We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Derivatives

Study concepts, example questions & explanations for Precalculus

Create An Account

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #5 : Rate Of Change Problems

Find the average rate of change of $y=\sqrt{x-3}$ between x=4 and x=7 .

Possible Answers:

$\frac{1}{3}$

$-\frac{1}{3}$

Correct answer:

$\frac{1}{3}$

Explanation:

The solution will be found by the formula $\frac{f(7)-f(4)}{7-4}$ .

Here f(7) gives us $\sqrt{7-3}=\sqrt{4}=2$ , and $f(4)=\sqrt{4-3}=\sqrt{1}=1$ .

Thus, we find that the average rate of change is $\frac{2-1}{7-4}=\frac{1}{3}$ .

Report an Error

Example Question #4 : Rate Of Change Problems

Find the average rate of change of y=2x^2+3x over the interval from x=1 to x=3 .

Possible Answers:

Correct answer:

Explanation:

The average rate of change will be $\frac{f(3)-f(1)}{3-1}$ .

f(3)=2(3)^2+3(3)=2(9)+9=18+9=27 .

f(1)=2(1)^2+3(1)=2(1)+3=2+3=5 .

This gives us $\frac{27-5}{3-1}=\frac{22}{2}=11$ .

Report an Error

Example Question #5 : Rate Of Change Problems

Find the average rate of change of y=-x^2+3x-6 over the interval from x=-2 to x=0 .

Possible Answers:

Correct answer:

Explanation:

The average rate of change will be $\frac{f(0)-f(-2)}{0-(-2)}$ .

Now f(0)=-(0)^2+3(0)-6=-0+0-6=-6 .

We also know f(-2)=-(-2)^2+3(-2)-6=-4-6-6=-16 .

So we have $\frac{-6-(-16)}{0-(-2)}=\frac{-6+16}{0+2}=\frac{10}{2}=5$ .

Report an Error

Example Question #6 : Rate Of Change Problems

Why can we make an educated guess that the average rate of change of f(x)=x^2 , between x=-1 and x=1 would be ?

Possible Answers:

We know f(x) is horizontal on that interval.

We know f(x) is a polynomial.

We know f(x) is vertical on that interval.

We know f(x) is symmetrical on that interval.

We know f(x) is odd on that interval.

Correct answer:

We know f(x) is symmetrical on that interval.

Explanation:

Because f(x) is symmetrical over the y axis, it increases exactly as much as it decreases on the interval from to . Thus the average rate of change on that interval will be .

Report an Error

Example Question #11 : Derivatives

If the average rate of change of f(x) between x=a and x=b , where b>a , is positive, then what can be said about f(x) on that interval?

Possible Answers:

f(x) is constant

f(x) is odd

f(x) is increasing

f(x) is decreasing

Correct answer:

f(x) is increasing

Explanation:

If the average rate of change is positive, then the formula gives us $\frac{f(b)-f(a)}{b-a}>0$ , so f(b)-f(a)>b-a . We know b>a because it is a given in the proble, so b-a>0 . Hence

f(b)-f(a)>0 and f(b)>f(a) . This shows that f(x) must increase over the interval from to .

Report an Error

Example Question #12 : Rate Of Change Problems

If the average rate of change of f(x) between x=a and x=b , where a<b , is negative, then what can be said about f(x) on that interval?

Possible Answers:

f(x) is decreasing

f(x) is negative

f(x) is an odd function

f(x) is constant

f(x) is increasing

Correct answer:

f(x) is decreasing

Explanation:

If the average rate of change is negative, then the function is changing in a negative direction overall. Hence, the graph of the function will be decreasing on that interval.

$\frac{f(b)-f(a)}{b-a}<0$

f(b)-f(a)<0

f(b)<f(a) and since a<b , f(x) is decreasing

Report an Error

Example Question #1 : Maximum And Minimum Problems

The profit of a certain cellphone manufacturer can be represented by the function

$\small \small p(x)=-2000x^3+12000x^2+126000x$

where $\small p$ is the profit in dollars and $\small x$ is the production level in thousands of units. How many units should be produced to maximize profit?

Possible Answers:

$\small 7$

$\small 3000$

$\small 42000$

$\small 7000$

$\small 21000$

Correct answer:

$\small 7000$

Explanation:

We can determine the production level that maximizes profit by taking the derivative of our function. This can be done using the power rule.

$\small p'(x)=3(-2000x^2)+2(12000x)+126000=-6000x^2+24000x+126000$

We then set this derivative equal to 0 and solve.

$\small 0=-6000x^2+24000x+126000$

We can factor out our greatest common factor and then divide it out.

$\small 0=-6000(x^2-4x-21)$

$\small 0=x^2-4x-21$

Next we can factor.

$\small 0=(x+3)(x-7)$

Therefore, we can solve

$\small x=-3\:or\:7$

We remember that these production levels are in thousands of units.

However, a factory cannot produce negative cellphones, so the maximum profit is obtained when seven thousand units are produced.

Report an Error

Example Question #2 : Maximum And Minimum Problems

What is the critical point of y=x^2 . Is it a max or a minimum?

Possible Answers:

Critical Point=(0,2)

Max

Critical Point=(0,0)

Max

Critical Point=(2,0)

Minimum

Critical Point=(0,0)

Minimum

Critical Point=(0,2)

Minimum

Correct answer:

Critical Point=(0,0)

Minimum

Explanation:

To find the critical points, we first take the first derivative using the Power Rule:

$y=x^n \rightarrow y'=nx^{n-1}$

Therefore,

y=x^2

y'=2x .

To find the critical point we need to set the derivative equal to zero and solve for x.

Doing so this gives the critical point at x=0 . To get the y value of the point plug x=0 back into the original equation.

y=0^2=0 thus the point is (0,0) .

Is it a max or min?

Take the second derivative

y''=2

It is a minimum since the second derivative is positive.

Report an Error

Example Question #13 : Derivatives

What is the critical point of y=-x^2 ? Is it a max or minimum?

Possible Answers:

Critical Point=(0,0)

Minimum

Critical Point=(2,0)

Max

Critical Point=(0,2)

Minimum

Critical Point=(0,2)

Max

Critical Point=(0,0)

Max

Correct answer:

Critical Point=(0,0)

Max

Explanation:

To find the critical points, we first take the first derivative using the Power Rule.

$y=x^n \rightarrow y'=nx^{n-1}$ .

Therefore,

y=-x^2

y'=-2x .

From here we need to set the derivative equal to zero and solve to x.

The critical point occurs when x=0 . Now to get the y value of the point we will need to plug in x=0 into the original equation.

y=-0^2=0 . This gives us a critical point at (0,0) .

To find if it is a max or min, we take the second derivative

y''=-2

Since the second derivative is negative, it is a maximum.

Report an Error

Example Question #4 : Maximum And Minimum Problems

Determine whether if there is a maximum or minimum, and location of the point for: y=3x^2-11x+5

Possible Answers:

$\textup{Maximum at: } \left(\frac{11}{6},\frac{61}{12}\right)$

$\textup{Minimum at: } \left(\frac{11}{6},-\frac{61}{12}\right)$

$\textup{Minimum at: } \left(\frac{11}{6},\frac{61}{12}\right)$

$\textup{Maximum at: } \left(\frac{11}{6},-\frac{61}{12}\right)$

$\textup{Minimum at: } \left(\frac{11}{6},5\right)$

Correct answer:

$\textup{Minimum at: } \left(\frac{11}{6},-\frac{61}{12}\right)$

Explanation:

Determine the derivative of this function.

y=3x^2-11x+5

y'=6x-11

Set the derivative function equal to zero and solve for .

0=6x-11

$x=\frac{11}{6}$

Since the parabola has a positive coefficient for 3x^2 , the parabola will open upwards, and therefore will have a minimum.

Substitute $x=\frac{11}{6}$ back into the original function and find the y-value of the minimum.

$y=3(\frac{11}{6})^2-11(\frac{11}{6})+5$

$= 3(\frac{121}{36})-\frac{121}{6}+5$

$=\frac{121}{12}-\frac{121}{6}+5$

$=\frac{121}{12}-\frac{242}{12}+\frac{60}{12}$

$y=-\frac{61}{12}$

The parabola has a minimum at the point $(\frac{11}{6},-\frac{61}{12})$ .

Report an Error

View Pre-Calculus Tutors

Dejuan
Certified Tutor

University of Illinois at Urbana-Champaign, Bachelor of Science, Electrical Engineering.

View Pre-Calculus Tutors

Leonard
Certified Tutor

Florida State University, Bachelor of Science, Environmental Science.

View Pre-Calculus Tutors

Yucheng
Certified Tutor

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

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

SSAT Tutors in Miami, Computer Science Tutors in Dallas Fort Worth, Spanish Tutors in New York City, GMAT Tutors in Atlanta, Computer Science Tutors in Los Angeles, Chemistry Tutors in Houston, ACT Tutors in Boston, Physics Tutors in Dallas Fort Worth, GMAT Tutors in Denver, SSAT Tutors in New York City

Popular Courses & Classes

GRE Courses & Classes in San Diego, SAT Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Seattle, Spanish Courses & Classes in Miami, GRE Courses & Classes in Houston, GRE Courses & Classes in Washington DC, ISEE Courses & Classes in Philadelphia, Spanish Courses & Classes in New York City, ISEE Courses & Classes in Phoenix, GRE Courses & Classes in Atlanta

Popular Test Prep

MCAT Test Prep in Chicago, LSAT Test Prep in Denver, LSAT Test Prep in Houston, GRE Test Prep in Houston, GRE Test Prep in New York City, ISEE Test Prep in Philadelphia, MCAT Test Prep in Atlanta, SSAT Test Prep in Houston, GRE Test Prep in San Diego, GRE Test Prep in Boston

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

Precalculus : Derivatives

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #5 : Rate Of Change Problems

Example Question #4 : Rate Of Change Problems

Example Question #5 : Rate Of Change Problems

Example Question #6 : Rate Of Change Problems

Example Question #11 : Derivatives

Example Question #12 : Rate Of Change Problems

Example Question #1 : Maximum And Minimum Problems

Example Question #2 : Maximum And Minimum Problems

Example Question #13 : Derivatives

Example Question #4 : Maximum And Minimum Problems

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: