We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus AB : Calculate Higher Order Derivatives

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

← Previous 1 2 Next →

Example Question #41 : Differentiating Functions

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 negative

f(x) is decreasing

Two or more of the other answers are correct.

f''(x) is decreasing

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 #41 : Differentiating Functions

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

Possible Answers:

$(-3,-\frac{4}{3})$

$(-\frac{1}{3}, \infty )$

$(-\infty ,-\frac{4}{3})$

$(-\frac{4}{3},\frac{1}{3})$

$(-\frac{4}{3}, \infty )$

Correct answer:

$(-\frac{4}{3},\frac{1}{3})$

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 $x=\frac{1}{3}$ . 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 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 $(-3,\frac{1}{3})$ .

Repeating the process for the second derivative,

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

The reader can verify that this equation hits at $-\frac{4}{3}$ . Thus, the intervals to test for the second derivative are

$(-\infty, -\frac{4}{3}),(-\frac{4}{3},\infty )$ . Plugging in and , 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:

$(-3,\frac{1}{3})\cap(-\frac{4}{3},\infty )=(-\frac{4}{3},\frac{1}{3})$

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

Thus, our final answer is $(-\frac{4}{3},\frac{1}{3})$

Report an Error

Example Question #43 : Differentiating Functions

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

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

← Previous 1 2 Next →

View Tutors

Mareda
Certified Tutor

New York University, Bachelor of Science, Biology, General.

View Tutors

Yucheng
Certified Tutor

The University of Texas at Austin, Bachelor of Science, Mechanical Engineering.

View Tutors

Sarah
Certified Tutor

Sarah Lawrence College, Bachelor, Language and Culture.

All Calculus AB Resources

45 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

ISEE Tutors in Phoenix, MCAT Tutors in Seattle, ACT Tutors in Miami, English Tutors in Atlanta, Chemistry Tutors in Phoenix, Computer Science Tutors in Miami, Spanish Tutors in Los Angeles, SSAT Tutors in Denver, ISEE Tutors in Los Angeles, Computer Science Tutors in Atlanta

Popular Courses & Classes

ISEE Courses & Classes in San Francisco-Bay Area, GMAT Courses & Classes in Chicago, SAT Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in Atlanta, SSAT Courses & Classes in Denver, MCAT Courses & Classes in Washington DC, GRE Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Boston, GMAT Courses & Classes in Los Angeles, SAT Courses & Classes in Boston

Popular Test Prep

GRE Test Prep in San Diego, SSAT Test Prep in Boston, MCAT Test Prep in Boston, SAT Test Prep in Chicago, ISEE Test Prep in Atlanta, SSAT Test Prep in Dallas Fort Worth, SAT Test Prep in San Diego, SAT Test Prep in Houston, ISEE Test Prep in Denver, ACT Test Prep in Denver

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 : Calculate Higher Order Derivatives

Study concepts, example questions & explanations for Calculus AB

All Calculus AB Resources

Example Questions

Example Question #41 : Differentiating Functions

Example Question #41 : Differentiating Functions

Example Question #43 : Differentiating Functions

All Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: