(a) In your own words, briefly state King's dream. (b) Based on this dream, what do you think was the reality of King's life?

A function f is defined over an interval [a, b]. Approximate the area A by partitioning [a, b] into eight subintervals of equal length and choosing u as the left endpoint of each subinterval.$f(x)=x^{2}-4, \quad[2,6]$

Air enters the compressor of a simple gas turbine at $p_1=14\ \mathrm{lbf} / \mathrm{in} .{ }^2, T_1=520^{\circ} \mathrm{R}$, and a volumetric flow rate of $10,000\ \mathrm{ft}^3 / \mathrm{min}$. The isentropic efficiencies of the compressor and turbine are $83$ and $87 \%$, respectively. The compressor pressure ratio is 14 and the temperature at the turbine inlet is $2500^{\circ} \mathrm{R}$. On the basis of an air-standard analysis, calculate (a) the thermal efficiency of the cycle. (b) the net power developed, in hp. (c) the rates at which entropy is produced within the compressor and turbine, each in $\mathrm{hp} /{ }^{\circ} \mathrm{R}$.

Reduce:

$$\frac { 5 \cdot 5 \cdot 5 \cdot 7 } { 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5 }$$

Find the area of the region enclosed by the graphs of the given equations by partitioning the x-axis. $y=e^{x}$, y=-x+1, x=2

Use a CAS to perform the following steps:

Subdivide the interval into $n=100,200,$ and $1000$ subintervals of equal length and evaluate the function at the midpoint of each subinterval. Compute the average value of the function values generated. Solve the equation $f(x)=$ (average value) for $x$ using the average value calculated above for the $n=1000$ partitioning. $f(x)=x \sin ^2 \frac{1}{x}$ on $\left[\frac{\pi}{4}, \pi\right]$

A clothing store recorded the length of time customers were in the store and the amount they spent.

$$\begin{matrix} \text{Time (min)} & \text{8} & \text{18} & \text{5} & \text{10} & \text{17} & \text{11} & \text{2} & \text{13} & \text{18} & \text{4} & \text{11} & \text{20} & \text{23} & \text{22} & \text{17}\\ \text{Money (€)} & \text{40} & \text{78} & \text{0} & \text{46} & \text{72} & \text{86} & \text{0} & \text{59} & \text{33} & \text{0} & \text{0} & \text{122} & \text{90} & \text{137} & \text{93}\\ \end{matrix}$$

Given the mean time $\approx 13.3$ mins, and the mean amount $\approx €57.07$, plot the mean point on your diagram and draw a line of best fit.

Find the exact value of the given expression. If an exact value cannot be given, give the value to the nearest ten-thousandth. If the exercise expression is not defined, simply respond with "Undefined." $\\$$\cos ^{-1}\left[\tan \left(-$$\frac{\pi}{3}\right)\right]$

Find the area of the region enclosed by the graphs of the given equations by partitioning the x-axis. $y=x^{2}, y=x^{4}$

A function f is defined over an interval [a, b]. Approximate the area A by partitioning [a, b] into four subintervals of equal length and choosing u as the left endpoint of each subinterval. $f(x)=x^{2}+2, \quad[0,4]$

A function f is defined over an interval [a, b]. Approximate the area A by partitioning [a, b] into four subintervals of equal length and choosing u as the left endpoint of each subinterval.$f(x)=x^{2}-4, \quad[2,6]$

Suppose that f(5)=1, f'(5)=6, g(5)=-3, and g'(5)=2. Find the values of (g/f)'(5).

Solve. $\sqrt[5]{x+60,000}-5=5$

Simplify, if possible. log$_{5} 5^{x - 5}$