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Calculus 1 : Spatial Calculus

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Example Questions

Example Question #991 : 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:

1.117

1.183

1.044

1.212

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 : 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 : 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

73.444

13.908

2.773

1.622

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 : 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:

-8.348

-10.444

-7.692

-11.666

-9.222

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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Example Question #995 : Calculus

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=(\frac{t}{2}-1)!$ p(2)=4 . Approximate p(14) using Euler's Method and three steps.

Possible Answers:

2988

112

Correct answer:

112

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 $v(t)=(\frac{t}{2}-1)!$ p(2)=4

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

$\Delta t = \frac{14-2}{3}=4$

p_0=4;t_0=2

$p_1=4+(4)(\frac{2}{2}-1)!=8$

$p_2=8+(4)(\frac{6}{2}-1)!=16$

$p_3=16+(4)(\frac{10}{2}-1)!=112$

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Example Question #996 : Calculus

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=\sqrt[3]{ln(x^{10})}$ p(1)=1 . Approximate p(10) using Euler's Method and three steps.

Possible Answers:

174.146

1014.633

283.515

73.034

599.008

Correct answer:

174.146

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 $v(t)=\sqrt[3]{ln(x^{10})}$ p(1)=1

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

$\Delta t = \frac{10-1}{3}=3$

p_0=1;t_0=1

$p_1=1+(3)\sqrt[3]{ln(1^{10})}=1$

$p_2=1+(3)\sqrt[3]{ln(4^{10})}=73.034$

$p_3=73.034+(3)\sqrt[3]{ln(7^{10})}=174.146$

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Example Question #997 : Calculus

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

Possible Answers:

57.064

8.457

10.115

27.891

42.587

Correct answer:

57.064

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 $v(t)=tan(\sqrt{t!})$ p(1)=7

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=7;t_0=1

$p_1=7+(2)tan(\sqrt{1!})=10.115$

$p_2=10.115+(2)tan(\sqrt{3!})=8.457$

$p_3=8.457+(2)tan(\sqrt{5!})=57.064$

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Example Question #998 : Calculus

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=4^{t!}$ p(0)=0 . Approximate p(3) using Euler's Method and three steps.

Possible Answers:

4116

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 $v(t)=4^{t!}$ p(0)=0

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=0;t_0=0

$p_1=0+(1)4^{0!}=4$

$p_2=4+(1)4^{1!}=8$

$p_3=8+(1)4^{2!}=24$

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Example Question #145 : Position

The velocity function of a particle and a position of this particle at a known time are given by v(t)=t^5+5 p(0)=1 . Approximate p(0.6) using Euler's Method and three steps.

Possible Answers:

4.00315

4.00028

4.00211

4.00199

4.00441

Correct answer:

4.00211

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^5+5 p(0)=1

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

$\Delta t = \frac{0.6-0}{3}=0.2$

p_0=1;t_0=0

p_1=1+(0.2)(0^5+5)=2

p_2=2+(0.2)(0.2^5+5)=3.00006

p_3=3.00006+(0.2)(0.4^5+5)=4.00211

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Example Question #146 : Position

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=(sin(t^2))^{\frac{1}{8}}$ p(0)=2 . Approximate p(2) using Euler's Method and two steps.

Possible Answers:

3.133

2.572

2.979

2.708

Correct answer:

2.979

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 $v(t)=(sin(t^2))^{\frac{1}{8}}$ p(0)=2

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

$\Delta t = \frac{2-0}{2}=1$

p_0=2;t_0=0

$p_1=2+(1)(sin(0^2))^{\frac{1}{8}}=2$

$p_2=2+(1)(sin(1^2))^{\frac{1}{8}}=2.979$

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Calculus 1 : Spatial Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #991 : Calculus

Example Question #992 : Calculus

Example Question #993 : Calculus

Example Question #994 : Calculus

Example Question #995 : Calculus

Example Question #996 : Calculus

Example Question #997 : Calculus

Example Question #998 : Calculus

Example Question #145 : Position

Example Question #146 : Position

All Calculus 1 Resources

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