The equation of the tangent and parallel to the line

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 3.1, Problem 66E
To determine

To find:The equation of the tangent and parallel to the line

Expert Solution

The equation of the line is y=2x+1

Explanation of Solution

Given:

The tangent to the curve

y=x4+ax3+bx2+cx+d

Parallel to the line

y=2x+1

Concept used:

The equation is in slope −intercept form, y=mx+c

An equation for the line through the point (x1,y1) with slope m is

yy1=m(xx1)

Calculation:

The function

y=x4+ax3+bx2+cx+d.......................(1)

The derivative of a function

y=f(x)y=f(x)=dydx=m

Differentiating the equation (1) with respect to x

y=x4+ax3+bx2+cx+dy=4x3+3ax2+2bx+c........................(2)

The derivative is slope of the tangent line so in order to the slope of the tangent line

The derivative of constant is zero

y=2x+1y=2.............................(3)

The equation is in slope −intercept form

The slope of the line is =2

The coordinates of points

y=2x+1x=0y=1y=8

An equation for the line through the point (x1,y1) with slope m is

yy1=m(xx1)y1=2(x0)y1=2xy=2x+1

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