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Calculus 1 : Calculus

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Example Questions

Example Question #91 : How To Find Velocity

What is the average velocity of an object from t=2 to t=7 if the position of the function is described by y=2t+1 ?

Possible Answers:

Correct answer:

Explanation:

Write the average velocity equation given the initial and final velocities.

$v_f=\frac{y(t_f)-y(t_i)}{t_f-t_i}$

Solve for y(t_f) .

y(t_f=7)= 2(7)+1=15

Solve for y(t_i) .

y(t_i=2)= 2(2)+1=5

Substitute the knowns into the average velocity equation.

$v_f=\frac{y(t_f)-y(t_i)}{t_f-t_i}=\frac{15-5}{7-2}=\frac{10}{5}=2$

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Example Question #91 : Calculus

Find the instantaneous velocity at t=1 if the position function is described by s(t)= 5ln(t)+4t .

Possible Answers:

Correct answer:

Explanation:

Do not confuse average velocity with instantaneous velocity. To determine the instantaneous velocity, take the derivative of the position function to obtain the velocity function.

s(t)= 5ln(t)+4t

$v(t)= \frac{5}{t}+4$

Substitute t=1 .

$v(t)= \frac{5}{1}+4=9$

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Example Question #92 : How To Find Velocity

For this question, keep in mind that velocity is defined as $\frac{position}{time}$ .

If a particle's position is given by the equation y = 2x, what is the particle's velocity at x = 4 ?

Possible Answers:

x^2

Correct answer:

Explanation:

Velocity can be viewed at the derivative of position, i.e. the rate of change of a position function.

So, we can find velocity by finding the derivative of the position function.

The position function is given as y = 2x , so the derivative of the position function, using the power rule is,

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

$y=2x\rightarrow y'=1\cdot 2x^{1-1}=2x^0=2$ , a constant, and the velocity function is thus always .

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Example Question #92 : Calculus

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

Possible Answers:

Correct answer:

Explanation:

By definition, velocity is the first derivative of a position function.

Therefore, v(t)=p'(t) .

For this particular problem we will use the power rule to find the derivative. The power rule states,

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

Since

$p(t)=7t^{2}-2t+5$ , then applying the power rule we find

v(t)=p'(t)=14t-2 .

Plugging in t=3 gets us,

v(t)=p'(t)=14(3)-2=42-2=40 .

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Example Question #91 : Calculus

A tortoise's position is defined by the equation $p(t)=4t^{2}+9t+13$ . What is the velocity of the tortoise at t=4 ?

Possible Answers:

Correct answer:

Explanation:

By definition, velocity is the first derivative of a position function.

Therefore, v(t)=p'(t) .

For this particular problem we will use the power rule to find the derivative. The power rule states,

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

Since $p(t)=4t^{2}+9t+13$ , then applying the power rule we find v(t)=p'(t)=8t+9 .

Plugging in t=4 gets us v(t)=p'(t)=8(4)+9=32+9=41 .

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Example Question #93 : Calculus

A speedboat's position is defined by the equation $p(t)=8t^{2}-6t+5$ . What is the velocity of the speedboat at t=7 ?

Possible Answers:

107

106

103

105

104

Correct answer:

106

Explanation:

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

For this particular problem we will use the power rule to find the derivative.

The power rule states,

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

Since $p(t)=8t^{2}-6t+5$ , then applying the power rule we find v(t)=p'(t)=16t-6 .

Plugging in t=7 gets us v(t)=p'(t)=16(7)-6=112-6=106 .

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Example Question #94 : Calculus

What is the velocity of an object with a position function $p(t)=4t^{2}+9t-7$ at t=7 ?

Possible Answers:

Correct answer:

Explanation:

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

Given,

$p(t)=4t^{2}+9t-7$ we can use the power rule which states,

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

Applying this rule we can deduce that,

v(t)=p'(t)=8t+9 .

Swapping in t=7 , we get

v(7)=8(7)+9=56+9=65 .

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Example Question #95 : Calculus

What is the velocity of a car with a position function $p(t)=6t^{2}-7t+12$ at t=4 ?

Possible Answers:

Correct answer:

Explanation:

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

Given,

$p(t)=6t^{2}-7t+12$ ,

we can use the power rule which states,

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

Applying this rule we can deduce that,

v(t)=p'(t)=12t-7 .

Swapping in t=4 , we get

v(4)=12(4)-7=48-7=41 .

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Example Question #96 : Calculus

What is the velocity of a speedboat with a position function $p(t)=9t^{2}+14t+15$ at t=5 ?

Possible Answers:

104

105

Correct answer:

104

Explanation:

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

Given,

$p(t)=9t^{2}+14t+15$ ,

we can use the power rule which states,

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

Applying the power rule we can deduce that,

v(t)=p'(t)=18t+14 .

Swapping in t=5 , we get

v(5)=18(5)+14=90+14=104 .

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Example Question #97 : Calculus

Given a particle's position as a function of time, determine its velocity at a time of 4 seconds.

$p(t) = 10 m + 25 (m/s^2 ) \cdot t^2$

Possible Answers:

25 m/s

100 m/s

410 m/s

200 m/s

Correct answer:

200 m/s

Explanation:

Since velocity is a rate change of position with respect to time, we need to take the derivative of the position function with respect to time:

$v(t) = x'(t) = 2 \cdot (25 m/s^2 ) \cdot t$

t = 4 s

$v(4 s) = (50 m/s^2) \cdot 4 s = 200 m/s$

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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #91 : How To Find Velocity

Example Question #91 : Calculus

Example Question #92 : How To Find Velocity

Example Question #92 : Calculus

Example Question #91 : Calculus

Example Question #93 : Calculus

Example Question #94 : Calculus

Example Question #95 : Calculus

Example Question #96 : Calculus

Example Question #97 : Calculus

All Calculus 1 Resources

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