. h I B. Average velocity can be interpreted as the slope of a tangent line. I C. The difference quotient program finds the exact value of f ( a)
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1 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) = or or average of te two.675 =.4 (ii) T (.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 (v) Te rate of cange of T( x ) at x. T () = 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.
2 ( ) Gs (). Find G () using an algebraic approac. s G( ) G() G lim 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 Te values of te derivative F ( x) are given below: x.4 3 F ( x) 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 ( )
3 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 ) and f ( A ) (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 Average rate of cange [,5] = Average rate of cange [,5] =./5 =.4 III. 4 IV Average rate of cange [,5] = 5/5 = Average rate of cange [,5]= m = /5 =.
4 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 ln( ) ln( ) f '( ) lim lim f '( ) lim lim (next page for te tables)
5 ( ) ln( ) 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 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) At 3 days a new employee s percentage of defective part rate is decreasing.34 percent per day.
6 Students 4. (a) f '( ) 6 (b) f '() (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 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.
7 C. Estimate coordinates of te inflection point. Explain te significance of tis point in terms of te problem cange in concavity 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, , 76 (b) N '(4) 88,856. Function is increasing, etc. 8, 97, , 67 (c) N '(6) 77, ,856 77,335 (d) N "(6),88. Concave up rate of cange is increasing (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
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