1. Find the maximum and minimum of on [0,3]
First, we need to find the zeros of the derivative to determine where there are critical values.
This has zeros at x = 1 and x = -1. Now, we just need to test the values.
Therefore, there is a max at x = 3 and a min at x = 1
2. Find a number c that satisfies the mean value theorem for on [-1,1].
First, we assert that f(x) is differentiable and continous on [-1,1], since it is a polynomial. Therefore, we know there is a point c such that
In this problem, we are given that b = -1 and a = 1. From here we can just plug the values into the equation and solve it.
Now, we just need to find where .
3. Each side of a square is increasing at a rate of 6 cm/s. At which rate is the area of the square increasing when the area is ?
Let’s define the length of the square as x. We know that the area of a square is .
We also know that each side of the square is increasing 6 cm/s. Therefore we know that
We want to know the rate of the area increase when the area is . To start, we know that the area is when x = 4, since .
So, we get the following formula:
From here, we just need to put in what we know. Since and x = 4, we get:
Start by checking the value of the function at x = 0.
This is indeterminate form, so we can apply L’Hoptial’s rule:
This is still indeterminate form, so we can apply L’Hopital’s rule again:
5. Use Newton’s Method to estimate the root of on [1,2]
From here we could keep calculating values, but it looks like the root trends to around 1.2, therefore x = 1.2 is a good estimate towards to root.