Find each product. $4(-10)(-3)$

The recursive definition, a(n+1)=ra(n), $a(0)=a_{0}$, is called a first-order difference equation and generates the sequence $a_{0}, r a_{0}, r\left(r a_{0}\right), r\left(r\left(r a_{0}\right)\right)$, .... A little simplification shows that the nth term of this sequence is $a(n)=a_{0} r^{n}$. Now suppose that I represents the annual interest rate, but the interest is awarded in discrete packets, m times per year. Then the rate awarded during each compounding period is I/m. Consequently, if the initial investment is $P_{0}$, the balance is $P_{0}(1+I / m)$ at the end of the first compounding period, $P_{0}(1+I / m)^{2}$ at the end of the second compounding period, and so on. (a) Give a first-order difference equation with an initial condition that generates a sequence describing the balance in the account at the end of each compounding period. (b) Find a formula for the nth term of the sequence generated by the first-order difference equation created in part (a).

Find the indefinite integral. $\int 2\left(x^{2}-1\right)^{9} x d x$

Let f(x) be a monic polynomial of degree n with roots

$$\alpha_{1}, \alpha_{2}, \ldots,\alpha_{n}.$$

(a) Show that the discriminant D of f(x) is the square of the Vandermonde determinant

$$\left|\begin{array}{ccccc} 1 & \alpha_{1} & \alpha_{1}^{2} & \dots & \alpha_{1}^{n-1} \\ 1 & \alpha_{2} & \alpha_{2}^{2} & \dots & \alpha_{2}^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \alpha_{n} & \alpha_{n}^{2} & \dots & \alpha_{n}^{n-1} \end{array}\right|=\prod_{i>j}\left(\alpha_{i}-\alpha_{j}\right)$$

(b) Taking the Vandermonde matrix above, multiplying on the left by its transpose and taking the determinant show that one obtains

$$\boldsymbol{D}=\left|\begin{array}{ccccc} p_{0} & p_{1} & p_{2} & \dots & p_{n-1} \\ p_{1} & p_{2} & p_{3} & \dots & p_{n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ p_{n-1} & p_{n} & p_{n+1} & \dots & p_{2 n-2} \end{array}\right|$$

where

$$p_{i}=\alpha_{1}^{i}+\cdots+\alpha_{n}^{i}$$

is the sum of the $i^{\text {th }}$ powers of the roots of f(x), which can be computed in terms of the coefficients of f(x) using Newton's formulas above. This gives an efficient procedure for calculating the discriminant of a polynomial.

Let

$$\mathbf{u}=\left[\begin{array}{l} 0 \\ 1 \\ 3 \end{array}\right] \quad \mathbf{v}=\left[\begin{array}{r} 1 \\ -1 \\ 2 \end{array}\right]$$

$$\mathbf{w}=\left[\begin{array}{r} 1 \\ 1 \\ -3 \end{array}\right]$$

Compute the quantity. $\frac{\mathbf{u} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} \mathbf{w}$

A physical pendulum is made of a uniform disk of mass M and radius R suspended from a rod of negligible mass. The distance from the pivot to the center of the disk is $l.$ What value of $l$ makes the period a minimum?

For the following exercise, use the functions y = f(x) to find a. $\frac{d f}{d x}$ at x = a and b. $x=f^{-1}(y)$. c. Then use part b. to find $\frac{d f^{-1}}{d y}$ at y = f(a). $f(x)=2 x^{3}-3, x=1$

Let F be a field contained in the ring of n $\times$ n matrices over $\mathbb{Q}$. Prove that $[F: \mathbb{Q}] \leq n$. (Note that the ring of n $\times$ n matrices over $\mathbb{Q}$ does contain fields of degree n over $\mathbb{Q}$.

Having received a large inheritance, a child's parents wish to establish a trust for the child's college education. If they need an estimated $120,000 7 years from now, how much should they set aside in trust now if they invest the money at 4.6% compounded quarterly? Continuously?

The amount (in billions of dollars) spent by the top 15 U.S. financial institutions on IT (information technology ) offshore outsourcing is approximated by $A(t)=0.92(t+1)^{0.61} \ (0 \leq t \leq 4)$ where t is measured in years, with t = 0 corresponding to 2004. a. Show that A is increasing on (0, 4), and interpret your result. b. Show that the graph of A is concave downward on (0, 4). Interpret your result.

Two surveys were independently conducted to estimate a population mean, $\mu$. Denote the estimates and their standard errors by $\bar{X}_{1} \text { and } \bar{X}_{2} \text { and } \sigma_{\bar{X}_{1}} \text { and } \sigma_{\bar{X}_{2}}$. Assume that $\bar{X}_{1} \text { and } \bar{X}_{2}$ are unbiased. For some $\alpha$ and $\beta$, the two estimates can be combined to give a better estimator: $X=\alpha \bar{X}_{1}+\beta \bar{X}_{2}$ a. Find the conditions on $\alpha$ and $\beta$ that make the combined estimate unbiased. b. What choice of $\alpha$ and $\beta$ minimizes the variances, subject to the condition of unbiasedness?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If $y=a x+b$ where a and b are constants, then $\Delta y=d y$.

Fill in the blanks. A function F is an antiderivative of f on an interval I if ____ for all x in I.

Let $\zeta_{n}$ denote a primitive $n^{\text {th }}$ root of unity and let $K=\mathbb{Q}\left(\zeta_{n}\right)$ be the associated cyclotomic field. Let a denote the trace of $\zeta_{n}$ from K to $\mathbb{Q}$. Prove that a = 1 if n = 1, a = 0 if n is divisible by the square of a prime, and $a=(-1)^{r}$ if n is the product of r distinct primes.