Calculus 1 : Meaning of Functions

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #61 : How To Find The Meaning Of Functions

Find the slope of the tangent line equation for  when .

Possible Answers:

Correct answer:

Explanation:

To find the slope, we need to differentiate the given function. By product rule and chain rule, we have  

Example Question #61 : How To Find The Meaning Of Functions

Function

What is a function?

Possible Answers:

A function is an equation with at least two unknowns such as x and y

A function is a relationship that can produce multiple output values for each single input value

A function is a matematical equation with one or, more variables

None of the above

A function is a relationship that assigns to each input value a single output value

Correct answer:

A function is a relationship that assigns to each input value a single output value

Explanation:

You could think of a function as a machine that takes in some number, performs an operation on it, and then spits out another number

Example Question #63 : Meaning Of Functions

Identify Function

Which one of the following is not a function?

Possible Answers:

Correct answer:

Explanation:

For any value of  in , there will be two values of .  So it is not a function.

Example Question #62 : How To Find The Meaning Of Functions

Type of Functions

What type of function is this:

Possible Answers:

Linear

Rational

Quadratic

Polynomial

Piecewise

Correct answer:

Piecewise

Explanation:

Piecewise functions are a special type function in which the formula changes for different values.

Example Question #2814 : Calculus

Find the domain of the function 

Possible Answers:

 or 

 or 

Correct answer:

Explanation:

 is defined when , and  is defined when . Since the radical part is in the denominator, it cannot be . Therefore, we need  Combining this domain with , we get

Example Question #66 : Meaning Of Functions

What is the slope of this curve at ?

Possible Answers:

Correct answer:

Explanation:

To find the slope of a curve at a certain point, you must first find the derivative of that curve. 

 

The derivative of this curve is .

Then, plug in  for  into the derivative and you will get  as the slope of  at .

 

 

Example Question #64 : How To Find The Meaning Of Functions

What are the critical points of ?

Possible Answers:

Correct answer:

Explanation:

To find the critical points of a function, you need to first find the derivative and then set that equal to 0. To find the derivative, multiply the exponent by the leading coefficient and then subtract 1 from the exponent. Therefore, the derivative is . Set that equal to 0 and then factor so that you get: . Solve for x in both expressions so that your answer is: .

Example Question #63 : How To Find The Meaning Of Functions

Finding when the second derivative is positive tells us what about a function?

Possible Answers:

When the function is decreasing.

When the function is increasing.

When the function is concave down.

When the function is concave up.

Correct answer:

When the function is concave up.

Explanation:

The first derivative of a function tells us about the slope and the second derivative tells us about concavity. Thus, when the second derivative is positive, it tell us that the concavity is upward. Therefore, a postive second derivative means the function is concave up.

Example Question #64 : How To Find The Meaning Of Functions

Given the position function below, what does evaluating it at  tell us?

Possible Answers:

It tells us that at , the position of our object is .

It cannot be determined.

It tells us that at , the time our object has been moving is .

It tells us that at , the slope of our function is .

Correct answer:

It tells us that at , the position of our object is .

Explanation:

When asked to evaluate the function, you must realize that you are going to be plugging in a value for t. Therefore, we need to figure out what the output value is telling us when t=1. Since we are simply given the position function, adn now asked to integrate it or differentiate it, our output will simply tell us the position of our object when t=1. Thus,

Example Question #2822 : Calculus

Evaluate the following limit:

Possible Answers:

Does not exist.

Correct answer:

Explanation:

When evaluating the limit through direct substitution, one ends with the indeterimate form 0/0, therefore another approach must be followed. Two approaches would work here:

1.

One could use L'Hospital's Rule, which states that if a function is indeterminate when evaluating, one should then take the derivative of the top function over the derivative of the bottom. 

In this case one would find 

 

2.

Alternatively, one could simply factor both the numerator and the denomenator, removing common factors.

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