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Calculus 1 : Velocity

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Example Questions

Example Question #101 : Velocity

The position of a ball attached to the end of a spring is given by the following function:

$y(t) = e^{-2t}sin(4t)$

At what time will the velocity of the ball first be exactly zero?

Possible Answers:

0.277 seconds

0. 116 seconds

The velocity will never be zero.

6.642 seconds

Correct answer:

0.277 seconds

Explanation:

Since we are given a position function, our first step will be to find the velocity function. This is found by taking the derivative of position with respect to time, using the product rule for derivatives:

f(t)g(t) = f'(t)g(t) + f(t)g'(t)

where in this case

$h(t) = e^{-2t}$ and $h'(t) = -2e^{-2t}$

g(t) = sin(4t) and g'(t) = 4cos(4t)

yielding the following derivative, our velocity function:

$v(t)=4e^{-2t}cos(4t)-2e^{-2t}sin(4t)$

To find when velocity is equal to 0, we can set the right side of the equation of zero, and proceed to solve for our time, t

$0=4e^{-2t}cos(4t)-2e^{-2t}sin(4t)$

$4e^{-2t}cos(4t) = 2e^{-2t}sin(4t)$

4cos(4t) = 2sin(4t)

$2 = \frac{sin(4t)}{cos(4t)}=tan(4t)$

From here, we can take the arctangent of both sides of the equation and solve for t:

$t = \frac{arctan(2)}{4} = 0.277 seconds$

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Example Question #102 : Velocity

If the position function of an object is y=cot(t) , what is the velocity when $t=\frac{\pi}{2}$ ?

Possible Answers:

$\frac{1}{4}$

$\textup{The answer is undefined.}$

Correct answer:

Explanation:

To obtain the velocity function from the position function, take the derivative.

The derivative of y=cot(t) is:

y'=-csc^2(t)

Substitute $t=\frac{\pi}{2}$ and solve for the velocity.

$y'=-csc^2\left(\frac{\pi}{2}\right)= -\frac{1}{sin(\frac{\pi}{2})} \cdot \frac{1}{sin(\frac{\pi}{2})}=-1$

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Example Question #103 : Velocity

The position of a bicycle is defined by the equation $p(t)=3t^{2}+7t-4$ . What is its velocity at t=5 ?

Possible Answers:

Correct answer:

Explanation:

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

Thus, given

$p(t)=3t^{2}+7t-4$ and using the power rule

$\frac{d}{dt}t^{n}=nt^{n-1}$

where $n\neq0$ ,

v(t)=p'(t)=6t+7 .

Swapping in t=5 ,

v(t)=p'(t)=6(5)+7=30+7=37 .

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Example Question #104 : Velocity

The position of an atom is defined by the equation $p(t)=17t^{2}+9t+86$ . What is its velocity at t=7 ?

Possible Answers:

246

247

245

243

244

Correct answer:

247

Explanation:

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

Thus, given

$p(t)=17t^{2}+9t+86$ and using the power rule

$\frac{d}{dt}t^{n}=nt^{n-1}$

where $n\neq0$ ,

v(t)=p'(t)=34t+9 .

Swapping in t=7 ,

v(t)=p'(t)=34(7)+9=238+9=247 .

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Example Question #105 : Velocity

The position of a bird is defined by the equation $p(t)=3t^{2}+4t+12$ . What is its velocity at t=2 ?

Possible Answers:

Correct answer:

Explanation:

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

Thus, given

$p(t)=3t^{2}+4t+12$ and using the power rule

$\frac{d}{dt}t^{n}=nt^{n-1}$

where $n\neq0$ ,

v(t)=p'(t)=6t+4 .

Swapping in t=2 ,

v(t)=p'(t)=6(2)+4=12+4=16 .

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Example Question #106 : Velocity

A given object has a position that is defined by the equation $p(t)=5t^{2}-4t+3$ . What is its velocity at t=2 ?

Possible Answers:

Correct answer:

Explanation:

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

Given,

$p(t)=5t^{2}-4t+3$ , then using the power rule which states

$f(x)=x^n \rightarrow f'(x)=nx^{n-1}$ we find v(t)=p'(t)=10t-4 .

Plugging in t=2 ,

v(2)=p'(2)=10(2)-4=20-4=16 .

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Example Question #107 : Velocity

A baseball has a position that is defined by the equation $p(t)=9t^{2}+7t-10$ . What is its velocity at t=3 ?

Possible Answers:

Correct answer:

Explanation:

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

Given,

$p(t)=9t^{2}+7t-10$ , we can apply the power rule which states,

$f(x)=x^n \rightarrow f'(x)=nx^{n-1}$ to find v(t)=p'(t)=18t+7 .

Plugging in t=3 ,

v(3)=p'(3)=18(3)+7=54+7=61 .

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Example Question #108 : Velocity

A bird has a position that is defined by the equation $p(t)=6t^{2}-3t+12$ . What is its velocity at t=5 ?

Possible Answers:

Correct answer:

Explanation:

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

Given,

$p(t)=6t^{2}-3t+12$ , then we can use the power rule which states,

$f(x)=x^n \rightarrow f'(x)=nx^{n-1}$ to find v(t)=p'(t)=12t-3 .

Plugging in t=5 ,

v(5)=p'(5)=12(5)-3=60-3=57 .

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Example Question #109 : Velocity

f(x)=ln(x^2+x+1)

Find f'(x) .

Possible Answers:

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

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

$\frac{2x+1}{x^2+x+1}$

$\frac{x^2+x+1}{2x+1}$

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

Correct answer:

$\frac{2x+1}{x^2+x+1}$

Explanation:

To find f'(x) , we need to use the chain rule and power rule.

Remember the chain rule is,

$[f(g(x))]'=f'(g(x))\cdot g'(x)$ .

The power rule is,

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

Also, recall that the natural log is a function who has a special derivative.

$f(x)=ln(x)\rightarrow f'(x)=\frac{1}{x}\cdot dx$

Applying these rules we get the following solution.

f(x)=ln(x^2+x+1)

$\\ f'(x)=\frac{1}{(x^2+x+1)}\cdot(2x+1)\\ \\=\frac{2x+1}{x^2+x+1}$

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Example Question #110 : Velocity

$\\ f(t)=e^{t^2-2t+1}$

What is f'(t) at t=1 ?

Possible Answers:

{}-1

{}0

{}2

{}1

{}3

Correct answer:

{}0

Explanation:

We get this by using the definition of the derivative of an exponential function. Remember that the derivative of an expotential function is:

$[e^{f(x)}]'=f'(x)\cdot e^{f(x)}$

Applying this rule to the function we are able to find the first derivative.

$\\ f(t)=e^{t^2-2t+1} \\ f'(t)=(2t-2)\cdot e^{t^2-2t+1} \\$

Now we simply plug in 1.

$\\ f'(1)=(2\cdot 1-2)\cdot e^{1^2-2\cdot 1+1}=0\cdot e^0=0\cdot 1=0$

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Calculus 1 : Velocity

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #101 : Velocity

Example Question #102 : Velocity

Example Question #103 : Velocity

Example Question #104 : Velocity

Example Question #105 : Velocity

Example Question #106 : Velocity

Example Question #107 : Velocity

Example Question #108 : Velocity

Example Question #109 : Velocity

Example Question #110 : Velocity

All Calculus 1 Resources

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