# Test you Knowledge: Applications of Derivatives

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:

4. Solve

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.