BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 3.4, Problem 48E

(a)

To determine

To sketch: The slope of the function f(x) by using the graph of f(x).

Expert Solution

Explanation of Solution

The value of the derivative of the function at any point x can be estimated by drawing the tangent line at any point (x,f(x)) and then obtain the slope.

Mark the slope of the tangent as a point in y-axis and the value of x as a point in x-axis in the graph of f(x).

Proceed in the similar way at several points and obtain the rough graph of the f(x) as shown in Figure 1.

Graph:

The rough sketch of the derivative function, f(x) by using the graph of the function f(x) is shown below in Figure 1.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 3.4, Problem 48E , additional homework tip  1

(b)

To determine

To calculate: The derivative of function f(x).

Expert Solution

Answer to Problem 48E

The derivative of f(x)=sin(x+sin2x) is f(x)=cos(x+sin2x)(1+2cos2x)_.

Explanation of Solution

Result used: Chain Rule

If h is differentiable at x and g is differentiable at h(x), then the composite function F=gh defined by F(x)=g(h(x)) is differentiable at x and F is given by the product,

F(x)=g(h(x))h(x) (1)

Calculation:

Obtain the derivative.

f(x)=ddx(f(x))=ddx(sin(x+sin2x))

Let h(x)=x+sin2x and g(u)=sinu  where u=h(x).

Apply the chain rule as shown in equation (1),

f(x)=g(h(x))h(x) (2)

The derivative of g(h(x)) is computed as follows,

g(h(x))=g(u)=ddu(g(u))=ddu(sinu)=cosu

Substitute u=x+sin2x in the above equation,

g(h(x))=cos(x+sin2x)

Thus, the derivative is g(h(x))=cos(x+sin2x).

The derivative of h(x) is computed as follows,

h(x)=ddx(x+sin2x)=ddx(x)+ddx(sin2x)=(1x11)+cos2x2=1+2cos2x

Thus the derivative is h(x)=1+2cos2x.

Substitute cos(x+sin2x) for g(h(x)) and 1+2cos2x for h(x) in equation (2),

F(x)=cos(x+sin2x)1+2cos2x=(1+2cos2x)cos(x+sin2x)

Therefore, the derivative of f(x)=sin(x+sin2x) is (1+2cos2x)cos(x+sin2x).

Graph:

Use the online graphing calculator to draw the graph of the derivative function f(x) as shown below in Figure 2.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 3.4, Problem 48E , additional homework tip  2

From Figure 2, it is observed that the given trigonometric function is differentiable everywhere and the rough sketch of the derivative f(x) in Figure 1 is same as the graph of the derivative f(x) in Figure 2.

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