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 \(\displaystyle v(t)=4sin^3(t)\) \(\displaystyle p(2)=2\)

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

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

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

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

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

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

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 \(\displaystyle v(t)=4e^{t^2}\) \(\displaystyle p(0)=5\)

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

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

\(\displaystyle p_0=5;t_0=0\)

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

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

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

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 \(\displaystyle v(t)=sec^3({\pi t})\) \(\displaystyle p(1)=4\)

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

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

\(\displaystyle p_0=4;t_0=1\)

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

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

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

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 \(\displaystyle v(t)=\frac{4}{tan(t)}\) \(\displaystyle p(1)=1\)

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

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

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

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

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

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

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 \(\displaystyle v(t)=5\sqrt{sin(\pi(t))}\)   \(\displaystyle p(0)=5\)

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

\(\displaystyle \Delta t = \frac{1.5-0}{3}=0.5\)

\(\displaystyle p_0=5;t_0=0\)

\(\displaystyle p_1=4+(0.5)5\sqrt{sin(\pi(0))}=4\)

\(\displaystyle p_2=4+(0.5)5\sqrt{sin(\pi(0.5))}=6.5\)

\(\displaystyle p_3=6.5+(0.5)5\sqrt{sin(\pi(1))}=6.5\)

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 \(\displaystyle v(t)=tan(cos(t^3))\) \(\displaystyle p(0)=4\)

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

\(\displaystyle \Delta t = \frac{1.5-0}{3}=0.5\)

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

\(\displaystyle p_1=4+(0.5)tan(cos(0^3))=4.779\)

\(\displaystyle p_2=4.779+(0.5)tan(cos(0.5^3))=5.545\)

\(\displaystyle p_3=5.545+(0.5)tan(cos(1^3))=5.845\)

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 \(\displaystyle v(t)=(-t)^{t}cos(\pi t)\) \(\displaystyle p(1)=-2\)

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

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

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

\(\displaystyle p_1=-2+(1)(-1)^{1}cos(\pi (1))=-1\)

\(\displaystyle p_2=-1+(1)(-2)^{2}cos(\pi (2))=3\)

\(\displaystyle p_3=3+(1)(-3)^{3}cos(\pi (3))=30\)

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 \(\displaystyle v(t)=\frac{t^3}{2+cos(\pi t)}\) \(\displaystyle p(0)=3\)

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

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

\(\displaystyle p_0=3;t_0=0\)

\(\displaystyle p_1=3+(1)\frac{0^3}{2+cos(\pi (0))}=3\)

\(\displaystyle p_2=3+(1)\frac{1^3}{2+cos(\pi (1))}=4\)

\(\displaystyle p_3=4+(1)\frac{2^3}{2+cos(\pi (2))}=\frac{20}{3}\)

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 \(\displaystyle v(t)=cos^3(\pi(t!))\) \(\displaystyle p(1)=5\)

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

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

\(\displaystyle p_0=5;t_0=1\)

\(\displaystyle p_1=5+(1)cos^3(\pi(1!))=4\)

\(\displaystyle p_2=4+(1)cos^3(\pi(2!))=5\)

\(\displaystyle p_3=5+(1)cos^3(\pi(3!))=6\)

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 \(\displaystyle v(t)=(2t)!\) \(\displaystyle p(1)=0\)

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

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

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

\(\displaystyle p_1=0+(1)(2(1))!=2\)

\(\displaystyle p_2=2+(1)(2(2))!=26\)

\(\displaystyle p_3=26+(1)(2(3))!=746\)