Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Meaning of Functions

Study concepts, example questions & explanations for Calculus 1

Create An Account

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Meaning Of Functions

Take the limit

$\lim_{x \to -2} \frac{x+\sqrt{x+6}}{x+2}$

Possible Answers:

$\frac{2}{3}$

$\frac{5}{4}$

$\frac{8}{3}$

Correct answer:

$\frac{5}{4}$

Explanation:

First, multiply the numerator and denominator by $x-\sqrt{x+6}$ and it turns into

$\lim_{x \to -2} \frac{(x+\sqrt{x+6})}{(x+2)}\frac{(x-\sqrt{x+6})}{(x-\sqrt{x+6})}$

$\lim_{x \to -2} \frac{x^2-x-6}{(x+2)(x-\sqrt{x+6})}$

Factor the numerator and then cancel out the 'x+2'

$\lim_{x \to -2} \frac{(x+2)(x-3)}{(x+2)(x-\sqrt{x+6})}$

$\lim_{x \to -2} \frac{(x-3)}{(x-\sqrt{x+6})}$

After taking the limit, the answer is $\frac{5}{4}$

Report an Error

Example Question #2 : Meaning Of Functions

If this limit is true, then what is the value of 'a'?

$\lim_{x \to 3} \frac{x^4-a}{x^2-\sqrt{a}}=18$

Possible Answers:

121

Correct answer:

Explanation:

Factor the numerator

$\lim_{x \to 3} \frac{(x^2-\sqrt{a})(x^2+\sqrt{a})}{x^2-\sqrt{a}}=18$

Cancel the $x^2-\sqrt{a}$ , plug in the limit and then solve for 'a'

$\lim_{x \to 3} x^2+\sqrt{a}=18$

$3^2+\sqrt{a}=18$

$a=81$

Report an Error

Example Question #3 : Meaning Of Functions

We have a line described as $y=-\frac{1}{2}x+2$ . Find the minimum distance between the origin and a point on that line.

Possible Answers:

$\frac{16}{5}$

$d=\frac{\sqrt{5}}{4}$

$d=\sqrt{\frac{4}{5}}$

$1$

$d=\frac{4}{\sqrt{5}}$

Correct answer:

$d=\frac{4}{\sqrt{5}}$

Explanation:

We have the origin (0,0) and a point (x,y) located on the line. That point represents the minimum distance to the orgin. Apply the distance formula to these two points,

$d=\sqrt{(x-0)^2+(y-0)^2}=\sqrt{x^2+y^2}$

Plug in the line equation, take the derivative, set it equal to zero, and solve for x.

$d=\sqrt{x^2+(-\frac{1}{2}x+2)^2}=\sqrt{\frac{5}{4}x^2-2x+4}$ ${d}'=\frac{\frac{5}{2}x-2}{2\sqrt{\frac{5}{4}x^2-2x+4}}=0$

$x=\frac{4}{5}$

Use this value to find

$y=-\frac{1}{2}\left ( \frac{4}{5} \right )+2=\frac{8}{5}$

So we have the point $(\frac{4}{5},\frac{8}{5})$ , which is closest to the origin. We can now find its distance from that origin.

$d=\sqrt{\left (\frac{4}{5} \right )^2+\left ( \frac{8}{5} \right )^2}$

$d=\sqrt{\frac{16}{5}}=\frac{4}{\sqrt{5}}$

Report an Error

Example Question #4 : Meaning Of Functions

We have the following,

$\int_{-5}^{5}\frac{x^2+(6+c)x+6c}{x+6}dx=-60$

What is c?

Possible Answers:

Correct answer:

Explanation:

First, factor the numerator of the integrand.

$\int_{-5}^{5}\frac{(x+6)(x+c)}{x+6}dx$

Cancel out x+6

$\int_{-5}^{5}\frac{(x+6)(x+c)}{x+6}dx=-60$

$\int_{-5}^{5}(x+c)dx=-60$

Perform the integral and then solve for

$\frac{1}{2}(5)^2+5c-\left ( \frac{1}{2}(-5)^2+(-5)c \right )=-60$

$10c=-60$

$c=-6$

Report an Error

Example Question #5 : Meaning Of Functions

If $h(x)=\int_{0}^{x}sin(u)du$ , find $h^{'}(x)$

Possible Answers:

$h^{'}(x) = sin(x)$

$h^{'}(x) = sin(0.5x^2)$

$h^{'}(x) = cos(x)$

$h^{'}(x) = -cos(x)$

Correct answer:

$h^{'}(x) = sin(x)$

Explanation:

Taking the derivative of an integral yields the original function, but because we have a different variable in the integration limits, the variable switches

Report an Error

Example Question #6 : Meaning Of Functions

Evaluate $\int(14x^7 +\frac{x^6}{(2)}+sin(x)-csc^2(x))dx$

Possible Answers:

$\frac{14x^8}{8}+\frac{x^7}{14}-cos(x)+cot(x)+C$

$\frac{14x^6}{6}+\frac{x^5}{14}+cos(x)-cot(x)+C$

$\frac{14x^8}{8}+\frac{x^7}{14}+cos(x)-cot(x)+C$

98x^6+3x^5+cos(x)+cot(x)

Correct answer:

$\frac{14x^8}{8}+\frac{x^7}{14}-cos(x)+cot(x)+C$

Explanation:

using integration identities: $\int{x^n dx}= \frac{x^{n+1}}{n+1} and \int{sin(x) dx}= - cos(x) and \int{csc^2(x) dx}= -cot (x)$

Report an Error

Example Question #7 : Meaning Of Functions

Evaluate

$\int_{1}^{4}(4x^2+2x-8)dx$

Possible Answers:

66.33

55.33

Correct answer:

Explanation:

$\int_{1}^{4}(4x^2+2x-8)dx$

$\frac{4x^3}{3} + \frac{2x^2}{2} -8x$

$\frac{4x^3}{3} + x^2 -8x$ evaluate at $\left [ 1,4 \right ]$

$\left(\frac{4(4)^3}{3} + (4)^2 -8(4)\right) - \left(\frac{4(1)^3}{3} + (1)^2 -8(1)\right)$

=75

Report an Error

Example Question #8 : Meaning Of Functions

Evaluate $\int_{0}^{3}(2x(x^2+1)^3)dx$

Possible Answers:

$\frac{9999}{4}$

$\frac{81}{4}$

$\frac{10001}{4}$

$\frac{10000}{4}$

Correct answer:

$\frac{9999}{4}$

Explanation:

$\int_{0}^{3}(2x(x^2+1)^3)dx$

Intergation by substitution

u= x^2+1

du=2xdx

new endpoints:

u=(3)^2+1=10

u=(0)^2+1=1

New Equation:

$\int_{1}^{10}(u^3)du$

$\frac{u^4}{4}$ at $\left [ 1,10 \right ]$

$= \frac{(10)^4}{4} - \frac{(1)^4}{4}$

$= \frac{(10000)}{4} - \frac{(1)}{4}$

$= \frac{9999}{4}$

Report an Error

Example Question #9 : How To Find The Meaning Of Functions

What is $\lim_{x \to 0^+} ~~x \cdot \ln(x)$ ?

Possible Answers:

undefined

$-\infty$

-1

Correct answer:

Explanation:

The relationship between ln(x) and x is an exponential relationship; is going to exponentially faster than ln(x) is going to $-\infty$ . One way to prove this is to write x = 1/(1/x) and use L'Hôpital's rule:

$\lim_{x \to 0^+} \frac{\ln x}{1/x} = \lim_{x \to 0^+} \frac{\ln'x}{(1/x)'} = \lim_{x \to 0^+} \frac{1/x}{-1/x^2} = \lim_{x \to 0^+} \frac{-x^2}{x} = -0 = 0$

Report an Error

Example Question #9 : Meaning Of Functions

Where is $\frac{4x^3 - 21 x + 10}{x^2 - 5x + 6}$ discontinuous? Are those discontinuities removable?

Possible Answers:

Removable discontinuities at x=-2 and x=-3 .

Removable discontinuities at x=2 and x=3 .

Removable discontinuity at x=2 , essential discontinuity at x=3 .

Essential discontinuities at x = -2 and x = -3 .

Removable discontinuity at x=-2 , essential discontinuity at x=-3 .

Correct answer:

Removable discontinuity at x=2 , essential discontinuity at x=3 .

Explanation:

The rational function $\frac{4x^3 - 21 x + 10}{x^2 - 5x + 6}$ has a denominator with two roots, x=2 and x=3 . These are discontinuities.

Factoring both top and bottom and canceling a term x-2 tells us that this function is equal to

$\frac{(2x - 1) (2x + 5)}{x - 3}$

except at x=2 . This point is a removable discontinuity. x=3 is therefore an essential discontinuity where the ratio goes to $\pm \infty$ .

Report an Error

View Calculus Tutors

Jonathan
Certified Tutor

University of Maryland-Baltimore County, Bachelor of Science, Mathematics.

View Calculus Tutors

James
Certified Tutor

St. Mary of the Plains College, Bachelors, Mathematics. Baker University, Masters, Business Administration and Management.

View Calculus Tutors

Satweka
Certified Tutor

The University of Texas at Dallas, Master of Science, Biomedical Engineering.

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

ISEE Tutors in Washington DC, GRE Tutors in Boston, Math Tutors in Phoenix, Spanish Tutors in Phoenix, GMAT Tutors in Philadelphia, Computer Science Tutors in Los Angeles, MCAT Tutors in Washington DC, LSAT Tutors in Seattle, Reading Tutors in San Diego, Chemistry Tutors in Miami

Popular Courses & Classes

MCAT Courses & Classes in San Diego, SSAT Courses & Classes in Chicago, ACT Courses & Classes in Washington DC, MCAT Courses & Classes in Atlanta, SAT Courses & Classes in Phoenix, GMAT Courses & Classes in Seattle, LSAT Courses & Classes in Houston, MCAT Courses & Classes in Los Angeles, GMAT Courses & Classes in San Diego, SAT Courses & Classes in Seattle

Popular Test Prep

ISEE Test Prep in Los Angeles, GMAT Test Prep in Seattle, MCAT Test Prep in San Francisco-Bay Area, ACT Test Prep in Houston, ISEE Test Prep in Chicago, SAT Test Prep in Washington DC, SAT Test Prep in Philadelphia, LSAT Test Prep in Denver, MCAT Test Prep in Washington DC, GRE Test Prep in Houston

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 1 : Meaning of Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #1 : Meaning Of Functions

Example Question #2 : Meaning Of Functions

Example Question #3 : Meaning Of Functions

Example Question #4 : Meaning Of Functions

Example Question #5 : Meaning Of Functions

Example Question #6 : Meaning Of Functions

Example Question #7 : Meaning Of Functions

Example Question #8 : Meaning Of Functions

Example Question #9 : How To Find The Meaning Of Functions

Example Question #9 : Meaning Of Functions

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: