At t = 0, a battery is connected to a series arrangement of a resistor and an inductor. If the inductive time constant is 37.0 ms, at what time is the rate at which energy is dissipated in the resistor equal to the rate at which energy is stored in the inductor's magnetic field?

Let $T:R^3\to R^2$ and $U:R^2\to R^3$ defined by

$$\begin{equation*}T\begin{pmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}\end{pmatrix}= \begin{bmatrix}2x_1+x_3\\x_1+x_2-x_3\end{bmatrix}\end{equation*}$$

and

$$\begin{equation*}U\begin{pmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}\end{pmatrix}= \begin{bmatrix}3x_1-x_2\\x_2\\x_1+x_2\end{bmatrix}\end{equation*}$$

Determine the matrices A and B of T and U, respectively.

In which blood vessel is arterial blood pressure commonly measured?

What will the following code display?

$$\begin{matrix} \text{dct = {'Monday':1, 'Tuesday':2, 'Wednesday':3}}\\ \text{print(dct['Tuesday'])}\\ \end{matrix}$$

Let $A$ be an $n\times n$ symmetric matrix with spectral decomposition $A=\lambda_1P_1+ \lambda_2P_2+\dots+\lambda_nP_n$. Assume that $\mu_1,\mu_2,\dots,\mu_k$ are all the distinct eigenvalues of $A$ and that $Q_j$ denotes the sum of all the $P_i$'s that are associated with $\mu_j$. Prove $Q_jQ_j=Q_j$ for all $j$, and $Q_iQ_j=O$ if $i\neq j$.

At t = 0, a battery is connected to a series arrangement of a resistor and an inductor. If the inductive time constant is 60.0 ms, at what time is the rate at which energy is dissipated in the resistor equal to the rate at which energy is stored in the inductor's magnetic field?

In an oscillating LC circuit, L = 3.00 mH and

$$C = 3.90 \mu F$$

. At t = 0 the charge on the capacitor is zero and the current is 1.75 A. (a) What is the maximum charge that will appear on the capacitor? (b) At what earliest time t > 0 is the rate at which energy is stored in the capacitor greatest, and (c) what is that greatest rate?

In an oscillating $L C$ circuit, $L=8.00 \mathrm{~mH}$ and $C=1.40 ~\mu \mathrm{F}$. At time $t=0$, the current is maximum at $12.0 \mathrm{~mA}$. At what earliest time $t>0$ is the rate of change of energy in the capacitor maximum?

In an oscillating LC circuit, L = 3.00 mH and C = 2.70 $\mu$F. At t = 0 the charge on the capacitor is zero and the current is 2.00 A. At what earliest time $t > 0$ is the rate at which energy is stored in the capacitor greatest?

Decide whether the statement is true or false. Give an explanation for your answer. If f(x) < g(x) for all x and a < b, then $\int _ { a } ^ { b } f ( x ) d x < \int _ { a } ^ { b } g ( x ) d x.$

A market research firm used a sample of individuals to rate the purchase potential of a particular product before and after the individuals saw a new television commercial about the product. The purchase potential ratings were based on a 0 to 10 scale, with higher values indicating a higher purchase potential. The null hypothesis stated that the mean rating “after” would be less than or equal to the mean rating “before.” Rejection of this hypothesis would show that the commercial improved the mean purchase potential rating. Use $\alpha =$ .05 and the following data to test the hypothesis and comment on the value of the commercial.

$$\begin{matrix} \text{ } & \text{Purchase} & \text{Rating} & \text{ } & \text{Purchase} & \text{Rating}\\ \text{Individual} & \text{After} & \text{Before} & \text{Individual} & \text{After} & \text{Before}\\ \text{1} & \text{6} & \text{5} & \text{5} & \text{3} & \text{5}\\ \text{2} & \text{6} & \text{4} & \text{6} & \text{9} & \text{8}\\ \text{3} & \text{7} & \text{7} & \text{7} & \text{7} & \text{5}\\ \text{4} & \text{4} & \text{3} & \text{8} & \text{6} & \text{6}\\ \end{matrix}$$

Suppose A is an

$$m \times n$$

matrix and C is an

$$r \times s$$

matrix. (a) What must the dimensions of a matrix B be in order for the product ABC to be defined? (b) Write an expression for the (i, j) element of ABC in terms of the elements of A, B, and C.

The function is one-to-one. Find the inverse of the function and express it using

$$f ^ { - 1 } ( x )$$

notation.

$$f ( x ) = \sqrt [ 3 ] { x }$$

Explain why both autotrophs and heterotrophs depend on photosynthesis to obtain the energy they need for life processes.