Let f(x)=3x^3 +7.5x +1. 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.
First choose the interval of length .1 with 3 being the left endpoint of
the interval.
Then f'(3)≈ ______
Now let the interval have length .01. Then f'(3)≈ _______
Finally let the interval have length .001
Then f'(3)≈
**I am genuinely lost on where to even begin with this problem. I understood the rest of my homework completely, except this one. My homework will be due before someone can get to answer this, I am almost sure of that, so please - just work it out for me so that I can better understand it for the midterm.
To estimate the derivative of a function at a specific point, you can use the concept of the average rate of change over a small interval. In this case, you need to estimate f'(3) by computing the average rate of change of f(x) over different intervals.
To begin, let's start with the interval of length 0.1, where 3 is the left endpoint. This means the interval goes from 3 to 3.1.
Step 1: Find the values of f(x) at the endpoints of the interval.
Evaluate f(3) and f(3.1) using the given function:
f(3) = 3(3)^3 + 7.5(3) + 1 = 82
f(3.1) = 3(3.1)^3 + 7.5(3.1) + 1 ≈ 86.671
Step 2: Compute the average rate of change.
The average rate of change formula for a function over an interval is given by:
Average rate of change = (f(b) - f(a)) / (b - a)
In this case, a = 3 (left endpoint of the interval) and b = 3.1 (right endpoint of the interval). Substituting the values we found earlier:
Average rate of change = (f(3.1) - f(3)) / (3.1 - 3)
Average rate of change = (86.671 - 82) / 0.1
Average rate of change ≈ 46.71
So, f'(3) is approximately equal to 46.71 (rounded to three decimal places).
Now, let's repeat the above steps for intervals of length 0.01 and 0.001.
For the interval of length 0.01:
Step 1: Find f(3) and f(3.01).
Step 2: Compute the average rate of change.
For the interval of length 0.001:
Step 1: Find f(3) and f(3.001).
Step 2: Compute the average rate of change.
By repeating these steps, you can estimate f'(3) for different interval lengths.