# The derivative of the curve and the value of the c

### 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 68E
To determine

## To find:The derivative of the curve and the value of the c

Expert Solution

The value of c=6

### Explanation of Solution

Given:

The tangent to the curve

y=cx

Parallel to the line

y=32x+6

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=cx.......................(1)

The derivative of a function

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

Differentiating the equation (1) with respect to x

y=cxy=c2x12........................(2)

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

y=32x+6y=32.............................(3)

The equation is in slope −intercept form

The slope of the line is =32

From equation (2) and equation (3)

32=c2x12c=3x12

A system of three equations with the same number of unknowns

That system is

{c=3x12y=cxy=32x+6}

Substitute these equations

3x12x=32x+63x=32x+632x=6x=4

Therefore,

c=3(4)12c=3(2)c=6

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