# The equation of the tangent line to the curve at the point.

### Single Variable Calculus: Concepts...

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

### 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

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

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.

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

### Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!