Write the VHDL code for an S-R flip-flop with a rising-edge clock. Use standard logic, and output 'X' if S = R = '1' at a rising clock edge.

Write VHDL code for a T flip-flop with an active-low asynchronous clear.

Find a minimum two-level OR-AND circuit to simultaneously realize $F_{1}(a, b, c, d)=\Sigma m(2,3,8,9,14,15)$ $F_{2}(a, b, c, d)=\Sigma m(0,1,5,8,9,14,15)$ (minimum solution has eight gates)

Use determinants to decide if the matrix is invertible.

$$\left[\begin{array}{rr} 2 & -1 \\ -2 & 2 \end{array}\right]$$

Find a splitting field of $x^{4}-2$ (a) over $\mathbb{Q}$. (b) over $\mathbb{R}$.

Determine the location and value of the absolute extreme values of f on the given interval, if they exist. $f(x)=3 x^{5}-25 x^{3}+60 x \text { on }[-2,3]$

Find y' by using logarithmic differentiation. $y=\frac{\sqrt{1-x^{2}}}{1-2 x}$

Write the abbreviations that should be capitalized in the following sentence.

Recently, I saw "ak" after a piayer's name but couldn't remember whether it stood for Alaska or Arkansas.

Suppose 5% of all people fitting the long income tax form seek deductions that they know are illegal, and an additional 2% incorrectly list deductions because they are unfamiliar with income tax regulations. Of the 5% who are guilty of cheating, 80% will deny knowledge of the error if confronted by an investigator. If the filer of the long form is confronted with an unwarranted deduction and he or she denies the knowledge of the error, what is the probability that he or she is guilty?

If a = bc with $a \neq 0$ and b and c nonunits, show that a is not an associate of b.

Refer to the Early Childhood Education Journal (Mar. 2014) study of interactions in a children's museum. (Interactions by visitors to the museum included show-and-tell, learning, teaching, refocusing, participatory play, advocating, and disciplining interactions.) Recall that the researchers observed a sample of 170 meaningful interactions, of which 81 were led by children and 89 were led by adult caregivers. a. Give a point estimate of the true proportion of all meaningful interactions in a children's museum that are led by children. b. Form a 90% confidence interval around the proportion, part a. c. Interpret the interval, part b, in the words of the study. d. Suppose an educator claims that the true proportion of all meaningful interactions in a children's museum that are led by children is 35%. Make an inference about this claim.

Use the definition of the derivative to find each of the following. $\frac{d}{d x}(3-2 x)$

A modulo 8 counter cycles through the states $Q_{0} Q_{1} Q_{2} Q_{3}=1000$, 1100, 0100, 0110, 0010, 0011, 0001, 1001. The counter has eight outputs: $Z_{0}=1$ when the counter is in state 1000 and the CLK is 0 and $Z_{0}=0$ otherwise; $Z_{1}=1$ when the counter is in state 1100 and the CLK is 0 and $Z_{1}=0$ otherwise,...: $Z_{7}=1$ when the counter is in state 1001 and the CLK is 0 and $Z_{7}=0$ otherwise. The counter has an asynchronous active-low reset input ClrN. (a) Derive minimum equations for the counter outputs. (b) Assume the counter is implemented using D flip-flops. Find minimum input equations for the flip-flops. (c) Assume the counter is implemented using D-CE flip-flops. Find minimum input equations for the flip-flops. (d) Write a VHDL behavioral description of the counter. Assume the flip-flops are positive edge triggered. (e) Write a VHDL dataflow description of the counter using the equations from part (b). Simulate the counter for a cycle to verify your code. (f ) Write a VHDL dataflow description of the counter using the equations from part (c). Simulate the counter for a cycle to verify your code.

Prove that if r and s are relatively prime, then $\varphi(r s)=\varphi(r) \varphi(s).$