Consider the following table:

$$\begin{matrix} \text{Time (hr)} & \text{Thermocule} 1 ^\circ F & \text{Thermocule} 2 ^\circ F & \text{Thermocule} 3 ^\circ F\\ \text{0} & \text{84.3} & \text{90.0} & \text{86.7}\\ \text{2} & \text{86.4} & \text{89.5} & \text{87.6}\\ \text{4} & \text{85.2} & \text{88.6} & \text{88.3}\\ \text{6} & \text{87.1} & \text{88.9} & \text{85.3}\\ \text{8} & \text{83.5} & \text{88.9} & \text{80.3}\\ \text{10} & \text{84.8} & \text{90.4} & \text{82.4}\\ \text{12} & \text{85.0} & \text{89.3} & \text{83.4}\\ \text{14} & \text{85.3} & \text{89.5} & \text{85.4}\\ \text{16} & \text{85.3} & \text{88.9} & \text{86.3}\\ \text{18} & \text{85.2} & \text{89.1} & \text{85.3}\\ \text{20} & \text{82.3} & \text{89.5} & \text{89.0}\\ \text{22} & \text{84.7} & \text{89.4} & \text{87.3}\\ \text{24} & \text{83.6} & \text{89.8} & \text{87.2}\\ \end{matrix}$$

(a) Create a column vector named times going from 0 to 24 in 2-hour increments. (b) Your instructor may provide you with the thermocouple temperatures in a file called thermocouple. dat, or you may need to create a matrix named thermocouple yourself by typing in the data. (c) Combine the times vector you created in part (a) with the data from thermocouple to create a matrix corresponding to the table in this problem. (d) Both the max and min functions can return not only the maximum values in a column, but also the element number where those values occur. Use this capability to determine the values of times at which the maxima and minima occur in each column.

Write the answers to the following calculations in standard form:

a $50000 \times 2400$

b $\left(3.7 \times 10^6\right) \times\left(4.0 \times 10^4\right)$

c $\left(5.8 \times 10^7\right)+\left(9.3 \times 10^6\right)$

d $\left(4.7 \times 10^6\right)-\left(8.2 \times 10^5\right)$

On Monday, a restaurant made $\$ 545$ from selling $110$ hamburgers. If they sold $53$ hamburgers on Tuesday, how much did they make?

Find the inverse of each function. Then graph the function and its inverse.

$$f ( x ) = \frac { 1 } { 2 } x ^ { 2 } - 1$$

Identify the line(s) of symmetry for the conic section.

$$(x+5)^2/49+(y-2)^2/16=1$$

Find each quotient, or, if applicable, state that the expression is undefined.

$$\frac { - 40 } { 5 }$$

Test for symmetry with respect to the line

$$\theta = \frac { \pi } { 2 }.$$

$$r = 2 \cos 2 \theta$$

Graph the function f(x). Then graph its inverse, $f^{-1}(x)$, and write a rule for the inverse function.

$f(x)=\frac{1}{4} x^3$

Let A be an n × n matrix and let

$$||·||_M$$

be a matrix norm that is compatible with some vector norm on ℝⁿ. Show that if λ is an eigenvalue of A, then

$$|λ| ≤ ||A||_M.$$

In this exercise, find the inverse of the function. Then graph the function and its inverse.

$$f(x)=\dfrac{3}{4} x+\dfrac{5}{8}$$

Compute a corresponding unit vector (vector of norm 1) with respect to the l∞-norm. Show that

$$||x||_∞≤||x||_2≤||x||_1.$$

This problem is closely related to the spectacular impact of the comet Shoemaker-Levy with Jupiter in July 1994 : http://www.jpl.nasa.gov/sl9/sl9.html. A rock far outside our Solar System is initially moving very slowly relative to the Sun, in the plane of Jupiter's orbit around the Sun. The rock falls toward the Sun, but on its way to the Sun it collides with Jupiter. Calculate the rock's speed just before colliding with Jupiter. Explain your calculation and any approximations that you make.

$$\begin{array}{l}{M_{\text {Sun }}=2 \times 10^{30} \mathrm{kg}, M_{\text {Jupiter }}=2 \times 10^{27} \mathrm{kg}} \\ {\text { Distance, Sun to Jupiter }=8 \times 10^{11} \mathrm{m}} \\ {\text { Radius of Jupiter }=1.4 \times 10^{8} \mathrm{m}}\end{array}$$

Calculate $\left[\mathrm{H}_3 \mathrm{O}^{+}\right]$and $\left[\mathrm{OH}^{-}\right]$for each of the following.

a. $0.030 \mathrm{M} \mathrm{HCl}$ $\left[\mathrm{H}_3 \mathrm{O}^{+}\right]=$ ______ $\left[\mathrm{OH}^{-}\right]=$ _____

b. $5.0 \times 10^{-3} \mathrm{M} \mathrm{HNO}_3 \quad\left[\mathrm{H}_3 \mathrm{O}^{+}\right]=$ _____$\left[\mathrm{OH}^{-}\right]=$ _____

c. $0.010 \mathrm{M} \mathrm{Ca}(\mathrm{OH})_2$ $\left[\mathrm{H}_3 \mathrm{O}^{+}\right]=$ _____ $\left[\mathrm{OH}^{-}\right]=$_____

Find the inverse matrix, if it exists. Round entries to the nearest hundredth. If the inverse matrix does not exist, write no inverse.

$$\left[\begin{array}{rr}2 & -4 \\ -3 & 6\end{array}\right]$$