCOle CompObjhjfObjInfokEquation Native 9 FMicrosoft Equation 3.0 DS Equation Equation.39q$ [a,b] FMicrosoft Equation 3.0 DS Equation Equation.39q_1159628425gnFCCOle CompObjmofObjInfop F8 t FMicrosoft Equation 3.0 DS Equation Equation.39q,¯_dq s(4)="16(4) 2 +100(4)+6Equation Native )_1159698397sFCCOle CompObjrtfObjInfouEquation Native _1159698394lxFCCOle   "#$%&'(),/01234569<=>?@ABCDEHMPQRSTWZ[\]^adgjmpsvwxyz{|}~="256+400+6=150 ft FMicrosoft Equation 3.0 DS Equation Equation.39q,J s(4.1)="16(4.1) 2 +100(CompObjwyfObjInfozEquation Native _1159698411q}FCC4.1)+6="268.96+410+6=147.04 ft FMicrosoft Equation 3.0 DS Equation Equation.39q, s(4.01)="16(4.01) 2 +10Ole  CompObj|~ fObjInfo Equation Native 0(4.01)+6="257.2816+401+6=149.7184 ft FMicrosoft Equation 3.0 DS Equation Equation.39q,?h s(4.000_1159698418FCCOle CompObjfObjInfoEquation Native [_1159698427{FCCOle CompObjf1)="16(4.0001) 2 +100(4.0001)+6="256.01280016+400.01+6=149.99719984 ft FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo Equation Native !_1159699770FCCOle *,D Average velocity on thetime interval [4,4.1]t=4 to  t=4.1=s(4.1)"s(4)4.1"4=147.04"1500.1="2.960.1="29.6 ft/sec FMicrosoft Equation 3.0 DS Equation Equation.39q,@ Average velocity on tCompObj+fObjInfo-Equation Native .3_1159699785FCChetime interval [4,4.01]t=4 to  t=4.01=s(4.01)"s(4)4.01"4=149.7184"1500.01="0.28160.01="28.16 ft/secOle 7CompObj8fObjInfo:Equation Native ; 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