Capter Review Packet (questions - ) KEY. In eac case determine if te information or statement is correct (C) or incorrect (I). If it is incorrect, include te correction. f ( a ) f ( a) I A. represents te slope of te line between points a, f ( a) and a, f ( a ). I B. Average velocity can be interpreted as te slope of a tangent line. I C. Te difference quotient program finds te exact value of f ( a). C D. Te slope of a function gt () at t 3 can be expressed as g (3). C E. Instantaneous velocity can be positive, negative, or zero. f ( x ) f ( x) I F. f( a) lim. xa y C G. Te slope of a tangent line is expressed as x. I H.. (i) T (.4) =..6.85 or 3...5.4.8.4 or average of te two.675 =.4 (ii) T (.4) = 3..8.75.6..4 is not on te table so te best average is te smallest =. (iii) T(.4 ) T(.4) lim =.675 or any answer from (i) since it is te same question (iv) Te average rate of cange of T ( x) between x.4 and x.4. T.4 T.4.75.675.95.4.4 (v) Te rate of cange of T( x ) at x. T () =..6.85.4 te smallest =.4 (vi) Find te equation of te tangent line to T( x ) at x. Slope =.85 te point is (,.6) equation is y x.85.6 x 3. F( x). Estimate F () using a numerical approac (table of values). Estimate to 3 decimal places, may ave to go very small to acieve tis. See table on te next page If you only go to -. you can t see te value approacing a number to 3 decimal places, you aren t even sure up to decimal places, so you must go smaller As got very small te value seems to be approacing 3.59 so for 3 decimal places F 3.5 or 3.6 if you rounded to 3 decimal places. Make sure your cart sows te function value approacing tis value. You must ave at least 4 values tat ave te same value for 3 decimal places.
( ) -..567 -..768 -. 3.3 -. 3.58 -. 3.58 -. 3.59. 3.59. 3.59. 3.59. 3.6. 5.895 4. Gs (). Find G () using an algebraic approac. s G( ) G() G lim 4 4 4 4 lim lim lim 4 4 lim(4 ) 4 (4 ) lim 4 4 lim 4( ) 4 lim 4 4 Just for practice te equation of te tangent line at s = is 3 y x 4 4 5. Te values of te derivative F ( x) are given below: x.4 3 F ( x) 3 3.5 Use tis to estimate te values of te function missing in te following table Must consider wat te derivative means cange in y for a cange in x So at x = te rate of cange is, one unit tat x increases y increases by We are going only.4 not a wole unit; so ratio y so y.8.4 At.4 te rate of cange is 3; so 3 y so y.8.6 x.4 3 Fx ( ) 8 8.8.6
6. Sketc a grap of a function, f( x ), wit te following properties: Answers will vary general sape is given. f (3) 6, f (3), f (8) is undefined, lim f( x), as x ; f ( x), x f( x) for x, x 8, f( x ) is continuous and defined everywere. 7. f '( A). terefore y. y.(.5). x (a) f ( A ) 4. 4. and f ( A ) 4. 3.89 (b) smaller since f(x) is concave up. 8. Eac of te graps below sows te position of a particle moving in a line as a function of time. During te indicated time interval, wic particle as A) Constant velocity IV B) Greatest initial velocity I C) Greatest average velocity III D) Zero average velocity I E) Zero acceleration IV F) Positive acceleration III Velocity slopes of te distance function Acceleration concavity of distance function I. II.. 5-5 Average rate of cange [,5] = Average rate of cange [,5] =./5 =.4 III. 4 IV.. 5-5 Average rate of cange [,5] = 5/5 = Average rate of cange [,5]= m = /5 =.
9. Sketc te grap of x ( ) if it as te following caracteristics: for x in te interval, ( x) ( ) 3 for x in te interval, ( x) (orizontal function) (linear function were te slope is ).. Consider te function ln x x gx ( ) x.7 C x A. Determine te value of C so tat tis function is continuous at x. x lim (ln( x)) ln(). lim (.7 C).7 C To be continuous te function value at x= must x equal. C. 7 x B. Now determine if tis function is differentiable at x. Prove it. x x.7.7.7.7.7.7 ln( ) ln( ) f '( ) lim lim f '( ) lim lim (next page for te tables)
( ).7.7 -..98 -..94 -..967 -..96 ln( )..9995..9999995..999999995..999999999995 - Since limits are not approacing te same value, te limits of te difference quotient does not exist at x=. Tat means tere are different slopes as x approaces. Te function is not differentiable at x =... Let p ( ) be te pressure on a diver (in dynes per square cm) at a dept of meters below te surface of te ocean. Determine wat eac of te quantities below represent in practical terms. Include units. A. p () te pressure in dynes per square centimeter at meters below te surface B. p ( ) te pressure in dynes per square centimeter is te pressure at te divers meter plus meters below te surface. C. p (5) driver. te dept in meters below te surface were tere is 5 dynes per square cm of pressure on te D. p () te rate (in dynes per square centimeter per meter) at wic te pressure is canging wen te driver is at meters. 3. Suppose te percent P of defective parts produced by a new employee t days after te employee starts t 75 work can be modeled by Pt (). 5( t ) A. Find P(3) and interpret its meaning in terms of te problem. 3 75 P ( 3).5 A new employee after 3 days will ave.5% defective parts. 5(3 ) B. Estimate P (3) and interpret its meaning in terms of te problem. P(3 ) P(3) P'(3) lim (Use te difference quotient or evaluate function on calculator) P ( 3 ) P(3) -. -.3446 -. -.344. -.344. -.3446. -.3446 At 3 days a new employee s percentage of defective part rate is decreasing.34 percent per day.
Students 4. (a) f '( ) 6 (b) f '() 5. 6. (a) Use calculator. (b) Use calculator. (c) Noting. (d) move values to rigt by k. 7. Te registrar as put a counter on te RSVP registration telepone lines to count te total number of students registering during te day. A grap of Nt (), te total number of students wo ave registered during te t ours since noon, is given below. 3 5 5 5 4 6 8 Hours A. Estimate N() and give an interpretation. Make a tangent line at t=. Te points (I found) (3, 75) and (4, 5) so te slope is 5 At ours after noon or pm, te pone registration is increasing 5 calls per our. B. Estimate N () and give an interpretation. Find on te vertical and see were tat intersects te function (in 4.5 ours) At 4:3 pm tere ave been students wo ave registered.
C. Estimate coordinates of te inflection point. Explain te significance of tis point in terms of te problem. 3.75 cange in concavity. 3.75 ours since noon, at 3:45pm te students registering starts to slow down. (Te rate at wic registering students starts to decrease) 8. (a) abortions/year, 97, 66 586, 76 (b) N '(4) 88,856. Function is increasing, etc. 8, 97, 66 988, 67 (c) N '(6) 77,335 4 88,856 77,335 (d) N "(6),88. Concave up rate of cange is increasing. 4 9. (a) (i) ( 3, ) (.5, ) (ii) (, 3) (,.5) (iii) -3, -,.5 (b) (i) (, ) (,4) (ii) (,) (4, ) (iii) -,, 4 (c) (i) - (ii). t DNE t g t t t t t t '( ) 3