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Calculus 1 : Spatial Calculus

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Example Questions

Example Question #981 : Calculus

The velocity function of a particle and a position of this particle at a known time are given by v(t)=4sin^3(t) p(2)=2 . approximate p(5) using Euler's Method and three steps.

Possible Answers:

5.044

2.797

1.046

3.761

3.284

Correct answer:

3.284

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

$y_n=y_{n+1} +\Delta x f'(x_n,y_n)$

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told v(t)=4sin^3(t) p(2)=2

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{5-2}{3}=1$

p_0=2;t_0=2

p_1=2+(1)4sin^3(2)=5.007

p_2=5.007+(1)4sin^3(3)=5.018

p_3=5.018+(1)4sin^3(4)=3.284

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Example Question #128 : How To Find Position

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=4e^{t^2}$ p(0)=5 . approximate p(0.9) using Euler's Method and three steps.

Possible Answers:

14.008

7.513

6.200

32.916

9.233

Correct answer:

9.233

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

$y_n=y_{n+1} +\Delta x f'(x_n,y_n)$

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told $v(t)=4e^{t^2}$ p(0)=5

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{0.9-0}{3}=0.3$

p_0=5;t_0=0

$p_1=5+(0.3)4e^{0^2}=6.2$

$p_2=6.2+(0.3)4e^{0.3^2}=7.513$

$p_3=7.513+(0.3)4e^{0.6^2}=9.233$

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Example Question #129 : How To Find Position

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=sec^3({\pi t})$ p(1)=4 . approximate p(1.27) using Euler's Method and three steps.

Possible Answers:

4.395

4.111

3.808

3.658

3.910

Correct answer:

3.658

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

$y_n=y_{n+1} +\Delta x f'(x_n,y_n)$

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told $v(t)=sec^3({\pi t})$ p(1)=4

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{1.27-1}{3}=0.09$

p_0=4;t_0=1

$p_1=4+(0.09)sec^3({\pi (1)})=3.91$

$p_2=3.91+(0.09)sec^3({\pi (1.09)})=3.808$

$p_3=3.808+(0.09)sec^3({\pi (1.18)})=3.658$

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Example Question #981 : Spatial Calculus

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=\frac{4}{tan(t)}$ p(1)=1 . approximate p(7) using Euler's Method and three steps.

Possible Answers:

-27.818

-9.403

6.137

-49.985

-52.352

Correct answer:

-52.352

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

$y_n=y_{n+1} +\Delta x f'(x_n,y_n)$

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told $v(t)=\frac{4}{tan(t)}$ p(1)=1

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{7-1}{3}=2$

p_0=1;t_0=1

$p_1=1+(2)\frac{4}{tan(1)}=6.137$

$p_2=6.137+(2)\frac{4}{tan(3)}=-49.985$

$p_3=-49.985+(2)\frac{4}{tan(5)}=-52.352$

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Example Question #981 : Spatial Calculus

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=5\sqrt{sin(\pi(t))}$ p(0)=5 . approximate p(1.5) using Euler's Method and three steps.

Possible Answers:

2.5

6.5

Correct answer:

6.5

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

$y_n=y_{n+1} +\Delta x f'(x_n,y_n)$

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told $v(t)=5\sqrt{sin(\pi(t))}$ p(0)=5

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{1.5-0}{3}=0.5$

p_0=5;t_0=0

$p_1=4+(0.5)5\sqrt{sin(\pi(0))}=4$

$p_2=4+(0.5)5\sqrt{sin(\pi(0.5))}=6.5$

$p_3=6.5+(0.5)5\sqrt{sin(\pi(1))}=6.5$

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Example Question #132 : How To Find Position

The velocity function of a particle and a position of this particle at a known time are given by v(t)=tan(cos(t^3)) p(0)=4 . approximate p(1.5) using Euler's Method and three steps.

Possible Answers:

5.034

5.545

4.779

5.845

5.111

Correct answer:

5.845

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

$y_n=y_{n+1} +\Delta x f'(x_n,y_n)$

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told v(t)=tan(cos(t^3)) p(0)=4

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{1.5-0}{3}=0.5$

p_0=4;t_0=0

p_1=4+(0.5)tan(cos(0^3))=4.779

p_2=4.779+(0.5)tan(cos(0.5^3))=5.545

p_3=5.545+(0.5)tan(cos(1^3))=5.845

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Example Question #982 : Spatial Calculus

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=(-t)^{t}cos(\pi t)$ p(1)=-2 . Approximate p(4) using Euler's Method and three steps.

Possible Answers:

Correct answer:

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

$y_n=y_{n+1} +\Delta x f'(x_n,y_n)$

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told $v(t)=(-t)^{t}cos(\pi t)$ p(1)=-2

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{4-1}{3}=1$

p_0=-2;t_0=1

$p_1=-2+(1)(-1)^{1}cos(\pi (1))=-1$

$p_2=-1+(1)(-2)^{2}cos(\pi (2))=3$

$p_3=3+(1)(-3)^{3}cos(\pi (3))=30$

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Example Question #982 : Spatial Calculus

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=\frac{t^3}{2+cos(\pi t)}$ p(0)=3 . Approximate p(3) using Euler's Method and three steps.

Possible Answers:

$\frac{8}{3}$

$\frac{20}{3}$

$\frac{10}{3}$

Correct answer:

$\frac{20}{3}$

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

$y_n=y_{n+1} +\Delta x f'(x_n,y_n)$

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told $v(t)=\frac{t^3}{2+cos(\pi t)}$ p(0)=3

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{3-0}{3}=1$

p_0=3;t_0=0

$p_1=3+(1)\frac{0^3}{2+cos(\pi (0))}=3$

$p_2=3+(1)\frac{1^3}{2+cos(\pi (1))}=4$

$p_3=4+(1)\frac{2^3}{2+cos(\pi (2))}=\frac{20}{3}$

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Example Question #984 : Calculus

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=cos^3(\pi(t!))$ p(1)=5 . Approximate p(4) using Euler's Method and three steps.

Possible Answers:

Correct answer:

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

$y_n=y_{n+1} +\Delta x f'(x_n,y_n)$

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told $v(t)=cos^3(\pi(t!))$ p(1)=5

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{4-1}{3}=1$

p_0=5;t_0=1

$p_1=5+(1)cos^3(\pi(1!))=4$

$p_2=4+(1)cos^3(\pi(2!))=5$

$p_3=5+(1)cos^3(\pi(3!))=6$

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Example Question #985 : Calculus

The velocity function of a particle and a position of this particle at a known time are given by v(t)=(2t)! p(1)=0 . Approximate p(4) using Euler's Method and three steps.

Possible Answers:

746

720

733

772

759

Correct answer:

746

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

$y_n=y_{n+1} +\Delta x f'(x_n,y_n)$

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told v(t)=(2t)! p(1)=0

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{4-1}{3}=1$

p_0=0;t_0=1

p_1=0+(1)(2(1))!=2

p_2=2+(1)(2(2))!=26

p_3=26+(1)(2(3))!=746

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Calculus 1 : Spatial Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #981 : Calculus

Example Question #128 : How To Find Position

Example Question #129 : How To Find Position

Example Question #981 : Spatial Calculus

Example Question #981 : Spatial Calculus

Example Question #132 : How To Find Position

Example Question #982 : Spatial Calculus

Example Question #982 : Spatial Calculus

Example Question #984 : Calculus

Example Question #985 : Calculus

All Calculus 1 Resources

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