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Calculus 1 : How to find position

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Example Questions

Example Question #975 : Spatial Calculus

The velocity function of a particle and a position of this particle at a known time are given by $\displaystyle v(t)=sin(t^2)$ p(0)=4 $\displaystyle p(0)=4$. approximate p(0.6) $\displaystyle p(0.6)$ using Euler's Method and three steps.

Possible Answers:

4.032 $\displaystyle 4.032$

4.016 $\displaystyle 4.016$

4.008 $\displaystyle 4.008$

4.024 $\displaystyle 4.024$

4.040 $\displaystyle 4.040$

Correct answer:

4.040 $\displaystyle 4.040$

Explanation:

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

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

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

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

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

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

For this problem, we are told $\displaystyle v(t)=sin(t^2)$ p(0)=4 $\displaystyle p(0)=4$

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

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

$\displaystyle p_0=4;t_0=0$

$\displaystyle p_1=4+(0.2)sin(0^2)=4$

$\displaystyle p_2=4+(0.2)sin(0.2^2)=4.008$

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

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Example Question #976 : Spatial Calculus

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

Possible Answers:

5.849 $\displaystyle 5.849$

5.153 $\displaystyle 5.153$

5.666 $\displaystyle 5.666$

6.013 $\displaystyle 6.013$

5.408 $\displaystyle 5.408$

Correct answer:

5.408 $\displaystyle 5.408$

Explanation:

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

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

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

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

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

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

For this problem, we are told $\displaystyle v(t)=csc^2(t)$ p(2)=5 $\displaystyle p(2)=5$

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

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

$\displaystyle p_0=5;t_0=2$

$\displaystyle p_1=5+(0.1)csc^2(2)=5.121$

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

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

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Example Question #977 : Spatial Calculus

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

Possible Answers:

6.131 $\displaystyle 6.131$

5.933 $\displaystyle 5.933$

6.410 $\displaystyle 6.410$

6.912 $\displaystyle 6.912$

7.885 $\displaystyle 7.885$

Correct answer:

6.131 $\displaystyle 6.131$

Explanation:

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

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

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

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

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

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

For this problem, we are told $v(t)=e^{2^t}$ $\displaystyle v(t)=e^{2^t}$ $\displaystyle p(0.5)=3$

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

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

$\displaystyle p_0=3;t_0=0.5$

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

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

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

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Example Question #978 : Spatial Calculus

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

Possible Answers:

2.240 $\displaystyle 2.240$

46.169 $\displaystyle 46.169$

33.552 $\displaystyle 33.552$

0.378 $\displaystyle 0.378$

37.208 $\displaystyle 37.208$

Correct answer:

46.169 $\displaystyle 46.169$

Explanation:

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

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

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

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

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

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

For this problem, we are told $\displaystyle v(t)=tan^3(t^2)$ p(1)=0 $\displaystyle p(1)=0$

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

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

$\displaystyle p_0=0;t_0=1$

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

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

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

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Example Question #979 : Spatial Calculus

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

Possible Answers:

1.550 $\displaystyle 1.550$

2.540 $\displaystyle 2.540$

2.199 $\displaystyle 2.199$

1.765 $\displaystyle 1.765$

2.304 $\displaystyle 2.304$

Correct answer:

2.304 $\displaystyle 2.304$

Explanation:

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

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

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

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

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

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

For this problem, we are told $\displaystyle v(t)=cos(t^2+t+1)$ p(0)=2 $\displaystyle p(0)=2$

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

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

$\displaystyle p_0=2;t_0=0$

$\displaystyle p_1=2+(1)cos(0^2+0+1)=2.540$

$\displaystyle p_2=2.540+(1)cos(1^2+1+1)=1.550$

$\displaystyle p_3=1.550+(1)cos(2^2+2+1)=2.304$

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Example Question #980 : Spatial Calculus

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

Possible Answers:

27.608 $\displaystyle 27.608$

5.927 $\displaystyle 5.927$

1.196 $\displaystyle 1.196$

22.377 $\displaystyle 22.377$

15.332 $\displaystyle 15.332$

Correct answer:

22.377 $\displaystyle 22.377$

Explanation:

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

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

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

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

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

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

For this problem, we are told $v(t)=(tan(t))^{t}$ $\displaystyle v(t)=(tan(t))^{t}$ $\displaystyle p(0.7)=0$

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

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

$\displaystyle p_0=0;t_0=0.7$

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

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

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

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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 $\displaystyle v(t)=4sin^3(t)$ p(2)=2 $\displaystyle p(2)=2$. approximate p(5) $\displaystyle p(5)$ using Euler's Method and three steps.

Possible Answers:

2.797 $\displaystyle 2.797$

1.046 $\displaystyle 1.046$

3.761 $\displaystyle 3.761$

3.284 $\displaystyle 3.284$

5.044 $\displaystyle 5.044$

Correct answer:

3.284 $\displaystyle 3.284$

Explanation:

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

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

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

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

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

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

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

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

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

$\displaystyle p_0=2;t_0=2$

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

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

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

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Example Question #122 : How To Find Position

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

Possible Answers:

32.916 $\displaystyle 32.916$

9.233 $\displaystyle 9.233$

14.008 $\displaystyle 14.008$

7.513 $\displaystyle 7.513$

6.200 $\displaystyle 6.200$

Correct answer:

9.233 $\displaystyle 9.233$

Explanation:

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

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

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

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

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

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

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

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

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

$\displaystyle p_0=5;t_0=0$

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

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

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

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Example Question #123 : How To Find Position

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

Possible Answers:

3.658 $\displaystyle 3.658$

4.395 $\displaystyle 4.395$

3.910 $\displaystyle 3.910$

3.808 $\displaystyle 3.808$

4.111 $\displaystyle 4.111$

Correct answer:

3.658 $\displaystyle 3.658$

Explanation:

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

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

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

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

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

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

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

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

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

$\displaystyle p_0=4;t_0=1$

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

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

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

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Example Question #124 : How To Find Position

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

Possible Answers:

-52.352 $\displaystyle -52.352$

-9.403 $\displaystyle -9.403$

-49.985 $\displaystyle -49.985$

-27.818 $\displaystyle -27.818$

6.137 $\displaystyle 6.137$

Correct answer:

-52.352 $\displaystyle -52.352$

Explanation:

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

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

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

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

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

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

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

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

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

$\displaystyle p_0=1;t_0=1$

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

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

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

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Calculus 1 : How to find position

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #975 : Spatial Calculus

Example Question #976 : Spatial Calculus

Example Question #977 : Spatial Calculus

Example Question #978 : Spatial Calculus

Example Question #979 : Spatial Calculus

Example Question #980 : Spatial Calculus

Example Question #121 : How To Find Position

Example Question #122 : How To Find Position

Example Question #123 : How To Find Position

Example Question #124 : How To Find Position

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