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 3.3, Problem 23E

(a)

To determine

To find: The equation of the tangent line to the curve at the point.

Expert Solution

Answer to Problem 23E

The equation of the tangent line to the curve y=2xsinx at (π2,π) is y=2x_.

Explanation of Solution

Given:

The equation of the curve is y=2xsinx and the point is (π2,π).

Derivative rules:

(1) Constant Multiple Rule: ddx[cf(x)]=cddxf(x)

(2) Power Rule: ddx(xn)=nxn1

(3) Product Rule: ddx(f(x)g(x))=f(x)ddx(g(x))+g(x)ddx(f(x))

Formula used:

The equation of the tangent line at (x1,y1) is, yy1=m(xx1) (1)

where, m is the slope of the tangent line at (x1,y1) and m=dydx|x=x1.

Calculation:

The derivative of y is dydx, which is obtained as follows,

dydx=ddx(y) =ddx(2xsinx) 

Apply the product rule (3),

dydx=2xddx[sinx]+sinxddx[2x]

Apply the constant multiple rule (1),

dydx=2xddx[sinx]+2sinxddx[x]

Apply the power rule (2) and simplify the expressions,

dydx=2x(cosx)+2sinx(1x11)=2xcosx+2sinx=2(xcosx+sinx)

Therefore, the derivative of the function y=2xsinx is 2(xcosx+sinx)_.

The slope of the tangent line at (π2,π) is,

m=dydx|x=π2=2(π2cosπ2+sinπ2)    {Qcosπ2=0, sinπ2=1}=2(0+1)=2

Thus, the slope of the tangent line at (π2,π) is m=2_.

Substitute (π2,π) for (x1,y1), and 2 for m in equation (1),

(yπ)=2(xπ2)yπ=2xπy=2x

Therefore, the equation of the tangent line to the curve y=2xsinx at (π2,π) is y=2x_.

(b)

To determine

To sketch: The given curve and the tangent line at the given point (π2,π).

Expert Solution

Explanation of Solution

Given:

The curve is y=2xsinx and the tangent line at (π2,π) is y=2x.

Graph:

Use the online graphing calculator to draw the graph of the curve and the tangent line as shown below in Figure 1.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 3.3, Problem 23E

From Figure 1, it is observed that the equation of the tangent line touches the curve y=2xsinx at the point (π2,π).

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