Chapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1
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1 Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. (b Determine the -coordinte of ech criticl vlue of g. Show the work tht leds to your nswer. (c Find the coordintes of the bsolute mimum of g. Justify your nswer.
2 Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. The slope of the tngent line t is given by g ( e. Furthermore, g( e. So the eqution of the tngent line is y y m( ( ( ( y ( e ( e + ( e y ( e y e e (b Determine the -coordinte of ech criticl vlue of g. Show the work tht leds to your nswer. ( g e 0 e e ln 0 (c Find the bsolute mimum of g. Justify your nswer. We must evlute g t the criticl vlues nd end points. g( e ( g(0 e ( 0 g( e The coordintes of the bsolute mimum re (, Slope is g ( e + g( e y e + ( + Set g ( 0 + e Evlute g t,, nd criticl vlue from prt (b + (,5.389 re the coordintes of the bsolute mimum
3 Chpter 5 3. Let f( nd g ( ln. ( Describe the grphicl reltionship between leds to your conclusion. f ( d nd g( g (. Show the work tht (b Which epression hs the gretest vlue: nswer. f(, f( d, g(3, or g (? Justify your (c The function f is its own inverse. Tht is, ( ( for 0. f f f. Show tht f ( f( ( f(
4 Chpter 5. Let f( nd g ( ln. ( Describe the grphicl reltionship between Show the work tht leds to your conclusion. f ( d is the re between the grph of f nd the -is on the intervl [, ]. g( g ( is the verticl distnce between the points, (, g (. Since ( g nd ( f ( d g( + C, f ( d g( g (. Both vlues equl ln. (b Which epression hs the gretest vlue: g (? Justify your nswer. f ( 0.5 f( d ln Of these, g (3 hs the gretest vlue. f ( d nd g ( g (. f(, f( d, g(3, or g(3 ln g ( 0.5 (c The function f is its own inverse. Tht is, ( ( f f f( f ( ( f( for 0. f ( f( f ( f f. Show tht f ( f( f ( + Correct grphicl eplntion of f ( d + Correct grphicl eplntion of g( g ( + Reltionship between the two + g (3 + Supporting work + f ( + f ( f( f + f f( (
5 Chpter Let f ( e. ( At wht -vlues does f hve reltive etrem? (b Which epression hs the lest vlue: f ( d or f (? Justify your nswer. 0 (c Given ( ( f d f, find.
6 Chpter Let f ( e. ( At wht -vlues does f hve reltive etrem? f ( e 0 e e 0 Since f ( < 0 nd f ( > 0, reltive minimum occurs t 0. (b Which epression hs the lest vlue: your nswer. f ( d e.350 e f ( d or f (? Justify + f ( e Justify etrem + f ( + Supporting work ( ( f e.78 f hs the lest vlue. 0 (c Given ( ( 0 0 f d f, find. f ( d e d e 0 e ( f e e 0 ( f ( d f e f( d e f e + ( + 0
7 Chpter 5 7 e + e. Let f( cosh(. ( At wht -vlues does f hve reltive etrem? (b Find the ect vlue for f ( d. (c Show tht f does not hve ny inflection points.
8 Chpter 5 8 e + e. Let f( cosh(. ( At wht -vlues does f hve reltive etrem? f f ( ( e e ( < 0 f ( ( e e f ( > 0 0 e e A reltive minimum occurs t e 0. e e ln 0 0 (b Find the ect vlue for f ( d. F( ( e e f( d F( F( F( ( e e e e F( ( e e (c Show tht f does not hve ny inflection points. f ( f ( Since e > 0 nd e > 0 for ll vlues of, f( ( e + e > 0 for ll vlues of. Thus f ( > 0 f concve up everywhere. Therefore, f does not hve ny inflection points Justify etrem + f ( ( e e + F( ( e e + Limits + e e + f ( f ( + f ( 0 > everywhere + f concve up everywhere
9 Chpter Let f ( cosh( + sinh( nd g ( cosh( sinh(. ( Use clculus to show tht f ( is n incresing function. (b Define h( f ( g(. Determine the criticl vlues of h.. (c Write the epressions in order from the smllest vlue to the lrgest vlue: h(0, h ( 0, f ( Show the work tht leds to your conclusion.
10 Chpter Let f ( cosh( + sinh( nd g ( cosh( sinh(. ( Use clculus to show tht f ( is n incresing function. f e e e e e f e > 0 for ll ( ( + + ( ( > for ll, f is incresing. Since f ( 0 (b Define h( f ( g(. Determine the criticl vlues of h. h ( cosh sinh ( ( cosh ( sinh ( sinh ( cosh h 0 Since h ( 0 for ll vlues of, every -vlue is criticl vlue. (c Write the epressions in order from the smllest vlue to the lrgest vlue: h(0, h ( 0, f (. Show the work tht leds to your conclusion. f (0 g(0 h(0 f(0 g(0 ( h 0 < h 0 < f. So ( ( ( From (b, ( 0 0 h. From (, f e.78 ( + f ( e + f ( > 0 + Justify incresing + h ( 0 + All vlues of re criticl vlues + h ( 0 + h ( f ( e + correct order
11 Chpter 5 6. Let f(. ( Determine f ( d (b Wht is the re of the region bounded by f, the -is, the y-is, nd the line? (c Determine the vlue of tht mkes the following equlity true: f ( d f(. 0
12 6. Let Chpter 5 f( ( Determine f ( d. f ( d rcsin + C + ( rcsin f d + constnt of integrtion (b Wht is the re of the region bounded by f, the -is, the y-is, nd the line? f( d rcsin rcsin ( f ( d 0 π + Fundmentl Theorem of 6 Clculus 0.5 π (c Determine the vlue of tht mkes the following equlity true: f ( d f(. 0 f( d rcsin rcsin π rcsin 6 π π rcsin 6 6 π rcsin 3 f ( d f( 0 + π f( d rcsin 6 π + rcsin 3 π + sin π sin 3 π sin
13 Chpter Let f ( rccos ( nd g( rcsin (. ( At wht -vlues does f hve reltive etrem? (b Use technology to determine the re of the region bounded by f nd g. (c Define h ( f( g (. Write h ( in terms of f (.
14 Chpter 5 7. Let f ( rccos ( nd g( rcsin (. ( At wht -vlues does f hve reltive etrem? f f ( ( 0.5 > 0 f ( f ( 0.5 < 0 A reltive mimum occurs 0 t 0. (b Use technology to determine the re of the region bounded by f nd g. Using technology, we determine points of intersection t ± ( ( rccos( rcsin d.85 (c Define h ( f( g ( ( f. ( ( ( f ( f ( f ( h f g. Write h ( ( in terms of Justify etrem + f ( + ± ( rccos( rcsin( d + h ( f ( g ( + g ( f ( + h ( f (
15 Chpter Price infltion in the United Sttes is typiclly bout 3% per yer. Assuming n nnul infltion rte of 3%, do the following: ( Crete model tht gives the future price of cndy br tht costs $0.75 tody. (b Determine when the price of the cndy br is epected to rech $.50. (c When the price reches $.50, t wht instntneous rte (in dollrs per yer will the price of the cndy br be chnging?
16 Chpter Price infltion in the United Sttes is typiclly bout 3% per yer. Assuming n nnul infltion rte of 3%, do the following: ( Crete model tht gives the future price of cndy br tht costs $0.75 tody. p( t 0.75(.03 t + eponentil function + initil vlue is growth fctor is.03 (b Determine when the price of the cndy br is epected to rech $.50. ( t t.03 ln t ln.03 ln t ln.03 t 3.50 yers from now (c When the price reches $.50, t wht instntneous rte (in dollrs per yer will the price of the cndy br be chnging? p ( t 0.75ln (.03(.03 ( p dollrs per yer t (.03 t + Correctly log both sides + t 3.50 yers + p ( t ( ( 0.75ln t dollrs per yer
17 Chpter Two towers of equl height re spced 366 feet prt. A cble suspended between the two forms centenry whose height bove the ground is given by f( 5cosh where 0 t the point on the ground hlfwy between the two towers. ( Wht is the height of ech tower (rounded to the nerest foot? (b Wht is the verge height of the cble? (c At the point where the height of the cble is equl to the verge height of the cble nd the cble is rising, wht is the slope of the cble?
18 Chpter Two towers of equl height re spced 366 feet prt. A cble suspended between the two forms centenry whose height bove the ground is given by f( 5cosh where 0 t the point on the ground hlfwy between the two towers. ( Wht is the height of ech tower (rounded to the nerest foot? Since feet, the towers re t 83 nd 83. f ( feet (b Wht is the verge height of the cble? 83 f( d 86, verge height f ( d , feet (c At the point where the height of the cble is equl to the verge height of the cble nd the cble is rising, wht is the slope of the cble? We set f( nd solve for using technology f ( 5sinh f verticl foot per horizontl foot ( + f ( feet f( d 86,56.55 verge height feet f ( f ( d 366 5sinh f (
19 Chpter The function f ( 0 hs inverse function g( log ( Show tht g (. f ( g ( with > 0. (b Wht is the re of the region between f nd g on the intervl [, ]? (c Wht is the eqution of the tngent line of g t?
20 Chpter The function f ( 0 hs inverse function g( log with > 0. ( Show tht g (. f ( ln0( 0 g( ( g( ln0( 0 log ln0( 0 f f ( g ( g ( ln 0 f ( ( g ( ( ln0 (b Wht is the re of the region between f nd g on the intervl [, ]? ( f( g( d (c Wht is the eqution of the tngent line of g t? g ( y + b ( ln0 ( ln0 g ( log ( + b ( ln0 ( ln b log ln y + log ( ln0 ln f ( ln0( 0 + f ( g ( ( ln0 + Show g ( f ( g ( + limits of integrtion + integrnd g ( m g ( + b 0.68
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