A lens with a power of +10 D is used as a simple microscope. (a) For the image of an object to be seen clearly, can the object be placed infinitely close to the lens, or is there a limit on how close it can be? Explain. (b) Calculate how close an object can be brought to the lens. (c) What is the angular magnification at this point?

Test for symmetry with respect to each axis and to the origin. $y=\left|x^{3}+x\right|$

Find any intercepts and test for symmetry. Then sketch the graph of the equation. $y=|6-x|$

Find each limit. $\lim _{x \rightarrow 3}\left(\frac{x^{2}}{x-3}-\frac{3 x}{x-3}\right)$

Let C be a piecewise $C^1$, simple, closed curve, oriented counterclockwise, enclosing a region D in the plane. Let n be the outward unit normal vector to D. If f (x, y) and g(x, y) are functions of class $C^2$ on D, establish Green’s first identity: $\iint_{D}\left(f \nabla^{2} g+\nabla f \cdot \nabla g\right) d A=\oint_{C} f \nabla g \cdot \mathbf{n} d s.$

Evaluate the integrals. $\int_{1 / 2}^{1} \frac{\sin ^{-1} x}{x^{2}} d x$

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. Assume a, b, A, B, $P_{0}$, h, and k are constants. $\int \frac{e^{x}}{2+e^{x}} d x$

When a man's face is in front of a concave mirror of radius 100 cm, the lateral magnification of the image is +1.5. What is the image distance?

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 $\theta=2, b=1, \varepsilon=1, \text { and } \mu=1.$

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

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

Verify Stokes’s Theorem by evaluating $\int_{C} \mathbf{F} \cdot d \mathbf{r}$ as a line integral and as a double integral. $\mathbf{F}(x, y, z)=x y z \mathbf{i}+y \mathbf{j}+z \mathbf{k}$, $S: 6 x+6 y+z=12$, first octant