For the function f(x)=x^3−8x+3,

a) Determine a SIMPLIFIED expression for [f(5+h)−f(5)] / h. What does this expression represent?
b) Using your simplified expression from part (a), evaluate if h = 0.001. What does this value represent?

you will definitely need your calculator for this

for a) follow the steps in the answer to your post from 9:24 pm

b)
[f(5+h)−f(5)] / h
= (5.001^3 - 8(5.001) + 3 - (5^3 - 8(5) + 3) )/.001
= ......

let me know what you get

I got 67.015

good job, the real value using Calculus would be 67

To determine the expression [f(5+h) - f(5)] / h, we first need to find the values of f(5+h) and f(5).

a) Finding f(5+h):
We substitute x = 5+h into the function f(x) = x^3 - 8x + 3:
f(5+h) = (5+h)^3 - 8(5+h) + 3
= (125 + 30h + 3h^2 + h^3) - (40 + 8h) + 3
= h^3 + 3h^2 + 22h + 88

b) Finding f(5):
We substitute x = 5 into the function f(x) = x^3 - 8x + 3:
f(5) = 5^3 - 8(5) + 3
= 125 - 40 + 3
= 88

Now, we can substitute the values of f(5+h) and f(5) into the expression [f(5+h) - f(5)] / h:
[f(5+h) - f(5)] / h = [h^3 + 3h^2 + 22h + 88 - 88] / h
= [h^3 + 3h^2 + 22h] / h
= h^2 + 3h + 22

So, the simplified expression for [f(5+h) - f(5)] / h is h^2 + 3h + 22.

Now, let's evaluate the expression for h = 0.001:

[h^2 + 3h + 22]_{h=0.001} = (0.001)^2 + 3(0.001) + 22
= 0.001 + 0.003 + 22
= 22.004

The value 22.004 represents the slope of the function at x = 5, as h approaches 0. This value is close to the instantaneous rate of change of the function at that point.