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Calculus 1 : Position

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Example Questions

Example Question #131 : Position

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:

6.5

2.5

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 : 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:

5.545

5.034

5.111

4.779

5.845

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 #133 : Position

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 #134 : Position

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 #135 : Position

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 #136 : Position

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:

720

733

772

759

746

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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Example Question #991 : Spatial Calculus

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=sin(\frac{\pi}{tan(t^3)})$ p(0.7)=1 . Approximate p(1.3) using Euler's Method and three steps.

Possible Answers:

1.212

1.117

1.183

1.044

Correct answer:

1.183

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 $v(t)=sin(\frac{\pi}{tan(t^3)})$ p(0.7)=1

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

$\Delta t = \frac{1.3-0.7}{3}=0.2$

p_0=1;t_0=0.7

$p_1=1+(0.2)sin(\frac{\pi}{tan(0.7^3)})=1.117$

$p_2=1.117+(0.2)sin(\frac{\pi}{tan(0.9^3)})=1.044$

$p_3=1.044+(0.2)sin(\frac{\pi}{tan(1.1^3)})=1.183$

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Example Question #992 : Spatial Calculus

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=3cos^2(\pi(t^2+1))$ p(0)=4 . Approximate p(6) 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 $v(t)=3cos^2(\pi(t^2+1))$ p(0)=4

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

$\Delta t = \frac{6-0}{3}=2$

p_0=4;t_0=0

$p_1=4+(2)3cos^2(\pi(0^2+1))=10$

$p_2=10+(2)3cos^2(\pi(2^2+1))=16$

$p_1=16+(2)3cos^2(\pi(4^2+1))=22$

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Example Question #993 : Spatial Calculus

The velocity function of a particle and a position of this particle at a known time are given by v(t)=ln(t^8) p(1)=0 . Approximate p(2.5) using Euler's Method and three steps.

Possible Answers:

4.395

1.622

2.773

13.908

73.444

Correct answer:

4.395

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)=ln(t^8) p(1)=0

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

$\Delta t = \frac{2.5-1}{3}=0.5$

p_0=0;t_0=1

p_1=0+(0.5)ln(1^8)=0

p_2=0+(0.5)ln(1.5^8)=1.622

p_3=1.622+(0.5)ln(2^8)=4.395

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Example Question #994 : Spatial Calculus

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=tan(\pi e^{t!})$ p(0)=-8 . Approximate p(3) using Euler's Method and three steps.

Possible Answers:

-9.222

-8.348

-10.444

-11.666

-7.692

Correct answer:

-7.692

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 $v(t)=tan(\pi e^{t!})$ p(0)=-8

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=-8;t_0=0

$p_1=-8+(1)tan(\pi e^{0!})=-9.222$

$p_2=-9.222+(1)tan(\pi e^{1!})=-10.444$

$p_3=-10.444+(1)tan(\pi e^{2!})=-7.692$

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Calculus 1 : Position

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #131 : Position

Example Question #132 : Position

Example Question #133 : Position

Example Question #134 : Position

Example Question #135 : Position

Example Question #136 : Position

Example Question #991 : Spatial Calculus

Example Question #992 : Spatial Calculus

Example Question #993 : Spatial Calculus

Example Question #994 : Spatial Calculus

All Calculus 1 Resources

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