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

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

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

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

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:

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

For this problem, we are told \(\displaystyle v(t)=cos(\sqrt[5]{t})\) \(\displaystyle p(0)=0\)

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

\(\displaystyle \Delta t = \frac{9-0}{3}=3\)

\(\displaystyle p_0=0;t_0=0\)

\(\displaystyle p_1=0+(3)cos(\sqrt[5]{0})=3\)

\(\displaystyle p_2=3+(3)cos(\sqrt[5]{3})=3.958\)

\(\displaystyle p_3=3.958+(3)cos(\sqrt[5]{6})=4.376\)

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

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

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

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

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:

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

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

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

\(\displaystyle \Delta t = \frac{1.6-1}{3}=0.2\)

\(\displaystyle p_0=2;t_0=1\)

\(\displaystyle p_1=2+(0.2)csc^3(1^2)=2.336\)

\(\displaystyle p_2=2.336+(0.2)csc^3(1.2^2)=2.541\)

\(\displaystyle p_3=2.541+(0.2)csc^3(1.4^2)=2.794\)

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

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

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

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

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:

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

For this problem, we are told \(\displaystyle v(t)=sin(4t^4)\) \(\displaystyle p(1.1)=0\)

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

\(\displaystyle \Delta t = \frac{1.4-1.1}{3}=0.1\)

\(\displaystyle p_0=0;t_0=1.1\)

\(\displaystyle p_1=0+(0.1)sin(4(1.1)^4)=-0.041\)

\(\displaystyle p_2=-0.041+(0.1)sin(4(1.2)^4)=0.049\)

\(\displaystyle p_3=0.049+(0.1)sin(4(1.3)^4)=-0.042\)

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

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

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

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

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:

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

For this problem, we are told \(\displaystyle v(t)=5sin(t^3-t^2)\) \(\displaystyle p(1)=0\)

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

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

\(\displaystyle p_0=0;t_0=1\)

\(\displaystyle p_1=0+(2)(5sin(1^3-1^2))=0\)

\(\displaystyle p_2=0+(2)(5sin(2^3-2^2))=-7.568\)

\(\displaystyle p_3=-7.568+(2)(5sin(4^3-4^2))=-15.251\)

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

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

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

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

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:

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

For this problem, we are told \(\displaystyle v(t)=cos(t^4-t)\) \(\displaystyle p(0)=0\)

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

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

\(\displaystyle p_0=0;t_0=0\)

\(\displaystyle p_1=0+(1)cos(0^4-0)=1\)

\(\displaystyle p_2=1+(1)cos(1^4-1)=2\)

\(\displaystyle p_3=2+(1)cos(2^4-2)=2.137\)

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

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

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

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

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:

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

For this problem, we are told \(\displaystyle v(t)=1.2^{t^3-t^2}\) \(\displaystyle p(0)=4\)

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

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

\(\displaystyle p_0=4;t_0=0\)

\(\displaystyle p_1=4+(1)1.2^{0^3-0^2}=5\)

\(\displaystyle p_2=5+(1)1.2^{1^3-1^2}=6\)

\(\displaystyle p_3=6+(1)1.2^{2^3-2^2}=8.073\)

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

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

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

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

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:

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

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

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

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

\(\displaystyle p_0=0;t_0=0\)

\(\displaystyle p_1=0+(0.3)2(0)^3=0\)

\(\displaystyle p_2=0+(0.3)2(0.3)^3=0.0162\)

\(\displaystyle p_3=0.0162+(0.3)2(0.6)=0.1458\)

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

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

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

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

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:

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

For this problem, we are told \(\displaystyle v(t)=t!\) \(\displaystyle p(1)=8\)

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

\(\displaystyle \Delta t = \frac{4-1}{3}=1\)

\(\displaystyle p_0=8;t_0=1\)

\(\displaystyle p_1=8+(1)1!=9\)

\(\displaystyle p_2=9+(1)2!=11\)

\(\displaystyle p_3=11+(1)3!=17\)

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

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

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

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

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:

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

For this problem, we are told \(\displaystyle v(t)=cos(\pi t!)\) \(\displaystyle p(1)=0\)

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

\(\displaystyle \Delta t = \frac{10-1}{3}=3\)

\(\displaystyle p_0=0;t_0=1\)

\(\displaystyle p_1=0+(3)cos(\pi (1)!)=-3\)

\(\displaystyle p_2=-3+(3)cos(\pi (3)!)=0\)

\(\displaystyle p_3=0+(3)cos(\pi (6)!)=3\)

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

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

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

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

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:

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

For this problem, we are told \(\displaystyle v(t)=sin^2(\frac{\pi}{3}t!)\) \(\displaystyle p(1)=2\)

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

\(\displaystyle \Delta t = \frac{4-1}{3}=1\)

\(\displaystyle p_0=2;t_0=1\)

\(\displaystyle p_1=2+(1)sin^2(\frac{\pi}{3}(1)!)=\frac{11}{4}\)

\(\displaystyle p_2=\frac{11}{4}+(1)sin^2(\frac{\pi}{3}(2)!)=\frac{7}{2}\)

\(\displaystyle p_3=\frac{7}{2}+(1)sin^2(\frac{\pi}{3}(3)!)=\frac{7}{2}\)