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Calculus 1 : Spatial Calculus

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #121 : Calculus

Find f'(x) at x=0 for

$f(x)=e^{2x^2+4x}$ .

Possible Answers:

Correct answer:

Explanation:

First we need to find f'(x) before evaluating it at x=0 .

$f(x)=e^{2x^2+4x}$

$f'(x)=(4x+4)e^{2x^2+4x}$ .

This was found by using the definition of the derivative for exponential functions.

The definition is,

$f(x)=e^{g(x)}$

$f'(x)=g'(x)e^{g(x)}$ .

Now we can evaluate f'(x) at x=0

$f'(0)=(4\cdot0+4)e^{2\cdot (0)^2+4\cdot 0}$

$f'(0)=4\cdot e^{0}=4$

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Example Question #121 : Spatial Calculus

At time t=0 , a diver jumps from a cliff that is feet above the water. The cliff diver's position is represented by the following: s(t)=-36t^2+36t+72 , where is measured in feet and is measured in seconds.

What is the velocity of the cliff diver when he/she hits the ground?

Possible Answers:

-36f/s

-72f/s

-108 f/s

-144 f/s

Correct answer:

-108 f/s

Explanation:

First, set s(t) to equal .

0=-36(t^2-t-2)=-36(t-2)(t+1) .

Then solve for :

t-2=0 and t+1=0 .

Therefore, t=-1, 2 . Since represents time in seconds, we can rule out t=-1 .

This means that it takes the cliff diver 2 seconds to reach the water. Next, take the derivative of s(t) by using the Power Rule $x^n=nx^{n-1}$ and plug in for : $\\s'(t)=-72t+36\\ s'(2)=-72(2)+36=-108$ .

The cliff diver's velocity was -108 f/s when he/she hit the water.

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Example Question #121 : Calculus

Mark throws a tennis ball into the air at t=0 . It's position is represented by s(t)=-4t^2+2t-8 , where is in seconds and is in meters.

What is the velocity of the ball at t=2 ?

Possible Answers:

-14m/s

16m/s

2m/s

18m/s

Correct answer:

-14m/s

Explanation:

First, take the first derivative of the equation by using the Power Rule $x^n=nx^{n-1}$ :

s'(t)=-8t+2 .

Then, plug in for t=2 :

$\\s'(2)=-8(2)+2\\ s'(2)=-16+2=-14$ .

Therefore, the velocity is -14m/s .

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Example Question #122 : Spatial Calculus

Sara goes to a skate park and enters a bowl. The bowl at the park is represented by s(t)=10t^2-16t-2 , where represents distance in feet and represents time in seconds.

What is Sara's velocity at t=3 ?

Possible Answers:

20ft/s

60ft/s

44ft/s

16ft/s

Correct answer:

44ft/s

Explanation:

Take the first derivative of the equation by using the Power Rule $x^n=nx^{n-1}$ :

s'(t)=2(10)t-16=20t-16 .

Then, plug in t=3 :

20(3)-16=44 .

Therefore, the velocity at t=3 is 44ft/s .

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Example Question #122 : Calculus

A cannonball is shot from a cannon. Its position is represented by s(t)=-5t^2-3t+15 , where represents distance in meters and represents time in seconds.

What is the velocity of the cannonball at t=1 ?

Possible Answers:

-31m/s

-3m/s

-13m/s

-10m/s

Correct answer:

-13m/s

Explanation:

Take the first derivative by using the Power Rule $x^n=nx^{n-1}$ of s(t) ,

s'(t)=-5(2)t-3=-10t-3 .

Then, plug in t=1 ,

s'(1)=-10(1)-3=-13 .

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Example Question #123 : Calculus

Leela throws a football across a field. It's position is represeted by s(t)=-6t^2+8t-12 , where represents distance in feet and represents time in secomds.

What is the velocity at t=4 ?

Possible Answers:

4ft/s

-40ft/s

-4ft/s

-12ft/s

Correct answer:

-40ft/s

Explanation:

Take the first derivative by using the Power Rule $x^n=nx^{n-1}$ of s(t) ,

s'(t)=-6(2)t+8=-12t+8 .

Then, plug in for t=4 ,

$\\s'(4)=v(4)\\v(4)=-12(4)+8=-48+8\\ v(4)=-40$ .

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Example Question #127 : Calculus

Lucas tosses an orange into the air. Its position is represented by s(t)=-2t^2+6t-4 , where represents distance in feet and represents time in seconds.

What is the velocity of the orange at t=3 ?

Possible Answers:

-18ft/s

-12ft/s

6ft/s

-6ft/s

Correct answer:

-6ft/s

Explanation:

Take the first derivative by using the Power Rule $x^n=nx^{n-1}$ of s(t) ,

s'(t)=-2(2)t+6=-4t+6 .

Then, plug in t=3 :

-4(3)+6=-12+6=-6 .

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Example Question #125 : Spatial Calculus

A ball is dropped into a bowl. The position of the ball is represented by s(t)=8t^2-6t+4 , where represents distance in inches and represents time in seconds.

What is the velocity of the ball at t=2 ?

Possible Answers:

32in/s

26in/s

-6in/s

16in/s

Correct answer:

26in/s

Explanation:

Take the first derivative by using the Power Rule $x^n=nx^{n-1}$ of s(t) ,

s'(t)=8(2)t-6=16t-6 .

Then, plug in for t=2 ,

$\\s'(2)=v(2)\\ v(2)=16(2)-6\\v(2)=32-6=26$ .

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Example Question #121 : How To Find Velocity

A given arrow has a position defined by the equation $p(t)=12t^{2}+5(t)-10$ . 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 a position

$p(t)=12t^{2}+5(t)-10$ , we can use the power rule

$\frac{d}{dt}t^{n}=nt^{n-1}$ where $n\neq0$ to determine that

v(t)=p'(t)=24t+5 .

Therefore, at t=2 ,

v(2)=p'(2)=24(2)+5=48+5=53 .

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Example Question #121 : How To Find Velocity

Aaron's car has a position defined by the equation $p(t)=15t^{2}+17t-9$ . 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 a position

$p(t)=15t^{2}+17t-9$ , we can use the power rule

$\frac{d}{dt}t^{n}=nt^{n-1}$ where $n\neq0$ to determine that

v(t)=p'(t)=30t+17 .

Therefore, at t=2 ,

v(2)=p'(2)=30(2)+17=60+17=77 .

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Calculus 1 : Spatial Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #121 : Calculus

Example Question #121 : Spatial Calculus

Example Question #121 : Calculus

Example Question #122 : Spatial Calculus

Example Question #122 : Calculus

Example Question #123 : Calculus

Example Question #127 : Calculus

Example Question #125 : Spatial Calculus

Example Question #121 : How To Find Velocity

Example Question #121 : How To Find Velocity

All Calculus 1 Resources

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