Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Slope

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 : Slope

What is the slope of the tangent line of f(x) = 3x⁴ – 5x³ – 4x at x = 40?

Possible Answers:

743,996

331,841

684,910

None of the other answers

768,000

Correct answer:

743,996

Explanation:

The first derivative is easy:

f'(x) = 12x³ – 15x² – 4

The slope of the tangent line is found by calculating f'(40) = 12 * 40³ – 15 * 40² – 4 = 768,000 – 24,000 – 4 = 743,996

Report an Error

Example Question #1 : How To Find Slope By Graphing Functions

Find the slope of the line tangent to y(b) when is equal to .

y(b)=b^7-b^5+4b^2

Possible Answers:

3216

3160

320160

Correct answer:

320160

Explanation:

To find the slope of a tangent line, we need to find the first derivative of the function at that point. In other words, we need y'(6).

y(b)=b^7-b^5+4b^2

Taking the first derivative using the Power Rule $f(x)=x^n \rightarrow f'(x)=nx^{n-1}$ we get the following.

y'(b)=7b^6-5b^4+8b

Substituting in 6 for b and solving we get:

$y'(6)=7\cdot6^6-5\cdot6^4+8\cdot6=320160$ .

So our answer is 320160

Report an Error

Example Question #1 : How To Find Slope By Graphing Functions

Find function which gives the slope of the line tangent to j(x) .

$j(x)=\frac{4x^5}{5}-x^4+x^2$

Possible Answers:

4x^4-4x^3+2

3x^4-12x^3+2x

4x^4-4x^3+2x

16x^3-12x^2+2

x^4-x^3+x

Correct answer:

4x^4-4x^3+2x

Explanation:

To find the slope of a tangent line, we need the first derivative.

Recall that to find the first derivative of a polynomial, we need to decrease each exponent by one and multiply by the original number.

$j(x)=\frac{4x^5}{5}-x^4+x^2$

j'(x)=4x^4-4x^3+2x

Report an Error

Example Question #2 : How To Find Slope By Graphing Functions

Find the slope of the line tangent to b(s) at s=16 .

b(s)=14s^2+6s-4

Possible Answers:

144

454

343

545

544

Correct answer:

454

Explanation:

The slope of the tangent line can be found easily via derivatives. To find the slope of the tangent line at s=16, find b'(16) using the power rule on each term which states:

$f(x)=x^n \rightarrow f'(x)=nx^{n-1}$

Applying this rule we get:

b(s)=14s^2+6s-4

b'(s)=28s+6

$b'(16)=28\cdot 16+6=454$

Therefore, the slope we are looking for is 454.

Report an Error

Example Question #3 : How To Find Slope By Graphing Functions

Find the slope of f(x) at $x=\pi$ .

f(x)=ln(x^2 +2)-cos(17x)

Possible Answers:

0.529

10.229

17.529

0.629

Correct answer:

0.529

Explanation:

To find the slope of the line at that point, find the derivative of f(x) and plug in that point.

$f'(x)= \frac{2x}{x^2 +2}+17sin(17x)$

Remember that the derivative of cos(bx) = -bsin(bx) and the derivative of $ln(u) = \frac{u'}{u}$

Now plug in $\pi$

$f'(\pi )=\frac{2\pi}{\pi ^2 +2}=.529$

Report an Error

Example Question #3 : Slope

Find the slope of f(x) at x=2 given f''(x) . Assume the integration constant is zero.

f''(x)= x+2

Possible Answers:

Correct answer:

Explanation:

The first step here is to integrate f''(x) in order to get f'(x) .

$\int f''(x)dx =\int(x+2)dx=\frac{x^2}{2} +2x +C$

Here the problem tells us that the integration constant C=0 , so

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

Plug in x=2 here

f'(2)=6

Report an Error

Example Question #3 : Slope

Consider the curve

y=ln[x]+3x^2 .

What is the slope of this curve at x=1.5 ?

Possible Answers:

$\frac{17}{3}$

$\frac{29}{3}$

$\frac{34}{3}$

Correct answer:

$\frac{29}{3}$

Explanation:

The slope of a curve at any point is equal to the derivative of the curve at that point.

Remembering that the derivative of $ln(x)\rightarrow \frac{1}{x}$ and using the power rule on the second term we find the derivative to be:

$\frac{dy}{dx}=\frac{1}{x}+6x$ .

Pluggin in x=1.5 we find that the slope is $\frac{29}{3}$ .

Report an Error

Example Question #2 : Slope

Find the line tangent to $y=e^{5x}$ at x=3 .

Possible Answers:

$y=e^{15}x+e^{15}$

$y=5e^{15}x-14e^{15}$

$y=5e^{15}x-2e^{15}$

$y=5e^{15}x-3$

Correct answer:

$y=5e^{15}x-14e^{15}$

Explanation:

Find the line tangent to $y=e^{5x}$ at x=3 .

First, we find :

$y=e^{5x}=e^{5\cdot3}=e^{15}$

Next, we find the derivative:

$\frac{dy}{dx}=5e^{5x}$

Therefore, the slope at x=3 is:

$\frac{dy}{dx}=5e^{5x}=5e^{5\cdot3}=5e^{15}$ .

Using point-slope form, we can write the tangent line:

$y=5e^{15}(x-3)+e^{15}$

Simplifying this gives us:

$y=5e^{15}x-14e^{15}$

Report an Error

Example Question #9 : Slope

An isosceles triangle has one point at (6,2) , one point at (6,10) and one point on the -axis. What is the slope of the line between the point on the -axis and (6,10) ?

Possible Answers:

5/3

2/3

1/2

3/2

Correct answer:

2/3

Explanation:

The other point of the triangle must be at (0,6) as it must be equidistant from the other two points of the triangle. Since all points on the y-axis are units away from the other points in the direction, the third point must be equidistant in the direction from both (6,2) and (6,10) . The distance between these points is , so the third point must have a y-value of 2+(8/2)=6 . The third point is now at (0,6) so the slope of the line from (0,6) to (6,10) is as follows.

$\\m=\frac{y_2-y_1}{x_2-x_1}\\ \\=\frac{10-6}{6-0}\\ \\ =\frac{4}{6}=\frac{2}{3}$

Report an Error

Example Question #5 : Slope

What is the slope of the line tangent to the graph of $f(x)=\sin(2x)$ at $x=\pi/2$ ?

Possible Answers:

-1/2

1/2

Correct answer:

Explanation:

We must take the derivative of the function using the chain rule yielding $f'(x)=2\cos(2x)$ .

The chain rule is $[f(g(x))]'=f'(g(x))\cdot g'(x)$ .

Also remember that the derivative of sin(x) is cos(x) .

Applying these rules we get the following.

$\\f(x)=\sin(2x)\\ \\ u=2x\\ \\ \frac{du}{dx}=2\\ \\ \frac{d}{dx}\sin(u)=\cos(u)\frac{du}{dx}=2\cos(u)\\ \\ f'(x)=2\cos(2x)$

Plugging in the value for we get $2\cos(\pi)$ which is .

Report an Error

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.

View Calculus Tutors

Anjali
Certified Tutor

Delhi University, Bachelor of Science, Physics. Kumuan University, Master of Science, Physics.

All Calculus 1 Resources

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

Popular Subjects

MCAT Tutors in San Francisco-Bay Area, Computer Science Tutors in Atlanta, GRE Tutors in Seattle, GRE Tutors in Philadelphia, Physics Tutors in San Francisco-Bay Area, Math Tutors in Washington DC, Math Tutors in Denver, French Tutors in Boston, GMAT Tutors in Boston, SAT Tutors in San Francisco-Bay Area

Popular Courses & Classes

Spanish Courses & Classes in Miami, SSAT Courses & Classes in Denver, SSAT Courses & Classes in Houston, SSAT Courses & Classes in Phoenix, ACT Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in Boston, SAT Courses & Classes in Chicago, SAT Courses & Classes in Houston, ISEE Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Miami

Popular Test Prep

GRE Test Prep in Dallas Fort Worth, MCAT Test Prep in New York City, MCAT Test Prep in Phoenix, SAT Test Prep in New York City, GRE Test Prep in Denver, LSAT Test Prep in Chicago, GMAT Test Prep in Miami, ISEE Test Prep in San Francisco-Bay Area, ISEE Test Prep in Denver, GMAT Test Prep in Atlanta

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 : Slope

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #1 : Slope

Example Question #1 : How To Find Slope By Graphing Functions

Example Question #1 : How To Find Slope By Graphing Functions

Example Question #2 : How To Find Slope By Graphing Functions

Example Question #3 : How To Find Slope By Graphing Functions

Example Question #3 : Slope

Example Question #3 : Slope

Example Question #2 : Slope

Example Question #9 : Slope

Example Question #5 : Slope

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: