10th Edition

Ron Larson + 1 other

ISBN: 9781285057095

Textbook Problem

Graphical Analysis In Exercises 85 and 86, graph f(x)/g(x) and f'(x)/g'(x) near x = 0. What do you notice about these ratios as x →0? How does this illustrate L’Hôpital’s Rule?

f ( x ) = sin 3 x , g ( x ) = sin 4 x

To determine

The graphs for f(x)/g(x) and f′(x)/g′(x) near x=0 and explain how it illustrates L’Hopital rule when the functions are f(x)=sin3x and g(x)=sin4x.

The functions are f(x)=sin3x and g(x)=sin4x.

Calculation:

Consider the function f(x)=sin3x and g(x)=sin4x.

Since, f(x)g(x)=sin3xsin4x,

sin3xsin4x=3sinx−4sin3x4sinxcosx(2cos2x−1)=3−4sin2x4cosx(2cos2x−1)=3−4sin2x8cos3x−4cosx

Let y1=3−4sin2x8cos3x−4cosx

Substitute π4 for x.

y1=3−4sin2(π4)8cos3(π4)−4cos(π4)=3−4(12)28(12)3−4(12)=3−4(12)24(12)−4(12)=3−4(−12)20

So, it will approach towards infinity.

Substitute −π4 for x.

y1=3−4sin2(−π4)8cos3(−π4)−4cos(−π4)=3−4(−12)28(12)3−4(12)=3−4(−12)24(12)−4(12)=3−4(−12)20

So, it will approach towards infinity.

Substitute 0 for x.

y1=3−4sin2(0)8cos3(0)−4cos(0)=3−4(0)8(1)3−4(1)=3−08−4=34

The table representing the values for y and x is:

x	−π4	π4	0
y	∞	∞	34

For f′(x)g′(x)=f′(sin3x)g′(sin4x)

Check out a sample textbook solution.

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Graphical Analysis In Exercises 85 and 86, graph f(x)/g(x) and f'(x)/g'(x) near x = 0. What do you notice about these ratios as x →0? How does this illustrate L’Hôpital’s Rule?

f ( x ) = sin 3 x , g ( x ) = sin 4 x

Chapter 8 Solutions

Additional Math Solutions

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Calculus

Solutions

Calculus

Graphical Analysis In Exercises 85 and 86, graph f(x)/g(x) and f'(x)/g'(x) near x = 0. What do you notice about these ratios as x →0? How does this illustrate L’Hôpital’s Rule? f ( x ) = sin 3 x , g ( x ) = sin 4 x

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Calculus Calculus 10th Edition

Graphical Analysis In Exercises 85 and 86, graph f(x)/g(x) and f'(x)/g'(x) near x = 0. What do you notice about these ratios as x →0? How does this illustrate L’Hôpital’s Rule?

f ( x ) = sin 3 x , g ( x ) = sin 4 x