Solve each system of equations. Check your answers.

$$\left\{\begin{aligned} x+y+z &=4 \\ 4 x+5 y \quad \quad &=3 \\ \quad \quad y-3 z &=-10 \end{aligned}\right.$$

Find the area of the region bounded by the graphs of the equations.

$$y=\sin ^2 \pi x, \quad y=0, \quad x=0, \quad x=1$$

Solve the equation. $A = l w \quad \quad \text { for } w.$

$$\frac { \text { the Find the values of } a \text { and } b \text { so the system shown has infinitely many solutions. } } { 2 x + 9 y = 3 }$$

$$\begin{array} { l } { 2 x + 9 y = 3 } \\ { \quad \quad \quad , \quad \quad 4 x + a y = b } \end{array}$$

$$\text { Find an isomorphism from } Z _ { 12 } \text { to } Z _ { 4 } \oplus Z _ { 3 }$$

Consider

$$\frac{d^2 u}{d x^2}=f(x) \quad \text { with } \quad u(0)=0 \quad \text { and } \quad u(L)=0 .$$

Write the system as a matrix system $A u=f$. What is $A$?

Use the Bisection method to find solutions, accurate to within $10^{-5}$ for the following problems.

$$\begin{array}{l}{\text { a. } \quad 3 x-e^{x}=0 \text { for } 1 \leq x \leq 2} \\ {\text { b. } \quad 2 x+3 \cos x-e^{x}=0 \quad \text { for } 0 \leq x \leq 1} \\ {\text { c. } \quad x^{2}-4 x+4-\ln x=0 \quad \text { for } 1 \leq x \leq 2 \quad \text { and } \quad 2 \leq x \leq 4} \\ {\text { d. } \quad x+1-2 \sin \pi x=0 \quad \text { for } 0 \leq x \leq 0.5 \quad \text { and } \quad 0.5 \leq x \leq 1}\end{array}$$

Evaluate: $\quad m=x y \quad$

if, $m=10, x=2$, and $y=3$

Renaldo said that a polynomial equation with rational coefficients that has zeros -1 +2i and

$$3 + \sqrt { 5 }$$

has a degree of 4. Is Renaldo correct? Explain.

If $(-3,2)$ lies on the graph of $f$, then $f(\underline{\quad})=\underline{\quad}$.

Use Euler’s method to approximate the solutions for each of the following initial-value problems.

$$\begin{array}{l}{\text { a. } \quad y^{\prime}=y / t-(y / t)^{2}, \quad 1 \leq t \leq 2, \quad y(1)=1, \text { with } h=0.1} \\ {\text { b. } \quad y^{\prime}=1+y / t+(y / t)^{2}, \quad 1 \leq t \leq 3, \quad y(1)=0, \text { with } h=0.2} \\ {\text { c. } \quad y^{\prime}=-(y+1)(y+3), \quad 0 \leq t \leq 2, \quad y(0)=-2, \text { with } h=0.2} \\ {\text { d. } \quad y^{\prime}=-5 y+5 t^{2}+2 t, \quad 0 \leq t \leq 1, \quad y(0)=\frac{1}{3}, \text { with} h=0.1}\end{array}$$

Find $f(x)$ given that: a. $f^{\prime}(x)=2 x-1 \quad \text { and } \quad f(0)=3$ b. $f^{\prime}(x)=3 x^{2}+2 x \text { and } f(2)=5$ c. $f^{\prime}(x)=e^{x}+\frac{1}{\sqrt{x}} \quad \text { and } \quad f(1)=1$ d. $f^{\prime}(x)=x-\frac{2}{\sqrt{x}} \quad \text { and } \quad f(1)=2$

Find the coordinates of the point(s) of intersection of the graphs with equations: a.

$$y = x ^ { 2 } - 2 x + 8 \quad \text { and } \quad y = x + 6$$

b.

$$y = - x ^ { 2 } + 3 x + 9 \quad \text { and } \quad y = 2 x - 3$$

c.

$$y = x ^ { 2 } - 4 x + 3 \quad \text { and } \quad y = 2 x - 6$$

d.

$$y = x ^ { 2 } - 3 x + 2 \text { and } y = x - 3$$

e.

$$y = - x ^ { 2 } + 4 x - 7 \text { and } y = 5 x - 4$$

f.

$$y = x ^ { 2 } - 5 x + 9 \quad \text { and } \quad y = 3 x - 7$$

Find the area of the region bounded by the graphs of the equations.
$y=3 x^2+1, \quad x=0, \quad x=2, \quad y=0$