For each of the sets of data that follows, use the least squares approximation to find the best fits with both (i) a linear function and (ii) a quadratic function. Compute the error $E$ in both cases. $\\$ (a)$\{(-3,9),(-2,6),(0,2),(1,1)\}$(b)$\{(1,2),(3,4),(5,7),(7,9),(9,12)\}$(c)$\{(-2,4),(-1,3),(0,1),(1,-1),(2,-3)\}$

eI Lheck ‘ ! Compare and cantrast substances and mixtures. Give WO exdmples of eacn

An object of mass 2.00 kg is oscillating freely on a vertical spring with a period of 0.600 s. Another object of unknown mass on the same spring oscillates with a period of 1.05 s. Find the unknown mass.

A potter’s wheel having a radius of 0.50 m and a moment of inertia of 12 $k g \cdot m ^ { 2 }$ is rotating freely at 50 rev/min. The potter can stop the wheel in 6.0 s by pressing a wet rag against the rim and exerting a radially inward force of 70 N. Find the effective coefficient of kinetic friction between the wheel and the wet rag.

The area of an oil spill is increasing at a rate of 25t

$$\mathrm { ft } ^ { 2 } / \mathrm { s }$$

seconds after the spill. Between times t=2 and t=4 the area of the spill increases by ___

$$ft^2$$

.

Describe two advantages of using credit cards.

What three variables influence the speed of stream flow?

Sketch the curves of the given functions by addition of ordinates. $y=3-2 \cos x$

Determine the longest interval in which the given initial value problem is certain to have a unique twice differentiable solution. Do not attempt to find the solution.

$$\begin{gather*} (x + 3)y'' + xy' + (\ln |x|)y = 0, \,\,\, y(1) = 0, \,\,\, y'(1) = 1 \end{gather*}$$

Find the minimal solution to each of the following systems of linear equations. $\\$ (a)$x+2 y-z=12$(b)$

$$\begin{aligned} x+2 y-z &=1 \\ 2 x+3 y+z &=2 \\ 4 x+7 y-z &=4 \end{aligned}$$

$(c)$

$$\begin{aligned} x+y-z &=0 \\ 2 x-y+z &=3 \\ x-y+z &=2 \end{aligned}$$

$(d)$

$$\begin{aligned} x+y+z-w &=1 \\ 2 x-y &+w=1 \end{aligned}$$

$

Variations in members of a population help ____.

For each of the following inner product, spaces V (over $F^{\prime} )$ and linear transformations $g : V \rightarrow F,$ find a vector $y$ such that $g(x)=\langle x, y\rangle$ for all $x \in V .$ $\\$ (a)$$\mathrm{V}$=$\mathrm{R}$^{3}, $\mathrm{g}$\left(a_{1}, a_{2}, a_{3}\right)=a_{1}-2 a_{2}+4 a_{3} $(b)$$\mathrm{V}$=$\mathrm{C}$^{2}, $\mathrm{g}$\left(z_{1}, z_{2}\right)=z_{1}-2 z_{2} $(c)$     $\mathrm{V}$=$\mathrm{P}${2}(R)$with$\langle f, h\rangle=\int{0}^{1} f(t) h(t) d t, $\mathrm{g}$(f)=f(0)+f^${\prime}$(1)$

Find the inner surface area A of a cylindrical 250.0 cubic cm cup as a function of the radius r of the base. Then if A = 175.0 square cm, solve for r. Write one or two paragraphs explaining your method for setting up the function, and how you used a graphing calculator to solve for r with the given value of A.

Prove that if $V=W \oplus W^{\perp}$ and $T$ is the projection on $W$ along $W^{\perp}$ , then $T=T^{*} .$ Ilint: Recall that $N(T)=W^{\perp} .$