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Calculus 1 : Position

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #121 : 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)=sin(t^2) p(0)=4 . approximate p(0.6) using Euler's Method and three steps.

Possible Answers:

4.024

4.032

4.016

4.040

4.008

Correct answer:

4.040

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) p(0)=4

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=4;t_0=0

p_1=4+(0.2)sin(0^2)=4

p_2=4+(0.2)sin(0.2^2)=4.008

p_3=4.008+(0.2)sin(0.4^2)=4.040

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Example Question #971 : Spatial Calculus

The velocity function of a particle and a position of this particle at a known time are given by v(t)=csc^2(t) p(2)=5 . approximate p(2.3) using Euler's Method and three steps.

Possible Answers:

5.153

6.013

5.408

5.849

5.666

Correct answer:

5.408

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)=csc^2(t) p(2)=5

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

$\Delta t = \frac{2.3-2}{3}=0.1$

p_0=5;t_0=2

p_1=5+(0.1)csc^2(2)=5.121

p_2=5.121+(0.1)csc^2(2.1)=5.255

p_3=5.255+(0.1)csc^2(2.2)=5.408

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Example Question #971 : Calculus

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=e^{2^t}$ p(0.5)=3 . approximate p(1.1) using Euler's Method and three steps.

Possible Answers:

6.912

6.410

7.885

6.131

5.933

Correct answer:

6.131

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 $v(t)=e^{2^t}$ p(0.5)=3

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

$\Delta t = \frac{1.1-0.5}{3}=0.2$

p_0=3;t_0=0.5

$p_1=3+(0.2)e^{2^{0.5}}=3.823$

$p_2=3.823+(0.2)e^{2^{0.7}}=4.838$

$p_3=4.838+(0.2)e^{2^{0.9}}=6.131$

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Example Question #972 : Spatial Calculus

The velocity function of a particle and a position of this particle at a known time are given by v(t)=tan^3(t^2) p(1)=0 . approximate p(1.3) using Euler's Method and three steps.

Possible Answers:

46.169

0.378

33.552

2.240

37.208

Correct answer:

46.169

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^3(t^2) p(1)=0

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

$\Delta t = \frac{1.3-1}{3}=0.1$

p_0=0;t_0=1

p_1=0+(0.1)tan^3(1^2)=0.378

p_2=0.378+(0.1)tan^3(1.1^2)=2.240

p_3=2.240+(0.1)tan^3(1.2^2)=46.169

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Example Question #125 : 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)=cos(t^2+t+1) p(0)=2 . approximate p(3) using Euler's Method and three steps.

Possible Answers:

2.540

1.550

1.765

2.199

2.304

Correct answer:

2.304

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(t^2+t+1) p(0)=2

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=2;t_0=0

p_1=2+(1)cos(0^2+0+1)=2.540

p_2=2.540+(1)cos(1^2+1+1)=1.550

p_3=1.550+(1)cos(2^2+2+1)=2.304

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Example Question #973 : Spatial Calculus

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=(tan(t))^{t}$ p(0.7)=0 . approximate p(1.9) using Euler's Method and three steps.

Possible Answers:

22.377

27.608

1.196

5.927

15.332

Correct answer:

22.377

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 $v(t)=(tan(t))^{t}$ p(0.7)=0

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

$\Delta t = \frac{1.9-0.7}{3}=0.4$

p_0=0;t_0=0.7

$p_1=0+(0.4)(tan(0.7))^{0.7}=0.355$

$p_2=0.355+(0.4)(tan(1.1))^{1.1}=1.196$

$p_3=1.196+(0.4)(tan(1.5))^{1.5}=22.377$

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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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Calculus 1 : Position

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #121 : How To Find Position

Example Question #971 : Spatial Calculus

Example Question #971 : Calculus

Example Question #972 : Spatial Calculus

Example Question #125 : How To Find Position

Example Question #973 : Spatial Calculus

Example Question #981 : Calculus

Example Question #128 : How To Find Position

Example Question #129 : How To Find Position

Example Question #981 : Spatial Calculus

All Calculus 1 Resources

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