Calculus 1 : Meaning of Functions

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #31 : Meaning Of Functions

Evaluate:

 

Possible Answers:

The limit does not exist.

Correct answer:

Explanation:

In order to evaluate the limit, lets factor the numerator.

Now we can simplify this expression to

.

Now we plug in 10.

Example Question #2782 : Calculus

Evaluate:

Possible Answers:

The limit does not exist.

Correct answer:

Explanation:

If we plug in , we get 

Since we get  , we can use L'Hopitals rule. 

L'Hopitals rule is if we have one of the following cases

 

where a is any real number, then

 

After applying L'Hopitals rule, we get

 

 

Now if we plug in , we get

Example Question #33 : Meaning Of Functions

Let  represent the growth rate of the city of Tucson at a year  where  represents the year . Give a practical interpretation of  .

 

Possible Answers:

The average number of people between  and .

The number of cats in people in .

The average rate of change of people between  and .

The total increase in people in Tucson from  to .

The rate of change in the number of people in .

Correct answer:

The total increase in people in Tucson from  to .

Explanation:

The integral of the rate of change of people gives the change in the number of people over the time interval. However, it does not tell you how many people are present because it contains no information on how many people were present initially. (Think that if you integrate  you get Here  denotes the number of people added and  denotes the initial number of people.)

Example Question #34 : Meaning Of Functions

Find the critical point(s) of .

Possible Answers:

 and 

  and 

 and 

Correct answer:

Explanation:

To find the critical point(s) of a function , take its derivative , set it equal to , and solve for .

Given ,  use the power rule

 to find the derivative. Thus the derivative becomes, .

Since :

The critical point  is 

Example Question #35 : Meaning Of Functions

Find the critical point(s) of .

Possible Answers:

 and 

 and 

 and 

Correct answer:

Explanation:

To find the critical point(s) of a function , take its derivative , set it equal to , and solve for .

Given , use the power rule  to find the derivative.

Thus the derivative becomes, .

Since :

The critical point  is 

Example Question #2781 : Calculus

Which of the following is not a function?

Possible Answers:

Correct answer:

Explanation:

A function is defined when each value of x yields a single value of y (ordered pairs).   The expression

is not a function because there are two values of f(x) for which a single value of x (7) creates.  f(7) can be either 3(7)+2=19 or 3(7)-2=21, and is therfore not a function.

Example Question #37 : Meaning Of Functions

Evaluate:

Possible Answers:

The limit does not exist

Correct answer:

Explanation:

To evaluate the limit, we can factor the numerator

Now we simplify and evaluate.

Example Question #2782 : Calculus

Evaluate:

Possible Answers:

Correct answer:

Explanation:

In order to evaluate the limit, we need to factor the numerator

Now we can simplify the expression to

.

Now we can plug in 1 to get

.

Example Question #2791 : Calculus

Evaluate the limit:

Possible Answers:

None of the other answers

Correct answer:

Explanation:

By applying L'Hôpital's rule, we can find the limit by evaluating

The function is now written as

Plugging in 0 gives us

Example Question #31 : How To Find The Meaning Of Functions

Which of the following statements is true regarding the behavior of functions with respect to its derivatives?

Possible Answers:

The natural log  is a decreasing function because its second derivative is always negative.

An example of a point of inflection is where the function goes from increasing at a decreasing rate to increasing at an increasing rate.

If at a point, the derivative is positive and the second derivative in negative, the function is decreasing at an increasing rate.

The second derivative tells you when the function changes from decreasing to increasing and vice versa.

Correct answer:

An example of a point of inflection is where the function goes from increasing at a decreasing rate to increasing at an increasing rate.

Explanation:

The statement

"An example of a point of inflection is where the function goes from increasing at a decreasing rate to increasing at an increasing rate."

is true. Point of inflection is when the function changes from concave up to concave down and vice versa. In the example above, the function changes from concave down (slopes are decreasing) to concave up (slopes are increasing). 

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