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?

A converging lens with a focal length of 20 cm is used to produce an image on a screen that is 2.0 m from the lens. What are the object distance and the lateral magnification of the image?

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

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

A certain farsighted person has a near point of 50 cm. (a) Which type of contact lenses, (1) converging, (2) diverging, or (3) bifocal, should an optometrist prescribe to enable the person to see clearly objects as close as 25 cm? Explain. (b) What is the power of the lenses, in diopters?

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. $d y / d x=2 y, \quad d z / d x=x-z+3$

In general, the eigenvalues of an upper triangular matrix are given by the entries on the diagonal. The same is true for a lower triangular matrix. Verify this for $2 \times 2 \text { and } 3 \times 3$ matrices.

Indicate whether the statement is true or false. $\int e^{x} d x=\frac{e^{x+1}}{x+1}+C$.

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

Fenchel’s theorem states that if C is a simple, closed $C^1$ curve in $R^3$, then $K \geq 2 \pi$ and, moreover, $K=2 \pi$ if and only if C is a plane convex curve. (A simple, closed curve C is convex if the line segment joining any two points of C lies entirely in the region enclosed by C.) Verify Fenchel’s theorem for the ellipse $x^{2} / a^{2}+y^{2} / b^{2}=1$.

Find the derivative of the functions. $g(x)=\left(4 x^{2}+1\right)^{7}$

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

An upper triangular matrix is a square matrix with all zeros below its diagonal. A lower triangular matrix is a square matrix with all zeros above its diagonal. For all $3 \times 3$ upper triangular matrices U, verify that $U^{2}$ is also upper triangular.

Find the integrals. Check your answers by differentiation. $\int \frac{e^{t}}{e^{t}+1} d t$