Let F(x) = -x^2+2x-3

Question

Three part Question:( a ) Find the average rate of change of the graph of f between x=c and c+h.

( b ) Find the slope of the tangent line of the graph of f(x) at x=c by the definition.

( c )Find f'(-1).

Answer 1

Hi Sam, "the average rate of change of the graph" just means the slope, or derivative in this case. You could just straight differentiate it and get an average rate of change of the graph to be y=-2x+2. But, that may not be what's being asked when given x=c and c+h. 

You may have to do a "rise over run" and obtain the derivative the long way.

The average rate of change of the graph = lim (h->0)[[[-(x+h)²+2(x+h)-3]-[-x²+2x-3]]/h]

= lim (h->0) [[(-x²-2hx-h²+2x=2h-3)-(-x²+2x-3)]/h]

= lim (h->0) [(-2hx+2h+h²)/h]

= lim (h->0) [-2x+2+h]

= -2x+2

f¹(-1) = -2(-1) +2 = 4

Answer 2

Since  x=c  , then   c+h = x+h

 

f(x) = -x²+2x-3

 

f(x+h) = -(x+h)² + 2(x+h) -3

 

For part a)

[f(x+h)-f(x)]/[(x+h)-x)]

 

 

Part b)

Use the difference quotient

 

lim [f(x+h) - f(x)]/h        as h approaches zero.

 

 

Part c)

Use the answer from Part b and substitute the value of -1.