Chapter Six Chapter Six

Transcription

1 Chaper Si Chaper Si

2 0 CHAPTER SIX ConcepTess and Answers and Commens for Secion.. Which of he following graphs (a) (d) could represen an aniderivaive of he funcion shown in Figure.? Figure. (a) (b) (c) (d) (d). Because he graph in Figure. is decreasing, he graph of he aniderivaive mus be concave down. Because he graph in Figure. is posiive for <, zero for =, and negaive for >, hen = is a local maimum. You can mimic his problem wih =.

3 CHAPTER SIX. Which of he following graphs (a) (d) could represen an aniderivaive of he funcion shown in Figure.? π/ π Figure. (a) (b) π/ π π/ π (c) (d) π/ π π/ π (c). Because he graph in Figure. is alwas posiive on his inerval, he aniderivaive mus be increasing for his inerval. You can mimic his problem wih = sin. You could also ask our sudens o draw oher possible aniderivaives of he funcion shown in Figure..

4 CHAPTER SIX. Which of he following graphs (a) (d) could represen an aniderivaive of he funcion shown in Figure.? Figure. (a) (b) (c) (d) (a) and (d). Because he graph in Figure. is coninuall increasing, he graph of is aniderivaive is concave up. Noice ha he graphs in (a) and (d) differ onl b a verical shif. You could also poin ou ha since he graph in Figure. is negaive from < < 0, hen he graph of an aniderivaive mus be decreasing on his inerval. Also, since he graph in Figure. is posiive for 0 < <, hen he graph of an aniderivaive mus be increasing on his inerval.

5 CHAPTER SIX. Consider he graph of f () in Figure.. Which of he funcions wih values from he Table. could represen f()? Table. 0 (a) g() (b) h() 7 7 (c) j() (d) k() f () Figure. (a), (b), (c), (d) You migh poin ou onl relaive values of funcions are imporan for his problem, no he acual values.

6 CHAPTER SIX. Graphs of he derivaives of four funcions are shown in (I) (IV). For he funcions (no he derivaive) lis in increasing order which has he greaes change in value on he inerval shown. (a) (I), (IV), (III), (II) (b) (I), (IV), (II), (III) (c) (I) = (II), (IV), (III) (d) (I) = (II), (III) = (IV) (e) (I) = (II) = (III) = (IV) (I) (II) (III) (IV) (e). The ordering will be given b he values of he area under he derivaive curves. These areas are (I) (/)()() =, (II) (/)()( + ) + (/)()() =, (III) () + () =, and (IV) () =. You can draw some oher graphs of derivaives and ask which funcion has he greaes change over a specified inerval.

7 CHAPTER SIX. Graphs of he derivaives of four funcions are shown in (I) - (IV). For he funcions (no he derivaive) lis in increasing order which has he greaes change in value on he inerval shown. (a) (I), (III), (IV), (II) (b) (I) = (III), (IV), (II) (c) (IV), (I) = (III), (II) (d) (I) = (III) = (IV), (II) (e) (I) = (II) = (III) = (IV) (I) (II) (III) (IV) (d). The ordering will be given b he values of he area under hese curves. These areas are (I) ()+(/)()() = 0, (II) () + (/)()() =, (III) same as (I), 0, and (IV) (/)()( + ) + () = 0. You could ask sudens how o change graph (II) o have area equal o 0 and no be idenical o an of he oher graphs.

8 CHAPTER SIX ConcepTess and Answers and Commens for Secion.. Graphs (I) (III) show veloci versus ime for hree differen objecs. Order graphs (I) (III) in erms of he disance raveled in four seconds. (Greaes o leas) (a) (I), (II), (III) (b) (III), (II), (I) (c) (II), (III), (I) (d) (II) = (III), (I) (e) (I) = (II) = (III) (I) veloci (II) veloci (III) veloci (sec) (sec) (sec) (d). The disance raveled in his siuaion is he area under he graph. These areas are (I) (/)()() =, (II) (/)()( + ) =, (III) (/)()( + ) =. You could mimic his problem using =, = /, and = +.

9 CHAPTER SIX 7. Figure. conains a graph of veloci versus ime. Which of he following could be an associaed graph of posiion versus ime? (a) (I) (b) (II) (c) (III) (d) (IV) (e) (I), (IV) (f) (II), (III) veloci Figure. (I) posiion (II) posiion (III) posiion (IV) posiion (e). Because he veloci is posiive for he inerval shown, he posiion versus ime graph mus be increasing for he inerval. Noice graphs (I) and (IV) differ b a verical shif and boh are possible. You could sar wih simpler problems using =, hen =, and finall =.

10 CHAPTER SIX. Figure. conains a graph of veloci versus ime. Which of (a) (d) could be an associaed graph of posiion versus ime? veloci Figure. (a) posiion (b) posiion (c) posiion (d) posiion (a). Because he veloci goes from a posiive value o a negaive value a., he posiion will be a maimum here. You could elaborae wh each of he oher choices has properies which eclude i.

11 ConcepTess and Answers and Commens for Secion. CHAPTER SIX 9. If he graph of f is given in Figure.7, hen which of (a) (d) is he graph of f() d for < <? Figure.7 (a) (b) (c) (d) (c). Because f() d = 0, he poin (, 0) is on he graph of and increasing, f() d will be increasing and concave up. Each choice could be eamined in deail o show wh i is no appropriae. f() d. Because he graph of f is posiive

12 0 CHAPTER SIX. If he graph of f is given in Figure., hen which of (a) (d) is he graph of f() d for 0? Figure. (a) (b) 0 0 (c) (d) 0 0 and (d). Because 0 f() d is posiive, hen f() d = 0, he poin (, 0) is on he graph of 0 f() d is negaive. This eample is useful showing he value of inerchanging limis. f() d. Because 0 f() d = 0 f() d

13 CHAPTER SIX. If he graph of f is given in Figure.9, hen which of (a) (d) is he graph of f() d for 0? 7 Figure.9 (a) (b) (c) (d) 0 (b). Because f() d = 0 f() d = 0, he graph of f() d, hen 0 Have sudens jusif he las saemen in he answer. f() d will conain he poin (, 0). Because f() is posiive and f() d < 0. f is a decreasing funcion, so f() d will be concave down.

14 CHAPTER SIX