Suppose we are given the following information about a continuous-time periodic signal with period 3 and Fourier coefficients

$$a_k$$

: 1.

$$a_k=a_{k+2}.$$

$$a_k=a_{-k}.$$

$$∫_{-0.5}^{0.5}x(t)dt=1.$$

4.(∫_{0.5}^{1.5}x(t)dt=2.

$$Determine x(t).$$

Sunrise was at 6:15 a.m. and sunset was at 5:45 p.m How many hours and minutes were there from sunrise to sunset?

Suppose we are given the following information about a casual and stable LTI system S with impulse response h(t) and a rational system function H(s). 1. H(1)=0.2, 2. When the input is u(t), the output is absolutely integrable. 3. When the input is tu(t), the output is not absolutely integrable. 4. The signal d²h(t)/dt²+2dh(t)/dy+2h(t) is of finite duration. 5. H(s) has exactly one zero at infinity. Determine H(s) and its region of convergence.

Let

$$x_c(t)$$

be a continuous-time signal whose Fourier transform has the property that

$$X_c(jω)=0$$

for |ω|≥2.000π. A discrete-time signal

$$x_d[n]=x_c(n(0.5x10^{-3}))$$

is obtained. For each of the following constraints on the Fourier transform

$$X_d(e^{jω})$$

of

$$x_d[n]$$

, determine the corresponding constraint on

$$X_c(jω)$$

; (a)

$$X_d(e^{jω})$$

is real. (b) The maximum value of

$$X_d(e^{jω})$$

over all ω is 1. (c)

$$X_d(e^{jω})=X_d(e^{j(ω-π)}). (d)$$

X_d(e^{jω})=X_d(e^{j(ω-π)}).

A strip of metal is reduced from $30 \mathrm{~mm}$ in thickness to $20 \mathrm{~mm}$ by cold working; a similar strip is reduced from $40$ to $30 \mathrm{~mm}$. Which of these cold-worked strips will recrystallize at a lower temperature? Why?

At $5.60 per pound, the cheddar cheese cost how many cents per ounce?

Let x(t) be a periodic signal with fundamental period T and Fourier series coefficients

$$a_k$$

. Derive the Fourier series coefficients of each of the following signals in terms of

$$a_k$$

: (a)

$$x(t-t_0)+x(t+t_0)$$

(b) Ev{x(t)}, (c) Re[x(t)}, (d) d²x(t)/dt², (e) x(3t-1) [for this part, first determine the period of x(3t-1)]

Use technology to find the following for a standard Normal distribution. Sketch a standard Normal curve and shade the area of interest. The number z such that 35% of all observations are greater than z

Suppose you are given the following information about a particular industry:

$$\begin{array}{ll} Q^D=6500-100 P & \text { Market demand } \\ Q^S=1200 P & \text { Market supply } \\ C(q)=722+\frac{q^2}{200} & \text { Firm total cost function } \\ M C(q)=\frac{2 q}{200} & \text { Firm marginal cost function } \end{array}$$

Assume that all firms are identical and that the market is characterized by perfect competition.

a. Find the equilibrium price, the equilibrium quantity, the output supplied by the firm, and the profit of each firm.

b. Would you expect to see entry into or exit from the industry in the long run? Explain. What effect will entry or exit have on market equilibrium?

c. What is the lowest price at which each firm would sell its output in the long run? Is profit positive, negative, or zero at this price? Explain.

d. What is the lowest price at which each firm would sell its output in the short run? Is profit positive, negative, or zero at this price? Explain.

Suppose you are given the following information about the default-free, coupon-paying yield curve:

$$\begin{array}{llcccc} \text{Maturity (years)} & 1 & 2 & 3 & 4 \\ \hline \text{Coupon rate (annual payments)} & 0.00 \% & 9.00 \% & 7.00 \% & 14.00 \\ \text{YTM} & 2.981 \% & 4.333 \% & 5.832 \% & 5.696 \\ \hline \end{array}$$

a. Use arbitrage to determine the yield to maturity of a two-year, zero-coupon bond.

b. What is the zero-coupon yield curve for years $1$ through $4$ ?

How do you decide if something like a computer is an asset or a liability?

An illusionist claims that she has the ability to toss a coin so that it turns up heads. Listed below are results from a test of her abilities.

$$\begin{matrix} \text{Quarter:} & \text{H} & \text{H} & \text{H} & \text{H} & \text{H} & \text{H} & \text{H} & \text{H} & \text{H}\\ \text{Penny:} & \text{H} & \text{H} & \text{T} & \text{H} & \text{T} & \text{H} & \text{H} & \text{T} & \text{T}\\ \end{matrix}$$

Using all of the results combined with a 0.01 significance level, test the claim that a coin can be tossed so that heads turns up more often than can be expected by chance.

Let x[n] be a periodic signal with period N=8 and Fourier series coefficients

$$a_y=-a_{k-4}$$

. A signal

$$y[n]=(1+(-1)^n/2)x[n-1]$$

with period N=8 is generated. Denoting the Fourier series coefficients of y[n] by

$$b_k$$

, find a function f[k[ such that

$$b_k=f[k]a_k.$$

Suppose we are given the following information about a periodic signal x[n] with period 8 and Fourier coefficients

$$a_k:$$

$$a_k=-a_{k-4}. 2.x[2n+1]=(-1)^n.$$

Sketch one period of x[n].