Please answer the following questions below on Characteristics of Functions:

Question

1. Given the function f(x)=x/x-2
2. determine the average rate of change on interval -4≤x≤-1
3. Is it possible to determine the instantaneous rate of change at x=2?
4. Given the function g(x)=x⁴-2x³+1
5. estimate the instantaneous rate of change at x=1

Answer 1

hello,

f(x ) = x / (x-2)   x is [-4,-1]

f'(x ) = [f(-1) - f(-4)] / ( -1 -(-4))   

f(-1) = -1/-3 = 1/3

f(-4) = -4/-6 = 2/3

f'(x) = [f(-1) - f(-4)] / ( -1 -(-4)) = [1/3 - 2/3 ] / 3 = -1/9

Is it possible to determine the instantaneous rate of change at x=2?

 No, because this function at x = 2 is not continuous and x = 2 is vertical asymptote.

Given the function g(x)=x4-2x3+1

 estimate the instantaneous rate of change at x=1

g'(x) = [g(x) - g(1)] / (x-1)

g(x)=x4-2x3+1

g(1) = 1-2+1 =0

g'(x) = [g(x) - g(1)] / (x-1) = [x4-2x3+1- 0 ] / (x-1) = (x-1) ( x^3 -x^2 -x -1 ) / (x-1) = x^3 -x^2 -x -1= -2

On the other  hand: g'(x) = 4x^3 -6x^2 ------> g'(1 ) = 4 - 6 = -2 the same answer

I hope it is useful,

Minoo