# Precalculus : Derivatives

## Example Questions

### Example Question #7 : Rate Of Change Problems

Find the average rate of change of  between  and .

Explanation:

The solution will be found by the formula .

Here  gives us , and .

Thus, we find that the average rate of change is .

### Example Question #8 : Rate Of Change Problems

Find the average rate of change of  over the interval from  to .

Explanation:

The average rate of change will be .

.

.

This gives us .

### Example Question #9 : Rate Of Change Problems

Find the average rate of change of  over the interval from  to .

Explanation:

The average rate of change will be .

Now.

We also know .

So we have .

### Example Question #10 : Rate Of Change Problems

Why can we make an educated guess that the average rate of change of , between  and  would be ?

We know  is a polynomial.

We know is horizontal on that interval.

We know  is odd on that interval.

We know  is symmetrical on that interval.

We know  is vertical on that interval.

We know  is symmetrical on that interval.

Explanation:

Because  is symmetrical over the y axis, it increases exactly as much as it decreases on the interval from  to . Thus the average rate of change on that interval will be .

### Example Question #11 : Rate Of Change Problems

If the average rate of change of  between  and , where , is positive, then what can be said about  on that interval?

is increasing

is odd

is decreasing

is constant

is increasing

Explanation:

If the average rate of change is positive, then the formula gives us , so . We know  because it is a given in the proble, so . Hence

and . This shows that  must increase over the interval from  to .

### Example Question #12 : Rate Of Change Problems

If the average rate of change of  between  and , where , is negative, then what can be said about  on that interval?

is decreasing

is increasing

is an odd function

is negative

is constant

is decreasing

Explanation:

If the average rate of change is negative, then the function is changing in a negative direction overall. Hence, the graph of the function will be decreasing on that interval.

and since  is decreasing

### Example Question #1 : Maximum And Minimum Problems

The profit of a certain cellphone manufacturer can be represented by the function

where  is the profit in dollars and  is the production level in thousands of units.  How many units should be produced to maximize profit?

Explanation:

We can determine the production level that maximizes profit by taking the derivative of our function.  This can be done using the power rule.

We then set this derivative equal to 0 and solve.

We can factor out our greatest common factor and then divide it out.

Next we can factor.

Therefore, we can solve

We remember that these production levels are in thousands of units.

However, a factory cannot produce negative cellphones, so the maximum profit is obtained when seven thousand units are produced.

### Example Question #2 : Maximum And Minimum Problems

What is the critical point of . Is it a max or a minimum?

Explanation:

To find the critical points, we first take the first derivative using the Power Rule:

Therefore,

To find the critical point we need to set the derivative equal to zero and solve for x.

Doing so this gives the critical point at . To get the y value of the point plug x=0 back into the original equation.

thus the point is .

Is it a max or min?

Take the second derivative

It is a minimum since the second derivative is positive.

### Example Question #3 : Maximum And Minimum Problems

What is the critical point of  ? Is it a max or minimum?

Explanation:

To find the critical points, we first take the first derivative using the Power Rule.

.

Therefore,

From here we need to set the derivative equal to zero and solve to x.

The critical point occurs when . Now to get the y value of the point we will need to plug in x=0 into the original equation.

. This gives us a critical point at .

To find if it is a max or min, we take the second derivative

Since the second derivative is negative, it is a maximum.

### Example Question #4 : Maximum And Minimum Problems

Determine whether if there is a maximum or minimum, and location of the point for:

Explanation:

Determine the derivative of this function.

Set the derivative function equal to zero and solve for .

Since the parabola has a positive coefficient for , the parabola will open upwards, and therefore will have a minimum.

Substitute back into the original function and find the y-value of the minimum.

The parabola has a minimum at the point .