p. 457

53.

a)

f[x_]  :=  (Sqrt[3  x  -  2])

Plot[ {f[x],  f'[x]},  {x, 2/3, 4},  
      PlotStyle -> {RGBColor[0, 0, 1],  RGBColor[0, 1, 0]}]

b)

Solve[ y  ==  f[x],  x]

g[y_]  :=  (1/3) (2 + (y^2))

c) Tangent line to y  =  f (x) at (3, f (3)):

{3, f[3]}

y  ==  f[3]  +  f'[3] (x  -  3)  // Simplify

d) Tangent line to y  =  g[x] at (f (3), 3): use slope 1/f' (3)

{f[3], 3}

1/f'[3]

y  ==  3  +  ((2 (Sqrt[ 7]))/3) (x  -  (Sqrt[7]))  // Simplify

e)

p  =  
      Plot[{f[x],  ((5 + 3 x)/(2 (Sqrt[7])))},  {x, 2/3, 4},  
        PlotStyle -> {RGBColor[0, 0, 1],  RGBColor[0,  .75, 1]}]

q  =  
      Plot[{g[x],  -(5/3) + ((2 (Sqrt[7]) x)/3)},  {x, 0, 3}, 
        PlotStyle -> {RGBColor[0, 1, 0],  RGBColor[0, 1,  .75]}]

The line bewteen (3, (Sqrt[7]))and ((Sqrt[7]), 3)is

y  == (Sqrt[7]) +  (((Sqrt[7]) -  3)/(3  -  (Sqrt[7]))) (x  -  3) // 
      Simplify

r  =  
      Plot[{x, 3 + (Sqrt[7]) - x},  {x, 0, 4},  PlotStyle -> GrayLevel[0.75], 
        AspectRatio -> Automatic]

Show[p, q, r,  AspectRatio -> Automatic]

55.

a)

f[x_]  :=  ((4  x)/((x^2) +  1))

Plot[{f[x], f'[x]}, {x, -1, 1},  
      PlotStyle -> {RGBColor[0, 0, 1],  RGBColor[0, 1, 0]}]

b)

Solve[ y  ==  f[x],  x]

Since the domain of f (x) is (-1, 1)and this interval should be the range of g (y) we should choose the first alternative for g (y):

g[y_]  :=  ((2 - (Sqrt[4 - (y^2)]))/y)

c) Tangent line to y  =  f (x) at ((1/2), f ((1/2))):

{(1/2), f[(1/2)]}

y  ==  f[(1/2)]  +  f'[(1/2)] (x  - (1/2))  // Simplify

d) Tangent line to y  =  g[x] at (f ((1/2)), (1/2)): use slope 1/f' ((1/2))

{f[(1/2)], (1/2)}

1/f'[(1/2)]

y  ==  (1/2) +  (25/48) (x  -  (8/5))  // Simplify

e)

p  =  
      Plot[{f[x], (16/25) (1 + 3 x)},  {x, -1, 1},  
        PlotStyle -> {RGBColor[0, 0, 1],  RGBColor[0,  .75, 1]}]

q  =  
      Plot[{g[x],  -(1/3) + ((25 x)/48)},  {x, -2, 2}, 
        PlotStyle -> {RGBColor[0, 1, 0],  RGBColor[0, 1,  .75]}]

The line bewteen ((1/2), (8/5))and ((8/5), (1/2))is

y  == 
        (8/5) +  (((8/5) -  (1/2))/((1/2)  -  (8/5))) 
            (x  -  (1/2)) // Simplify

r  =  
      Plot[{x, (21/10) - x},  {x, -2, 2},  PlotStyle -> GrayLevel[0.75], 
        AspectRatio -> Automatic]

Show[p, q, r,  AspectRatio -> Automatic]