Would you expect a company in a rapidly growing technology industry to have a high or low dividend payout ratio?

List the six problem-solving strategies you have learned in this chapter.

ExplaShow a segment that would measure 6 cm to the nearest centimeter but 6.5 cm to the nearest half-centimeter. How many possibilities are there?in why the statement "My brand-new pencil is exactly 7 1/2 inches long" is not completely true.

In this question you will find the value of the indefinite integral $t=\int \frac{1}{\sqrt{1+x^2}} \mathrm{~d} x$ using different methods. Suppose that, first, you are asked to find the value of a definite integral. You can do this using technology.

Hence, find the exact value of $\int_a^1 \frac{1}{\sqrt{1+x^2}} \mathrm{~d} x$ and verify that it agrees with the answer from part (a). (2 marks) We will now attempt to see if we can find the integral $l$ just by using the usual trigonometrical functions.

For the linear transformations $T$ as given below, do the following:

a. Find the function $f(t)$ defined in Exercise 47 and graph it for $0 \leq t \leq \frac{\pi}{2}$. You may use technology.

b. Find a number $c$, with $0 \leq c \leq \frac{\pi}{2}$, such that $f(c) = 0.$(In Problem 50, approximate $c$ to three significant digits, using technology.)

c. Find perpendicular unit vectors $\vec v_1$ and $\vec v_2$ in

$$\mathbb R^2$$

such that the vectors $T(\vec v_1)$ and $T(\vec v_2)$ are perpendicular as well. Draw a sketch showing $\vec v_1,\ \vec v_2, \ T(\vec v_1),$ and $T(\vec v_2)$.

$$T (\vec{x})=\begin{bmatrix} 1 & 1\\ 0& 1 \end{bmatrix}\vec{x}$$

(a) Classify vocabulary word listed below under one of the following themes: economics and technology, global interaction, religions and value systems, geography and history.

(b) Write a sentence explaining how that word relates to the theme.

island-hopping

Without using technology, sketch the graph of each function. Label all intercepts. (Hint: Factor.)

$$f(x)=-2 x^3+3 x^2+11 x-6$$

Use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile. $P_{15}$

The Yorkshire Clothing Company makes two styles of jacket, the ‘York’ and the ‘Wetherby’. The York requires 3 m of cloth and 3 h of labour, and makes a profit of £25. The Wetherby needs 4 m of cloth and 2 h of labour, and makes a profit of £30. The Yorkshire has 400 m of cloth available and 300 h of labour available each week. Advise the company on the number of each style it should produce in order to maximize profits. The company is prepared to buy more cloth to increase its profits, but it will not employ any more labour. Under this revised policy, is there a strategy that will increase its profits?

Use mathematical induction to prove that the formula is true for all natural numbers n. $\left(1+\frac{1}{1}\right)\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right) \cdots\left(1+\frac{1}{n}\right)=n+1$

Which of the following are legitimate mathematical operations: (a) $\vec{A} \cdot(\vec{B}-\vec{C})$; (b) $(\vec{A}-\vec{B}) \times \vec{C}$ (c) $\vec{A} \cdot(\vec{B} \times \vec{C})$; (d) $\vec{A} \times(\vec{B} \times \vec{C})$; (e) $\vec{A} \times(\vec{B} \cdot \vec{c})$? In each case, give the reason for your answer.

Let the matrix

$$\mathbf{P}=\left[ \begin{array}{cc}{2} & {1} \\ {-1} & {0}\end{array}\right].$$

b. Find $\mathbf{P}^{2}, \mathbf{P}^{3} \text { and } \mathbf{P}^{4}.$ b. Give a proposition (conjecture) for $P^n$ where $n \in \mathbb{Z}^{+}.$ c. Prove your proposition is true using mathematical induction.

Write out in full and verify the statements $S_{1}, S_{2}, S_{3}, S_{4}$, and $S_5$ for the following. Then use mathematical induction to prove that each statement is true for every positive integer n. $1+3+5+\cdots+(2 n-1)=n^{2}$

Write a mathematical expression to represent the situation. Then find the value of the expression to solve the problem.

The price of a stock was $\$ 325$ a share. The price of the stock went down $\$ 25$ each week for 6 weeks. What was the price of that stock at the end of 6 weeks?