# Calculus 1 : How to find the meaning of functions

## Example Questions

### Example Question #1744 : Functions

Find the derivative of .

Explanation:

To determine the derivative of this function: , it is necessary to use two chain rules after taking the derivatives of tangent and natural log functions.

Use brackets to identify the terms which require the chain rule.

### Example Question #1745 : Functions

Let  be the cost in dollars of renting a truck when you drive  miles. What is the interpretation of ?

If the distance driven increases by  miles, the cost increases by .

If the truck has already been driven  miles, then the cost of driving an additional mile is .

The cost of driving the truck  miles is .

The cost of driving the truck is  per mile.

The cost of driving the truck  miles is .

If the truck has already been driven  miles, then the cost of driving an additional mile is .

Explanation:

represents the rate of change of  when  In particular, this represents the change in cost of renting the truck when it is driven  miles. This change gives the approximate increase in cost if the truck is driven an additional mile. Notice that the units of the derivative are the units of  over the units of . Therefore the interpretation of this term must have units.

### Example Question #1746 : Functions

Consider the function  .  What must be true about the velocity?

The velocity will always be negative.

The velocity will always be positive.

The velocity is zero when .

The velocity is zero when .

The velocity is never zero.

The velocity is zero when .

Explanation:

To understand what's happening with the velocity given the acceleration curve, integrate the acceleration function to obtain the velocity curve.

Set the velocity equal to zero and solve for time.

Therefore, the velocity must be zero when .  All of the other statements are false.

### Example Question #1747 : Functions

If  describes the concavity of a function, which of the following statements must be true?

Explanation:

Concavity describes where on the original function the curve concaves up or down. Algebraically, concavity is the second derivative of the original function.

Set the concavity function equal to zero.

Test the values to the left and right of . Anywhere to the left of  will yield negative concavity, and the right of  will yield positive concavity. The original equation does not necessarily always concave up.

Take the antiderivative of the concavity function to obtain the derivative function, or the slope of the original function.

Since the derivative function has an  term, the slopes the original function are not necessarily constant values.

Set the derivative function equal to zero and solve for .

The slope of the original function must have zero slopes at the roots .

Therefore, the only true statement is:

### Example Question #1748 : Functions

Find:

Limit Does Not Exist

Explanation:

To find the limit of a function at a particular value we first want to simplify our function .

We can factor the numerator which results in the following.

This expression can be simplified to

.

Then we plug in  to get the final answer .

### Example Question #1749 : Functions

Evaluate:

Limit Does Not Exist

Explanation:

When finding the limit of a function at a particular point, you first want to see if direct substitution of the point results in an answer.

For the function,

if we plug in 0, we will get  , which is undefined. Since direct substitution did not succeed we can use L'Hospital's Rule.

L'Hospital's Rule states that if,

or  ,  where  is any real number then,

.

Let , and .

Now we find  and .

.

We can rewrite our limit as:

Evaluating the limit gives us

.

### Example Question #1750 : Functions

Evaluate:

Explanation:

If we plug in  , we will get  .

So we can use L'Hospital's Rule.

L'Hospital's Rule states that if

or  ,  where  is any real number then,

.

Let  , and  .

We now find , and .

Now we can rewrite our limit as:

After plugging in   , we get

### Example Question #2771 : Calculus

Evaluate:

Limit Does Not Exist

Explanation:

If we plug in 0, we get

So we can use L'Hospital's Rule.

L'Hospital's Rule states that if

or  ,  where  is any real number then,

.

Set , and  . Then find  and .

,  .

Now we can rewrite our limit as:

Now if we plug in 0, we get,

### Example Question #2781 : Calculus

Evaluate:

Limit Does Not Exist

Explanation:

If we plug in 10 we get, .

So we can use L'Hospital's Rule.

L'Hospital's Rule states that if

or  ,  where  is any real number then,

.

Set , and .

Then find , and .

We can now rewrite our limit as:

### Example Question #2782 : Calculus

Find  if .

Does not exist

Explanation:

When we sub in  for , we get , so we should try to factor the expression.

We can factor out an  in the numerator to get,

.

After canceling out the , the expression reduces to,

,

subbing in zero for  we get . That is our limit.