Suppose a microscope's resolution is diffraction limited. Which one of the following changes would provide the greatest improvement to its resolution? Observing at a longer wavelength through a smaller aperture.

Find each limit. Note that L'Hopital's rule does not apply to every problem, and some problems will require more than one application of L'Hopital's rule. Find $\lim _{x \rightarrow 0^{+}}(x \ln x)$.

A person in a boat floating in a small pond throws an anchor overboard. What happens to the level of the pond? (a) It rises. (b) It falls. (c) It remains the same.

Solve each equation. $4(-2 x+1)=6-(2 x-4)$

Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of y=f(x). $f(x)=e^{-2 x^{2}}$

Determine if $p^2+14p+49$ is a perfect square trinomial. Justify your answer.

Rewrite each expression in terms with no power greater than 1. $\frac{\sin ^{4} \theta}{\cos ^{2} \theta}$

From the following data for a PPP project, calculate the (a) conventional, and (b) modified benefit/cost ratios using an interest rate of 6% per year and an infinite project period.

$$\begin{array}{cccc} & \text { To the People } & \text { To the Government } \\ \hline \text { Benefits: } & \$ 100,000 \text { per year } & \text { Costs: } \$ 1.8 \text { million now } \\ & \text { beginning now } & \text { and } \$ 200,000 \\ & & \text { every } 3 \text { years } \\ \text { Disbenefits: } & \$ 60,000 \text { per year } & \text { Savings: } \$ 90,000 \text { per year } \\ \hline \end{array}$$

(A) Find the average value of each function over the indicated interval. (B) Use a graphing calculator to graph the function and its average value over the indicated interval in the same viewing window. $g(x)=\sqrt{x+1} ;[3,8]$

If $\hat{\theta}$ is sufficient for $\theta,$ show that any one-to-one function of $\hat{\theta}$ is also sufficient for $\theta$.

Use properties of exponents to write each function in the form $f(t)=k a^{t}$, where k is a constant. $f(t)=\left(\frac{1}{3}\right)^{1-2 t}$

Invent two different tile patterns that grow by adding 4 every time but have different $x \rightarrow y$ tables. Draw Figures 0, 1, 2, and 3 and find rules for each of your patterns. What is different about your rules? What is the same?

Jack has a tree in his backyard that he wants to cut down to ground level. He needs to know how tall the tree is, because when he cuts it, it will fall toward his fence. Jack measured the tree's shadow, and it measured 20 feet long. At the same time, Jack's shadow was 12 feet long. Jack is 5 feet tall. a. How tall is the tree? b. Will the tree hit the fence if the fence is 9 feet away?

At sea level, the weight of the atmosphere exerts a pressure of 14.7 pounds per square inch, commonly referred to as 1 atmosphere of pressure. As an object descends in water, pressure P and depth d are linearly related. In salt water, the pressure at a depth of 33 ft is 2 atms, or 29.4 pounds per square inch. (A) Find a linear model that relates pressure P (in pounds per square inch) to depth d (in feet). (B) Interpret the slope of the model. (C) Find the pressure at a depth of 50 ft. (D) Find the depth at which the pressure is 4 atms.