Average rate of change problem?!?!?

Question

Here's my problem: f(x) = x^2 + 1

I have to find the average rate of change between the points:

A.) (-1, f(-1)) and (3, f(3))

B.) (a, f(a)) and (b, f(b))

C.) (x, f(x)) and (x+h, f(x+h))

* I don't understand what I'm supposed to do. I thought maybe you just find the rate of change of each point, but then I realized that they gave you the problem for a reason?!?! Can somone please help me understand. Even if you can just show me how to get one of the answers then maybe I can do the rest. I promise 10 to the best answer. Thanks for your time.

Answer 1

f(x)=x²+1

Find the difference quotient:

(f(x+h)-f(x))/h=((x+h)²+1-x²-1)/h

=(x²+2hx+h²+1-x²-1)/h

=(2hx+h²)/h=2x+h

A) m=2*-1+(3+1)=-2+4=2

B) m=2*a+(b-a)

=2a+b-a=a+b

C) m=2*x+(x+h-x)

=2x+h

Basically what they're asking you to do is to find the slope.

Remember that slope=(y2-y1)/(x2-x1) so you can also do the problem by evaluating the y values but it takes longer.(at least for me it does) Just to clarify what I mean by slope=(y2-y1)/(x2-x1) I'll do problem A using that method.

A) (-1, f(-1)) and (3, f(3))

First find f(3), f(3)=3²+1=10

find f(-1), f(-1)=(-1)²+1=2

So y2=10 and y1=2, x2=3 and x1=-1

slope=(y2-y1)/(x2-x1) =(10-2)/(3--1)=(10-2)/(3+1)=8/4=2

Same answer we get by using the difference quotient.

Answer 2

Average rate of change is change in y/change in x, or delta y/delta x.

A. [f(3)-f(-1)]/(3-(-1))=(10-2)/4=2

B. [f(b)-f(a)]/(b-a)=[b^2 +1 -(a^2 +1)]/(b-a)= (b^2-a^2)/b-a

=(b+a)(b-a)/(b-a)

=b+a

C. [f(x+h)-f(x)]/(x+h-x) = [(x+h)^2 +1 -(x^2 +1)]/h

=[(x+h)^2 -x^2]/h

=[(x^2 +2hx + h^2)-x^2]/h

=(2hx+h^2)/h

=2x + h