Consider an experiment of throwing a fair die. Let X be the random variable (r.v.) which assigns 1 if the number that appears is even and 0 if the number that appears is odd. 1. What is the range of X? 2. Find P(X = 0) and P(X = 1).

E. If f is a continuous mapping of a metric space X into a metric space Y, prove that f(E) \subset \overline{f(E)} for every set EC X. (E denotes the closure of E.) Show, by an example, that f(E) can be a proper subset of f(E).

In Problems 1-6, identify the equation as separable, linear,exact, or having an integrating factor that is a function of either x alone or y alone. \text { 3. }(2 x+y) d x+(x-2 y) d y=0 \text { 5. }\left(x^{2} \sin x+4 y\right) d x+x d y=0

Consider the IVP y = x + 2y, y(0) = 1 A) (8 pts) Use Euler's method to obtain an approximation of y(0.5) usingh = 0.25 for the given IVP (Use four-decimal approximation) B) (12 pts) Use Euler's method to obtain an approximation of y(0.5) usingh = 0.1 for the given IVP (Use four-decimal approximation) A bacteria culture initially has 100 number of bacteria and doubles in size in 2hours. Assume that the rate of increase of the culture is proportional to the size. Write the initial value problem for the bacteria culture and solve it ts) How long will it take for the size to triple? Verify that y(x) = c1 cos(6x) + c2 sin(6x) is a solution of y" + 36y = 0 s) Either solve the boundary value problem y^{\prime \prime}+36 y=0, y(0)=0, y\left(\frac{2 \pi}{6}\right)=1 or else show that it has no solution

\text { 1. } \vec{x}^{\prime}=\left[\begin{array}{rr} -2 & -3 \\ 3 & 4 \end{array}\right] \vec{x} (a) Find the general solution. (b) Draw the phase portrait. (c) Classify the equilibrium solution with its stability. \text { 2. } \vec{x}^{\prime}=\left[\begin{array}{rr} -1 & 1 \\ -1 & -3 \end{array}\right] \vec{x} (a) Find the general solution. (b) Draw the phase portrait. (c) Classify the equilibrium solution with its stability.

. A thin wire coinciding with the x-axis on the interval [-L, L]is bent into the shape of a circle so that the ends xand x =L are joined. Under certain conditions the temperature u(x, t) in the wire satisfies the boundary-value problem k \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t},-L<x<L, t>0 u(-L, t)=u(L, t), t>0 \left.\frac{\partial u}{\partial x}\right|_{x=-L}=\left.\frac{\partial u}{\partial x}\right|_{x=L}, t>0 u(x, 0)=f(x),-L<x<L

\overrightarrow{x^{\prime}}=\left(\begin{array}{cc} 0 & 5 \\ -1 & 2 \end{array}\right) \vec{x}, \quad \vec{x}(0)=\left(\begin{array}{c} -3 \\ 1 \end{array}\right)

Find the steady state temperature u(r,0) in a quarter semi circular plate of radius r=2 subject to heat equation in polar coordinates. u\mleft(r,0\mright?=0,0<r<2, u\mleft(r,\frac{\pi}{8}\mright?=0,0<r<2, u\mleft(2,0\mright)=-9\sin \mleft(40\theta\mright),0<\theta<\frac{\pi}{8}

Your submission must be a single PDF and all pages must be oriented correctly (e.g. pages should not be upside down). If your submission does not follow these guidelines, then points may be deducted. Please answer each question fully, providing all reasoning. You will be graded based on both mathematical correctness and clarity of writing. See the Written Portion Rubric on D2L for more details. Let b denote the last nonzero digit of your UCID number. e.g. if your UCID is 9876543280 then b = 8. 1. Let C and D be constants. Consider the function f(x)=\left\{\begin{array}{ll} \sqrt{b x+1}+C, & \hat{x} \geq 0 \\ D x+x^{4} \sin \left(\frac{b}{x}\right), & x<0 \end{array}\right. Use the limit definition of the derivative f^{\prime}(a)=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a} to determine what C and D must equal in order for f(0) to exist. (You must use the limit definition of the derivative to receive credit for this question. If you do not, then you will not receive credit.) Hint: You will need to use one-sided limits.

46. For what values of the constant k does the Second Derivative Test guarantee that f(x, y) = x² + kxy + y^2 will have a saddle point at(0, 0)? A local minimum at (0, 0)? For what values of k is the Second Derivative Test inconclusive? Give reasons for your answers.