Find the output if the functions 'square' and 'subtract 4' act in succession on an input of (a) 2, (b) -5, (c) $\frac{1}{2}$, (d) x.

a) Search the Internet for data on the fraction of the Moon that is illuminated at midnight every day for the current month. b) Plot the data on a graph. c) Use the graph to determine a sinusoidal model for the data. d) Graph the model and compare it to the data. Comment on the accuracy of the fit.

An arithmetic progression has first term a and common difference 10. The sum of the first n terms of the progression is 10 000. Express a in terms of n, and show that the nth term of the progression is $\frac{10000}{n}+5(n-1)$. Given that the nth term is less than 500, show that $n^{2}-101 n+2000<0$ and hence find the largest possible value of n.

The domains of these functions are the set of all positive real numbers. Find their ranges. (a) f(x) = 2x + 7 (b) f(x) = -5x (c) f(x) = 3x - 1 (d) $f(x)=x^{2}-1$ (e) $f(x)=(x+2)^{2}-1$ (f) $f(x)=(x-1)^{2}+2$

A diode has $I_S = 0.2$ fA and n = 1. (a) What is the diode current if the diode voltage is 0.675 V? (b) What will be the diode voltage if the current increases by a factor of 3?

Are all even-degree polynomial functions even? Are all odd-degree polynomial functions odd? Explain.

The sodium nitrate (NaNO$_{3}$) solubility data is described nicely by the regression line y = 67.508 + 0.871x, where s = 0.959. Construct a 90% confidence interval for the y-intercept, $\beta_{0}$.

Recall the cereal company that was putting photos of athletes in boxes of cereal? It puts pictures of Serena Williams in half the boxes and pictures of either LeBron James or Dustin Johnson in the others. Hoping to get a complete set, you buy three boxes of cereal. a) Make a sample space showing the possible outcomes as you open the boxes one at a time. b) Why can't you use this sample space to find the probability that you get one picture of each athlete?

Differentiate the moment-generating function for a geometric random variable and verify the expressions given for E(X) and Var(X) in Theorem 4.4.1.

Find an expression, in terms of n, for the coefficient of x in the expansion $(1+4 x)+(1+4 x)^{2}+(1+4 x)^{3}+\ldots+(1+4 x)^{n}$.

State what the following abbreviations stand for: NOI, GI, $T_{e},$ NOPAT, TI, R, OE, EBIT.

Choose the best answer. Which is NOT true about geothermal energy? (a) It is only available in limited areas. (b) It cannot be used for cooling. (c) It can be locally depleted due to heavy use. (d) Ground source heat pumps require an additional source of energy.

The points A(-3, -4) and C(5, 4) are the ends of the diagonal of a rhombus ABCD. (a) Find the equation of the diagonal BD. (b) Given that the side BC has gradient $\frac{5}{3}$, find the coordinates of B and hence of D.

Suppose that

$$f_{X, Y}(x, y)=\lambda^{2} e^{-\lambda(x+y)}, 0 \leq x, 0 \leq y$$

Find E(X + Y).