Show that $x_{1}(t) \text { and } x_{2}(t)$ are solutions to the initial-value problem $d \mathbf{x} / d t=A \mathbf{x} \text { with } \mathbf{x}(0)=\mathbf{x}_{0}.$

$$A=\left[\begin{array}{rr} {1} & {-1} \\ {0} & {1} \end{array}\right] \quad \mathbf{x}_{0}=\left[\begin{array}{l} {2} \\ {1} \end{array}\right]$$

$x_{1}(t)=2 e^{t}-t e^{t}, x_{2}(t)=e^{t}$

Evaluate the integrals. Remember to include a constant of integration with the indefinite integrals. Your answers may appear different from thos e in the Answers section but may still be correct. For example, evaluating I = f sin x cos x dx using the substitution u = sin x leads to $l=\frac{1}{2} \sin ^{2} x+C$; using u = cos x leads to $l=-\frac{1}{2} \cos ^{2} x+C$; and rewriting $I=\frac{1}{2} \int \sin (2 x) d x$ leads to $I=-\frac{1}{4} \cos (2 x)+C$. These answers are all equal except for differ ent choices for the constant of integration $C: \frac{1}{2} \sin ^{2} x=-\frac{1}{2} \cos ^{2}+\frac{1}{2}=-\frac{1}{4} \cos (2 x)+\frac{1}{4}$. You can always check your own answer to an indefinite integral by differenti ating it to get back to the integrand. This is often easier than comparing your answer with the answer in the back of the book. You may find integrals that you can 't do, but you should not make mistakes in those you can do because the answer is so easily checked . (This is a good thing to remember during tests and exams.) $\int \csc ^{5} x \cot ^{5} x d x$

(a) Explain why the maximum emf produced by an ac generator occurs when the flux through its armature coil is zero. (b) Explain why the emf produced by an ac generator is zero when the flux through its armature coil is at its maximum.

Specify whether each system is autonomous or nonautonomous, and whether it is linear or nonlinear. If it is linear, specify whether it is homogeneous or nonhomogeneous. $d y / d t=3 y z-2 z, \quad d z / d t=2 z+5 y$

Indicate whether the statement is true or false. If S(t) is an income stream, then S(t) has units of dollars.

Consumer resource models often have the following general form $R^{\prime}=f(R)-g(R, C) \quad C^{\prime}=\varepsilon g(R, C)-h(C)$ where R is the number of individuals of the resource and C is the number of consumers. The function $f(R)$ gives the rate of replenishment of the resource, g(R, C) describes the rate of consumption of the resource, and h(C) is the rate of loss of the consumer. The constant $\varepsilon,$ where $0<\varepsilon<1,$ is the conversion efficiency of resources into consumers. Find all equilibria of the following examples and determine their stability properties. A chemostat is an experimental consumer-resource system. If the resource is not self-reproducing, then it can be modeled by choosing $f(R)=\theta, g(R, C)=b R C,$ and $h(C)=\mu C.$ Suppose $\theta=2, b=1, \varepsilon=1, \text { and } \mu=1.$

For some trigonometric expressions, multiplying by a conjugate helps to simplify the expression. Simplify $\frac{\sin \theta}{1+\cos \theta}$ by multiplying the numerator and the denominator by the conjugate of the denominator, $1-\cos \theta$. Describe how this process helps to simplify the expression.

In a Young's double-slit experiment using monochromatic light, if the slit spacing d increases, the interference maxima spacing will (a) decrease, (b) increase, (c) remain unchanged, (d) disappear.

A sound wave cannot be polarized. This is because sound is (a) a transverse wave, (b) a longitudinal wave, (c) none of the preceding.

Find r'(t). $\mathbf{r}(t)=3 \cos ^{3} t \mathbf{i}+2 \sin ^{3} t \mathbf{j}+\mathbf{k}$

The Casparian strip affects a. how water and minerals move into the vascular cylinder.b. vascular tissue composition.c. how soil particles function.d. cation exchange.e. Both a and c are correct.

Find the derivative of each function. $f(x)=\frac{x^{2}}{e^{x}}$

Find an antiderivative. $p(x)=x^{2}-6 x+17$

Solve the equation. Approximate the result to three decimal places. $2 \log _{3} 4 x=15$