A detective finds a murder victim at 9 am. The temperature of the body is measured at $90.3^{\circ} \mathrm{F}$. One hour later, the temperature of the body is $89.0^{\circ} \mathrm{F}$. The temperature of the room has been maintained at ta constant $68^{\circ} \mathrm{F}$. Assuming the temperature, T, of the body obeys Newton's Law of Cooling, write a differential equation for T.

Calculate the derivatives of the given functions. Simplify your answers whenever possible. $z=\frac{t^{2}+2 t}{t^{2}-1}$

Find (a) A+B, (b) A-B, (c) 4A, and (d) 5A-2B.

$$A=\left[\begin{array}{rr} {1} & {-4} \\ {0} & {-1} \\ {4} & {2} \end{array}\right], B=\left[\begin{array}{rr} {6} & {5} \\ {3} & {7} \\ {-1} & {4} \end{array}\right]$$

Solve the equation for d when c = 0. 10c + 4d = 40

True or False. In using the Ratio Test, if $\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=L<1$, then the sum of the series $\sum_{k=1}^{\infty} a_{k}$ equals L.

Without graphing, determine whether the equation represents a linear or nonlinear function. Explain. y = 3x

$$\begin{aligned} &\frac{d S}{d t}=-a S I\\ &\frac{d I}{d t}=a S I-b I \end{aligned}$$

The parameter a in $dS/dt=-aSI$ in the same as the parameter a in $dI/dt=aSI-bI$ because people are moving from one group to the other.

The length of the rectangle is 3 meters more than the width. The perimeter is 26 meters. Write and solve a system of equations that represents this situation. What are the dimensions of the rectangle?

Find the derivative. Assume that a, b, c, and k are constant s. $y=t e^{-t^{2}}$

By calculating the lengths of its three sides, show that the triangle with vertices at the points A(2 , 1), 8(6, 4), and C(5, -3) is isosceles.

If

$$A=\left[\begin{array}{rrr} {0} & {1} & {0} \\ {2} & {0} & {2} \\ {0} & {1} & {0} \end{array}\right]$$

, prove that

$$e^{A}=\left[\begin{array}{ccc} {x^{2}} & {x y} & {y^{2}} \\ {2 x y} & {x^{2}+y^{2}} & {2 x y} \\ {y^{2}} & {x y} & {x^{2}} \end{array}\right]$$

where $x=\cosh 1$ and $y=\sinh 1$.

Calculate the derivatives of the given functions. Simplify your answers whenever possible. $F(x)=(3 x-2)(1-5 x)$

Water flows at a depth of 8 in. with a velocity of 35 ft/s in a rectangular channel. (a) Is the flow subcritical or supercritical? (b) What is the alternate depth?

A 6 in. pipe abruptly expands to a 12 in. size. If the discharge of water in the pipes is 5 cfs, what is the head loss due to abrupt expansion?