Let f(x)=1x^3+2.5x+3. In this problem you will estimate f'(-3) by computing the average rate of change of f(x)over a small interval. Round your answers to three decimal places.
1. First choose the interval of length .1 with -3 being the left endpoint of the interval. Then f'(-3)=
2. Now let the interval have length .01. Then f'(-3)=
3. Finally let the interval have .001. Then f'(-3)=
To estimate f'(-3) by computing the average rate of change over a small interval, we need to use the following formula:
f'(x) = [f(x + h) - f(x)] / h
where h is the length of the interval.
1. First, let's choose the interval of length 0.1 with -3 being the left endpoint of the interval. Let's substitute the values into the formula:
f'(-3) = [f(-3 + 0.1) - f(-3)] / 0.1
Now, let's plug in the values into the function f(x):
f'(-3) = [f(-2.9) - f(-3)] / 0.1
Calculate f(-2.9):
f(-2.9) = 1*(-2.9)^3 + 2.5*(-2.9) + 3
Simplify the expression:
f'(-3) = [f(-2.9) - f(-3)] / 0.1
2. Now, let's choose the interval of length 0.01. Repeat the same process:
f'(-3) = [f(-3 + 0.01) - f(-3)] / 0.01
Plug in the values into the function f(x):
f(-3 + 0.01) = 1*(-2.99)^3 + 2.5*(-2.99) + 3
Calculate f(-2.99) and simplify the expression:
f'(-3) = [f(-2.99) - f(-3)] / 0.01
3. Finally, let's choose the interval of length 0.001. Repeat the same process:
f'(-3) = [f(-3 + 0.001) - f(-3)] / 0.001
Plug in the values into the function f(x):
f(-3 + 0.001) = 1*(-2.999)^3 + 2.5*(-2.999) + 3
Calculate f(-2.999) and simplify the expression:
f'(-3) = [f(-2.999) - f(-3)] / 0.001
From these calculations, you will find the estimates for f'(-3) for each interval length of 0.1, 0.01, and 0.001 by substituting the values into the formula and rounding the answers to three decimal places.