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Calculus 1 : Spatial Calculus

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #281 : How To Find Velocity

The position at a certain point is given by:

$f(x)=\frac{3}{5}x^{5} - 6x + 5$

What is the velocity at x=2 ?

Possible Answers:

Correct answer:

Explanation:

In order to find the velocity of a given point, you must first find the derivative of the position function.

In this case, the derivative of the position function is:

$f'(x)=3x^{4}-6$

Then, find f'(x) when x=2 .

The answer is:

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Example Question #282 : Spatial Calculus

The position at a certain point is given by:

$f(x)= x^{x-3}$

What is the velocity at x=4 ?

Possible Answers:

3.55

5.55

7.55

6.55

4.55

Correct answer:

6.55

Explanation:

In order to find the velocity of a given point, you must first find the derivative of the position function.

In this case, the derivative of the position function is:

$f'(x)= x^{x-3}(ln(x)+\frac{x-3}{x})$

Then, find f'(x) when x=4 .

The answer is: 6.55

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Example Question #283 : Spatial Calculus

The position at a certain point is given by:

$f(x)= ln(x-4) +\frac{1}{}6x^{2}$

What is the velocity at x=7 ?

Possible Answers:

$\frac{7}{3}$

$\frac{9}{3}$

$\frac{4}{3}$

$\frac{2}{3}$

$\frac{8}{3}$

Correct answer:

$\frac{8}{3}$

Explanation:

In order to find the velocity of a given point, you must first find the derivative of the position function.

In this case, the derivative of the position function is:

$f'(x)= \frac{x}{3} + \frac{1}{x-4}$

Then, find f'(x) when x=7 .

The answer is: $\frac{8}{3}$

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Example Question #284 : Spatial Calculus

The position at a certain point is given by:

$f(x)= x^{9} - e^{x} + 54$

What is the velocity at x=2 ?

Possible Answers:

564.17

3,000.61

2,296.61

984.09

1,842.32

Correct answer:

2,296.61

Explanation:

In order to find the velocity of a given point, you must first find the derivative of the position function.

In this case, the derivative of the position function is:

f'(x)= 9x^8-e^x

Then, find f'(x) when x=2 .

The answer is: 2,296.61

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Example Question #285 : Spatial Calculus

The position at a certain point is given by:

f(x)= (2x^4-4x^3+5x^2-2x)^4

What is the velocity at x=1 ?

Possible Answers:

Correct answer:

Explanation:

In order to find the velocity of a given point, you must first find the derivative of the position function.

In this case, the derivative of the position function is:

f'(x)= 4(8x^3-12x^2+10x-2)(2x^4-4x^3+5x^2-2x)^3

Then, find f'(x) when x=1 .

The answer is:

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Example Question #281 : Calculus

The position at a certain point is given by:

$f(x)=(\sin x)^{2} +2x^{3}$

What is the velocity at $x=\pi$ ?

Possible Answers:

$6\pi^2$

$\pi ^{2}$

$\pi$

$-6\pi ^{2}$

Correct answer:

$6\pi^2$

Explanation:

In order to find the velocity of a given point, you must first find the derivative of the position function.

In this case, the derivative of the position function is:

$f'(x)=2 \cos x \sin x +6x^{2}$

Then, find f'(x) when $x=\pi$ .

The answer is: $6\pi ^{2}$

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Example Question #287 : Spatial Calculus

The position of a particle is given by the following function:

$f(x) = 4x^{3} - 5x^{2} + e^{x}$

What is the velocity of the particle at x = 2 ?

Possible Answers:

19.39

45.38

99.43

35.39

12.49

Correct answer:

35.39

Explanation:

In order to find the velocity of a particle at a certain point, we must first find the derivative.

To find the derivative we must use the power rule which states,

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

$f'(x) = 12x^{2} - 10x + e^{x}$

Then find the value of f'(x) when x=2 :

$f'(2) = 12(2)^{2} - 10(2) + e^{(2)}$

Therefore, the answer is: 35.39

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Example Question #281 : Calculus

The position of a particle is given by the following function:

$f(x) = 3x^{6} - x^{7} + 12$

What is the velocity of the particle at x = 1 ?

Possible Answers:

Correct answer:

Explanation:

In order to find the velocity of a particle at a certain point, we must first find the derivative.

To find the derivative we must use the power rule which states,

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

$f'(x) = 18x^{}5 - 7x^{}6$

Then find the value of f'(x) when x=1 :

$f'(1) = 18(1)^{}5 - 7(1)^{}6$

Therefore, the answer is:

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Example Question #281 : How To Find Velocity

The position of a particle is given by the following function:

$f(x) = \sin x - \cos x + 12$

What is the velocity of the particle at x = 0 ?

Possible Answers:

Undefined

Correct answer:

Explanation:

In order to find the velocity of a particle at a certain point, we must first find the derivative.

To find the derivative we must use the trigonometric rules of differentiation for cosine and sine which states,

$f(x)=cos(x) \rightarrow f'(x)=-sin(x)$

$f(x)=sin(x) \rightarrow f'(x)=cos(x)$

Thus the derivative becomes:

$f'(x) = \cos x + \sin x$

Then find the value of f'(x) when x=0 :

$f'(0) = \cos (0) + \sin (0)$

Therefore, the answer is:

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Example Question #281 : Calculus

The position of a particle is given by the following function:

$f(t) = (t+2)^{3} - 2t$

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

Possible Answers:

Correct answer:

Explanation:

In order to find the velocity of a particle at a certain point, we must first find the derivative.

For this particular function we will need to apply the chain rule and power rule which state,

$f(g(x))'=f'(g(x))\cdot g'(x)$

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

Therefore our derivative becomes,

$f'(t) = 3(t+2)^{2} - 2$ .

Then find the value of f'(t) when x=2 :

$f'(2) = 3({\color{Blue} 2}+2)^{2} - 2$

Therefore, the answer is:

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Calculus 1 : Spatial Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #281 : How To Find Velocity

Example Question #282 : Spatial Calculus

Example Question #283 : Spatial Calculus

Example Question #284 : Spatial Calculus

Example Question #285 : Spatial Calculus

Example Question #281 : Calculus

Example Question #287 : Spatial Calculus

Example Question #281 : Calculus

Example Question #281 : How To Find Velocity

Example Question #281 : Calculus

All Calculus 1 Resources

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