Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Velocity

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 #111 : Velocity

$\\ f(x)=\frac{1}{x^{90}+ln(x)}$

Find f'(x)

Possible Answers:

${(x^{90}+ln(x))^2}$

$\frac{-90x^{89}+\frac{1}{x}}{x^{90}+ln(x)}$

$\frac{1}{(x^{90}+ln(x))^2}$

$\frac{-90x^{89}+\frac{1}{x}}{(x^{90}+ln(x))^2}$

$\frac{(x^{90}+ln(x))^2}{-90x^{89}+\frac{1}{x}}$

Correct answer:

$\frac{-90x^{89}+\frac{1}{x}}{(x^{90}+ln(x))^2}$

Explanation:

$\\ f(x)=\frac{1}{x^{90}+ln(x)}$

Rewrite f(x) as $f(x)=(x^{90}+ln(x))^{-1}$

Use the chain rule and power rule to obtain $\ f'(x)$ .

Remember that the chain rule is $[f(g(x))]'=f'(g(x))\cdot g'(x)$ and the power rule is,

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

Applying these rules we get,

$\\ \\ f'(x)=-1\cdot (x^{90}+ln(x))^{-2}\cdot \left(90x^{89}+\frac{1}{x}\right)$ .

This can be rewritten as

$\\ \\ f'(x)=\frac{-90x^{89}+\frac{1}{x}}{(x^{90}+ln(x))^2}$ .

Report an Error

Example Question #112 : Velocity

A ball is thrown upwards at a speed of 10m/s from a 100 m building. (Assuming gravity is 9.8m/s^2 ).

After one second, in which direction is the ball travelling?

Possible Answers:

The ball is not moving (at a minimum).

Down

The ball is moving horiztonally, not vertically.

The ball is not moving (at a maximum).

Correct answer:

Explanation:

The velocity function is v=-9.8t+10 , which is derived from the general formula for velocity,

$v(t)=a \cdot t+v_{0}$ .

Where the acceleration due to gravity and initial velocity are given by $a=-9.8, v_{0}=10$ .

Plugging in t=1 we arrive at a velocity of 0.2 m/s .

So after one second the ball is travelling upward.

Report an Error

Example Question #112 : Velocity

A ball is thrown upwards at a speed of 10 m/s from a 100m building. Assume gravity is 9.8m/s^2 .

What is the velocity of the ball after three seconds?

Possible Answers:

-19.6 m/s

-9.6 m/s

-13.2 m/s

-19.4 m/s

-4.4 m/s

Correct answer:

-19.4 m/s

Explanation:

We can integrate the constant acceleration of gravity by the general formula for integrating a constant, $\int a dt = at+c$ , where and are both constants.

This gives us a velocity function. Then plugging in our initial velocity, we arrive at an expression for the velocity of the ball over time: $v(t)=\int a(t)dt = \int -9.8 dt=-9.8t+v_0=-9.8t+10$ .

We then can plug in t=3 to get the velocity of the ball at seconds and arrive at our answer:

v(3)=-9.8(3)+10=-19.4 m/s .

Report an Error

Example Question #114 : Velocity

A given frisbee has a position defined by the equation $p(t)=3t^{2}+5t-2$ . What is its velocity at time t=2 ?

Possible Answers:

Correct answer:

Explanation:

By definition, velocity is the first derivative of position, or v(t)=p'(t) .

Given

$p(t)=3t^{2}+5t-2$ and the power rule

$\frac{d}{dt}t^{n}=nt^{n-1}$ where $n\neq0$ , then

v(t)=p'(t)=6t+5 .

Plugging in t=2 , we get

v(2)=6(2)+5=12+5=17 .

Report an Error

Example Question #115 : Velocity

A given horse has a position defined by the equation $p(t)=5t^{2}-7t+12$ . What is its velocity at time t=3 ?

Possible Answers:

Correct answer:

Explanation:

By definition, velocity is the first derivative of position, or v(t)=p'(t) .

Given

$p(t)=5t^{2}-7t+12$ and the power rule

$\frac{d}{dt}t^{n}=nt^{n-1}$ where $n\neq0$ , then

v(t)=p'(t)=10t-7 .

Plugging in t=3 , we get

v(3)=10(3)-7=30-7=23 .

Report an Error

Example Question #116 : Velocity

A given boat has a position defined by the equation $p(t)=10t^{2}-2t+19$ . What is its velocity at time t=4 ?

Possible Answers:

Correct answer:

Explanation:

By definition, velocity is the first derivative of position, or v(t)=p'(t) .

Given

$p(t)=10t^{2}-2t+19$ and the power rule

$\frac{d}{dt}t^{n}=nt^{n-1}$ where $n\neq0$ , then

v(t)=p'(t)=20t-2 .

Plugging in t=4 , we get

v(4)=20(4)-2=80-2=78 .

Report an Error

Example Question #111 : Spatial Calculus

Given and , find . Round to the nearest hundredth. a(t) and v(t) represent the acceleration and velocity functions, respectively.

Possible Answers:

153.41

136.91

148.41

159.91

Not enough information

Correct answer:

159.91

Explanation:

In order to solve for velocity at five seconds, we need to integrate a(t) to get v(t) .

Recall that,

$\int e^t=e^t$ . The power rule, $\int x^n=\frac{x^{n+1}}{n+1}$ , will be used to integrate the second term in the function.

Thus resulting in,

$\small v(t)=\int a(t)dt=e^t+t^2/2+c$ .

To find $\small c$ , we'll use our initial condition

$\small v(0)=0$ . $\small \small \small \small 0=e^{0}+0^2/2+c=1+c \therefore 0=1+c \rightarrow c=-1$ .

Now we can find $\small v(5)$ by simply plugging in $\small t=5$ and $\small c=-1$ into our $\small v(t)$ .

e^5+5^2/2-1=v(5) =e^5+23/2 .

In decimal form, this equals $\small 159.91$ . No units were mentioned in the problem, so leaving the answer unitless is acceptable.

Report an Error

Example Question #118 : Velocity

Given the equation of the position of a ball at time is S(t)=4.9t^2+6t-4 , find the velocity of the ball when t=4 .

Possible Answers:

46.2

44.2

45.2

None of the above.

Correct answer:

45.2

Explanation:

The velocity of an object is defined as the change in position of that object over the change in time. Since we are given the position equation in this problem, we simply take the derivative with respect to time of S(t) . To take the derivative of this equation, we must use the power rule,

$\frac{d}{dx}x^n=nx^{n-1}$ .

We also must remember that the derivative of an constant is 0. After taking the derivative of the function by applying the power rule, the funciton becomes

v(t)=9.8t+6 .

The questions states we must find the velocity of the ball when t=4 , therefore we simply plug in 4 for t and find that the velocity of the ball at that time is 45.2 .

Report an Error

Example Question #113 : Velocity

Boulder question

A boulder rolls off the side of a seaside cliff that is above the water. The boulder is traveling at directly to the right when it leaves the cliff.

What is the vertical velocity of the boulder when it is 200m above the water? Assume that gravity is the only force acting on the boulder, causing a downward acceleration of .

Possible Answers:

-117.1m/s

-70.0m/s

-62.6m/s

-132.8m/s

-99.0m/s

Correct answer:

-99.0m/s

Explanation:

The vertical acceleration is given by a(t)=-9.8 .

Initial vertical velocity is .

To integrate we do,

$\int x^n=\frac{x^{n+1}}{n+1}$ .

Integrating gives v(t)=-9.8t given our initial vertical velocity.

Integrating again gives a vertical position of y(t)= -4.9t^2+C .

Since we know the boulder starts at a height of 700 m , we determine that C=700 .

Solving 200=-4.9t^2+700 gives t=10.10

Evaluate the vertical velocity function when t=10.10

v(10.10)=-9.8(10.10)=-99.0m/s

Report an Error

Example Question #120 : Velocity

Let $f(t) = 4sin(\pi t)$ represent the position of a swinging marble.

Find the marble's velocity when t=3 .

Possible Answers:

$-4\pi$

$12\pi$

$4\pi$

Correct answer:

$-4\pi$

Explanation:

First, find the function representing velocity by differentiating the position function.

$f(t) = 4sin(\pi t)$

Recall the derivative of sin(u) is $cos(u)\frac{du}{dx}$ .

Applying this rule we get,

$f'(t)=4\pi cos(\pi t)$ .

Then, evaluate f'(t) at t=3 to get the answer.

$f'(3)=4\pi cos(3\pi)$

$=4\pi (-1)$

$=-4\pi$

Report an Error

View Calculus Tutors

Jose
Certified Tutor

State University of New York at Albany, Bachelor of Science, Applied Mathematics. Teachers College at Columbia University, Ma...

View Calculus Tutors

Abhisek
Certified Tutor

University of Florida, Bachelor of Engineering, Computer Science.

View Calculus Tutors

Anabel
Certified Tutor

UDELAR, Diploma, Chemistry. IPA, Bachelor of Education, Mathematics.

All Calculus 1 Resources

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

Popular Subjects

ACT Tutors in Washington DC, SAT Tutors in Boston, Reading Tutors in Houston, Math Tutors in Washington DC, MCAT Tutors in Seattle, GMAT Tutors in Seattle, English Tutors in Dallas Fort Worth, ISEE Tutors in Philadelphia, Math Tutors in New York City, Biology Tutors in Houston

Popular Courses & Classes

GMAT Courses & Classes in Washington DC, Spanish Courses & Classes in Chicago, Spanish Courses & Classes in San Diego, MCAT Courses & Classes in Miami, ISEE Courses & Classes in Phoenix, ACT Courses & Classes in San Francisco-Bay Area, LSAT Courses & Classes in Chicago, ACT Courses & Classes in Philadelphia, SAT Courses & Classes in Atlanta, SSAT Courses & Classes in San Diego

Popular Test Prep

ACT Test Prep in Boston, ISEE Test Prep in New York City, GMAT Test Prep in San Francisco-Bay Area, GMAT Test Prep in Philadelphia, ISEE Test Prep in Atlanta, ACT Test Prep in Denver, SAT Test Prep in San Francisco-Bay Area, ACT Test Prep in San Francisco-Bay Area, LSAT Test Prep in Miami, LSAT 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 1 : Velocity

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #111 : Velocity

Example Question #112 : Velocity

Example Question #112 : Velocity

Example Question #114 : Velocity

Example Question #115 : Velocity

Example Question #116 : Velocity

Example Question #111 : Spatial Calculus

Example Question #118 : Velocity

Example Question #113 : Velocity

Example Question #120 : Velocity

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: