   Chapter 3.1, Problem 80E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# The graph of any quadratic function f(x) = ax2 + bx + c is a parabola. Prove that the average of the slopes of the tangent lines to the parabola at the endpoints of any interval [p, q] equals the slope of the tangent line at the midpoint of the interval.

To determine

To prove: The average value of the slopes of the tangent lines to the parabola at the endpoints of [p, q] equals the slope of the tangent line at the midpoint of the interval.

That is, to prove f(p)+f(q)2=f(p+q2).

Explanation

Given:

Derivative rules:

(1) Constant function: ddx(c)=0

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

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

(4) Sum Rule: ddx[f(x)+g(x)]=ddx(f(x))+ddx(g(x))

(5) Difference Rule: ddx[f(x)g(x)]=ddx(f(x))ddx(g(x))

Proof:

Obtain the derivative of the function f(x)=ax2+bx+c.

f(x)=ddx(f(x))=ddx(ax2+bx+c)

Apply the sum rule (4) and the constant multiple rule (2).

Apply the derivative rule (1) and the power rule (3).

f(x)=a(2x21)+b(1x11)+0=2ax+b

Thus, the derivation of the function f(x) is f(x)=2ax+b

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### Intervals Graph the set. 65. (, 4) (4, )

Precalculus: Mathematics for Calculus (Standalone Book)

#### In Exercises 49-62, find the indicated limit, if it exists. 51. limx0x2xx

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

#### In Problems 83 and 84, simplify each complex fraction. x2x1x2+x1

Mathematical Applications for the Management, Life, and Social Sciences

#### True or False: limx6x3+8x29x+75x34x+30=limx6x35x3.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

#### For

Study Guide for Stewart's Multivariable Calculus, 8th 