# The x value halfway between the x -coordinate p and q .

### 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, Problem 3P
To determine

## To show: The x value halfway between the x-coordinate p and q.

Expert Solution

### Explanation of Solution

Given:

The equation of the parabola is y=ax2+bx+c.

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,y=y1

Proof:

Consider the function

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

Differentiate with respect to x,

f(x)=ddx(ax2+bx+c)=a(2x21)+b(1x11)+0=a(2x)+b(1)=2ax+b

Thus, the derivative of the equation is f(x)=2ax+b.

Substitute x=p and x=q in f(x),

f(p)=2ap+b  f(q)=2aq+b

Substitute x=p and x=q in f(x),

f(p)=ap2+bp+c f(q)=aq2+bq+c

That is, the slope of tangent to the equation is, dydx=2ap+b.

Substitute (p,ap2+bp+c) for (x1,y1) and slope m=2ap+b in equation (1),

y(ap2+bp+c)=2ap+b(xp)y=2ap+b(xp)+(ap2+bp+c)y=2apx2ap2+bxbp+ap2+bp+c

y=(2ap+b)xap2+c (2)

Substitute (q,aq2+bq+c) for (x1,y1) and slope m=2aq+b in equation (1),

y(aq2+bq+c)=2aq+b(xq)y=2aq+b(xq)+(aq2+bq+c)y=2aqx2aq2+bxbq+aq2+bq+c

y=(2aq+b)xaq2+c (3)

Equating equation (2) and equation (3),

(2ap+b)xap2+c=(2aq+b)xaq2+c2apx+bxap2+c=2aqx+bxaq2+ca(2pxp2)=a(2qxq2)2pxp2=2qxq2

Simplify further,

2px2qx=p2q22(pq)x=(pq)(p+q)2x=(p+q)x=p+q2

Hence the required proof is obtained.

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