Call Now to Set Up Tutoring:

(888) 888-0446

Calculus AB : Calculus AB

Study concepts, example questions & explanations for Calculus AB

Create An Account

All Calculus AB Resources

45 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Differentiate Inverse Functions

Let f (x) =6x^2 . Find $(f^{-1})'(x)$ .

Possible Answers:

$\frac{1}{12(\sqrt{}^x_6)}$

$\frac{1}{2x}$

$\frac{1}{12x^2}$

Correct answer:

$\frac{1}{12(\sqrt{}^x_6)}$

Explanation:

To find the derivative of the inverse of f (x) , it is useful to first solve for $f^{-1}$ .

This will help because $f^{-1}$ is needed in the derivative equation, $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$ .

$f (x) =6x^2 \rightarrow f^{-1} (x) =\sqrt[2]{^x_6}$

Next, the equation for the derivative of an inverse function can be evaluated.

$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}=\frac{1}{f'(\sqrt[2]{^x_6})}$

After substituting in $\sqrt[2]{^x_6}$ for $f^{-1} (x)$ , the derivative of f (x) , or f'(x) =12x , is applied:

$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}=\frac{1}{f'(\sqrt[2]{^x_6})}=\frac{1}{12(\sqrt[2]{^x_6})}$

Therefore, the correct answer is $(f^{-1})'(x)=\frac{1}{12(\sqrt{^x_6})}$

Report an Error

Example Question #2 : Differentiate Inverse Functions

Suppose the points in the table below represent the continuous function f(x) . The differentiable function g(x) is the inverse of the function f(x) . Find g'(2) .

Q9 table

Possible Answers:

$\frac{1}{6}$

$\frac{1}{2}$

Correct answer:

$\frac{1}{6}$

Explanation:

Below is the equation for the derivative of g(x) :

$g'(x) = \frac{1}{f'(f^{-1}(x))}$

$g'(2) = \frac{1}{f'(f^{-1}(2))}$

So, the value of $f^{-1}(2)$ must first be found.

Using the data from the table, $f^{-1}(2)=2$ since f(2)=2 .

Next, from the table the following can be obtained:

f'(2)=6

Now, the appropriate substitutions can be made to solve for g'(2) .

$g'(2)=\frac{1}{f'(f^{-1}(2))}=\frac{1}{f'(2)}=\frac{1}{6}$

Report an Error

Example Question #1 : Differentiate Inverse Functions

Find $(f^{-1})'(3)$ given f (x) =ln(3x) .

Possible Answers:

e^9

$\frac{1}{3}e^9$

e^3

$\frac{1}{3}e^3$

Correct answer:

$\frac{1}{3}e^3$

Explanation:

To find the derivative of the inverse of f (x) , it is useful to first solve for $f^{-1}$ .

This will help because $f^{-1}$ is needed in the derivative equation, $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$ .

$f (x) =ln(3x) \rightarrow f^{-1} (x) =\frac{1}{3}e^x$

Next, the equation for the derivative of an inverse function can be evaluated.

$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}=\frac{1}{f'(\frac{1}{3}e^x)}$

After substituting in $\frac{1}{3}e^x$ for $f^{-1} (x)$ , the derivative of f (x) , or $f'(x) =\frac{1}{x}$ , is found by taking the derivative of ln(3x) and applying chain rule.

$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}=\frac{1}{f'(\frac{1}{3}e^x)}=\frac{1}{\frac{1}{\frac{1}{3}e^x}}=\frac{\frac{1}{3}e^x}{1}=\frac{1}{3}e^x$

After finding the general term $(f^{-1})'(x)=\frac{1}{3}e^x$ , evaluate at x=3 .

$(f^{-1})'(3)=\frac{1}{3}e^3$

Therefore, the correct answer is $(f^{-1})'(3)=\frac{1}{3}e^3$ .

Report an Error

Example Question #1 : Calculate Higher Order Derivatives

f(x)=1

$f^{'}(x)=2$

g(x)=3

and $g^{'}(x)=4$ ,

then find $h^{'}(x)$ .

Possible Answers:

Correct answer:

Explanation:

We see the answer is when we use the product rule.

$h(x)=\frac{f+g}{f}$

$h'(x)=\frac{(f'(x)+g'(x))(f(x))-(f(x)+g(x))(f'(x))}{f^{2}}$

$h'(x)=\frac{(2+4)(1)-(1+3)(2)}{1^{2}} =-2$

Report an Error

Example Question #2 : Calculate Higher Order Derivatives

On a closed interval, the function f'(x) is decreasing. What can we say about f(x) and f''(x) on these intervals?

Possible Answers:

f''(x) is negative

f''(x) is decreasing

f(x) is decreasing

f(x) is negative

Two or more of the other answers are correct.

Correct answer:

f''(x) is negative

Explanation:

If f'(x) is decreasing, then its derivative is negative. The derivative of f'(x) is f''(x) , so this is telling us that f''(x) is negative.

For f(x) to be decreasing, f'(x) would have to be negative, which we don't know.

f(x) being negative has nothing to do with its slope.

For f''(x) to be decreasing, its derivative f'''(x) would need to be negative, or, alternatively f'(x) would have to be concave down, which we don't know.

Thus, the only correct answer is that f''(x) is negative.

Report an Error

Example Question #3 : Calculate Higher Order Derivatives

On what intervals is the function y = (x+3)^2(x-2) both concave up and decreasing?

Possible Answers:

(-4/3, 1/3)

$(1/3, \infty)$

$(-4/3, \infty)$

$(-\infty, -4/3)$

(-3, -4/3)

Correct answer:

(-4/3, 1/3)

Explanation:

The question is asking when the derivative is negative and the second derivative is positive. First, taking the derivative, we get

y' =2(x+3)(x-2) +(x+3)^2 = (x+3)(2(x-2) + (x+3)) = (x+3)(3x -1)

Solving for the zero's, we see hits zero at x = -3 and x = 1/3 . Constructing an interval test,

$(-\infty, -3), (-3, 1/3), (1/3, \infty)$ , we want to know the sign's in each of these intervals. Thus, we pick a value in each of the intervals and plug it into the derivative to see if it's negative or positive. We've chosen -5, 0, and 1 to be our three values.

y'(-5) = (-2)(-16) = 32

y'(0) = (3)(-1) = -3

y'(1) = (4)(2) = 8

Thus, we can see that the derivative is only negative on the interval (-3, 1/3) .

Repeating the process for the second derivative,

y'' = (3x-1) + 3(x+3) = 6x + 8

The reader can verify that this equation hits 0 at -4/3. Thus, the intervals to test for the second derivative are

$(-\infty, -4/3), (-4/3, \infty)$ . Plugging in -2 and 0, we can see that the first interval is negative and the second is positive.

Because we want the interval where the second derivative is positive and the first derivative is negative, we need to take the intersection or overlap of the two intervals we got:

$(-3, 1/3) \cap (-4/3, \infty) = (-4/3, 1/3)$

If this step is confusing, try drawing it out on a number line -- the first interval is from -3 to 1/3, the second from -4/3 to infinity. They only overlap on the smaller interval of -4/3 to 1/3.

Thus, our final answer is (-4/3, 1/3) .

Report an Error

Example Question #1 : Calculate Higher Order Derivatives

What does the second derivative of a function tell us?

Possible Answers:

The sign of the second derivative tells us if the function is concave up or down

The sign of the second derivative tells us if the first derivative is increasing or decreasing

All of the above are true

The sign of the second derivative tells us if the function is increasing or decreasing

Correct answer:

All of the above are true

Explanation:

The second derivative is essentially a derivative of the first derivative. Information we get from the first derivative is, if the first derivative is increasing, then the function is also increasing and concave up. If the second derivative has a positive sign, then the first derivative is increasing and the function is concave up. Similarly if the function is concave down, the first derivative is decreasing, and the second derivative is negative.

Report an Error

Example Question #2 : Calculate Higher Order Derivatives

What is the second derivative of f(x) = 5x^3 - 2x + 3 .

Possible Answers:

15x^2

30x

15x^2 - 2

30x -2

Correct answer:

30x

Explanation:

To find the second derivative we must first find the first derivative.

f(x) = 5x^3 - 2x + 3

f'(x) = 15x^2 - 2

Now that we have found the first derivative we will take the derivative of it.

f'(x) = 15x^2 - 2

f''(x) = 30x

And so the second derivative is f''(x) = 30x

Report an Error

Example Question #1 : Calculate Higher Order Derivatives

When considering a moving object, the second derivative is the ________ .

Possible Answers:

acceleration

distance

None of the above

velocity

Correct answer:

acceleration

Explanation:

If we think about the original function, this tells us the distance. We take the derivative of the position (we are finding the rate of change of the position) and we have the velocity. The rate of change of positions, gives us velocity. We would know which direction we are moving in. And now, to find the second derivative, we take the derivative of the velocity (finding the rate of change of the velocity) and we are left with the acceleration. Finding the rate of change of the velocity will tell us at what magnitude we are moving at.

Report an Error

Example Question #4 : Calculate Higher Order Derivatives

Find the second derivative of f(x) = sin(x) .

Possible Answers:

cos(x)

-sin(x)

sin(x)

-cos(x)

Correct answer:

-sin(x)

Explanation:

We begin by finding the first derivative.

f(x) = sin(x)

f'(x) = cos(x)

Now we will take the derivative of the first derivative.

f'(x) = cos(x)

f''(x) = -sin(x)

Report an Error

View Tutors

Norma
Certified Tutor

Queen's University, Bachelor in Arts, Psychology. Queen's University, Certificate, Writing, General.

View Tutors

Jozsef
Certified Tutor

University of Waterloo, Doctor of Philosophy, Applied Mathematics.

View Tutors

Neeta
Certified Tutor

Indian Institute Of Technology – Roorkee, Bachelor of Engineering, Telecommunications Engineering. Indira Gandhi National Ope...

All Calculus AB Resources

45 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

French Tutors in Miami, English Tutors in Chicago, French Tutors in Washington DC, MCAT Tutors in Seattle, Reading Tutors in Dallas Fort Worth, Reading Tutors in Washington DC, English Tutors in Denver, Spanish Tutors in Washington DC, SAT Tutors in Boston, GMAT Tutors in San Francisco-Bay Area

Popular Courses & Classes

ACT Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in San Francisco-Bay Area, Spanish Courses & Classes in Houston, GRE Courses & Classes in Washington DC, LSAT Courses & Classes in Denver, SAT Courses & Classes in Washington DC, SAT Courses & Classes in San Francisco-Bay Area, Spanish Courses & Classes in Atlanta, ACT Courses & Classes in Atlanta, LSAT Courses & Classes in Miami

Popular Test Prep

LSAT Test Prep in Chicago, LSAT Test Prep in Phoenix, SAT Test Prep in Seattle, GMAT Test Prep in Los Angeles, SSAT Test Prep in Miami, LSAT Test Prep in Atlanta, LSAT Test Prep in San Diego, ISEE Test Prep in Boston, MCAT Test Prep in Denver, LSAT Test Prep in Seattle

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Calculus AB : Calculus AB

Study concepts, example questions & explanations for Calculus AB

All Calculus AB Resources

Example Questions

Example Question #1 : Differentiate Inverse Functions

Example Question #2 : Differentiate Inverse Functions

Example Question #1 : Differentiate Inverse Functions

Example Question #1 : Calculate Higher Order Derivatives

Example Question #2 : Calculate Higher Order Derivatives

Example Question #3 : Calculate Higher Order Derivatives

Example Question #1 : Calculate Higher Order Derivatives

Example Question #2 : Calculate Higher Order Derivatives

Example Question #1 : Calculate Higher Order Derivatives

Example Question #4 : Calculate Higher Order Derivatives

All Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: