Precalculus : Derivatives

Study concepts, example questions & explanations for Precalculus

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Example Questions

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Example Question #1 : Derivatives

The function  is such that

 

When you take the second derivative of the function , you obtain 

What can you conclude about the function at ?

Possible Answers:

The point is an inflection point.

The point is a local minimum.

The point is a local maximum.

The point is an absolute minimum.

The point is an absolute maximum.

Correct answer:

The point is an inflection point.

Explanation:

We have a point at which . We know from the second derivative test that if the second derivative is negative, the function has a maximum at that point. If the second derivative is positive, the function has a minimum at that point. If the second derivative is zero, the function has an inflection point at that point.

Plug in 0 into the second derivative to obtain 

So the point is an inflection point.

Example Question #2 : Derivatives

Consider the function 

Find the maximum of the function on the interval .

Possible Answers:

Correct answer:

Explanation:

Notice that on the interval , the term  is always less than or equal to . So the function is largest at the points when . This occurs at  and .

Plugging in either 1 or 0 into the original function  yields the correct answer of 0.

Example Question #3 : Derivatives

In what -intervals are the relative minimum and relative maximum for the function below?

Possible Answers:

Correct answer:

Explanation:

A cubic function will have at most one relative minimum and one relative maximum.  We can determine the zeros be factoring at .  From then we only need to determine if the graph is positive or negative in-between the zeros. 

The graph is positive between  and  (plug in ) and negative between 0 and 4 (plug in ).  This can also be seen from the graph.

 

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Example Question #4 : Derivatives

What is the minimum of the function ?

Possible Answers:

Correct answer:

Explanation:

The vertex form of a parabola is:

where  is the vertex of the parabola.

The function for this problem can be simplified into vetex form of a parabola: 

 ,

with a vertex at .

Since the parabola is concave up, the minimum will be at the vertex of the parabola, which is at .

Example Question #1 : Derivatives

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

Possible Answers:

Correct answer:

Explanation:

The average rate of change will be found by .

Here, , and .

Now, we have .

Example Question #2 : Rate Of Change Problems

Let a function be defined by .

Find the average rate of change of the function over .

Possible Answers:

Correct answer:

Explanation:

We use the average rate of change formula, which gives us .

Now , and .

Therefore, the answer becomes .

Example Question #7 : Derivatives

Suppose we can model the profit, , in dollars from selling  items with the equation .

Find the average rate of change of the profit from  to .

Possible Answers:

Correct answer:

Explanation:

We need to apply the formula for the average rate of change to our profit equation. Thus we find the average rate of change is .

Since , and , we find that the average rate of change is .

Example Question #8 : Derivatives

Let the profit, , (in thousands of dollars) earned from producing  items be found by .

Find the average rate of change in profit when production increases from 4 items to 5 items.

Possible Answers:

Correct answer:

Explanation:

Since , we see that this equals. Now let's examine  which simplifies to .

Therefore the average rate of change formula gives us .

Example Question #9 : Derivatives

Suppose that a customer purchases  dog treats based on the sale price , where , where .

Find the average rate of change in demand when the price increases from $2 per treat to $3 per treat.

Possible Answers:

Correct answer:

Explanation:

Thus the average rate of change formula yields .

This implies that the demand drops as the price increases.

Example Question #10 : Derivatives

A college freshman invests $100 in a savings account that pays 5% interest compounded continuously. Thus, the amount  saved after  years can be calculated by .

Find the average rate of change of the amount in the account between  and , the year the student expects to graduate.

Possible Answers:

Correct answer:

Explanation:

.

.

Hence, the average rate of change formula gives us .

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