The grand unified theory would reduce the number of fundamental forces to (a) one, (b) two, (c) three, (d) four.

Find the intervals on which the functions f(x). $f(x)=x^{2}+4 x+3$

Find the open intervals on which the function is increasing or decreasing. $f(x)=(x-1)^{2}(2 x-5)$

An instructor gives regular 20-point quizzes and 100-point exams in a mathematics course. Average scores for six students, given as ordered pairs (x, y), where x is the average quiz score and y is the average exam score, are (18, 87), (10, 55), (19, 96), (16, 79), (13, 76), and (15, 82). (a) Use the regression capabilities of a graphing utility to find the least squares regression line for the data. (b) Use a graphing utility to plot the points and graph the regression line in the same viewing window. (c) Use the regression line to predict the average exam score for a student with an average quiz score of 17. (d) Interpret the meaning of the slope of the regression line. (e) The instructor adds 4 points to the average exam score of everyone in the class. Describe the changes in the positions of the plotted points and the change in the equation of the line.

Consider the tangent line j to the graph of f at any point (a, b) and the tangent line k to the graph of $f^{-1}$ at the point (b, a). Identify the line where the lines j and k intersect. Explain your reasoning.

(a) If the lifetime of excited state A is longer than that of another excited state B, then the width of a spectral line due to natural broadening for a transition from state A to the ground state will be (1) smaller than, (2) the same as, (3) greater than that for a transition from state B to the ground state. Why? (b) Calculate the ratio of the width of a spectral line due to natural broadening for a transition from an excited state with a lifetime of $10^{-12}\ \mathrm{s}$ to the ground state to that from a state with a lifetime of $10^{-8}\ \mathrm{s}$ to the ground state.

Show that the distance between the point $\left(x_{1}, y_{1}\right)$ and the line Ax+By+C=0 is $\text { Distance }=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}$.

Show that the functions f are one-to-one, and calculate the inverse functions $f^{-1}$. Specify the domains and ranges of f and $f^{-1}$. $f(x)=x^{2}, \quad x \leq 0$

Prove that $\ln (x / y)=\ln x-\ln y$, x>0, y>0.

Find the derivative of the function. $y=\log _{10} 2 x$

Mira uses algebra to rewrite the function $y=\frac{2-3 x}{x-7}$ in an equivalent form that she can graph by hand.

$$\begin{array}{l}{y=\frac{2-3 x}{x-7}} \\ {y=\frac{-3 x+2}{x-7}} \\ {y=\frac{-3 x-21+21+2}{x-7}} \\ {y=\frac{-3(x-7)+23}{x-7}} \\ {y=\frac{-3(x-7)}{x-7}+\frac{23}{x-7}} \\ {y=-3+\frac{23}{x-7}} \\ {y=\frac{23}{x-7}-3}\end{array}$$

a) Identify and correct any errors in Mira’s work. b) How might Mira have discovered that she had made an error without using technology? How might she have done so with technology?

The minimum angle of resolution of the diffraction patterns of two identical monochromatic point sources in a single-slit diffraction pattern is 0.0065 rad. If a slit width of 0.10 mm is used, what is the wavelength of the sources?

A ball is thrown vertically upward from a height of 6 ft with an initial velocity of 80 ft/s. The signed distance s (in feet) of the ball from the ground after t seconds is given by the position function $s=s(t)=6+80 t-16 t^{2}$. (a) What is the velocity of the ball after 2 s? (b) When will the ball reach its maximum height? (c) What is the maximum height the ball reaches? (d) What is the acceleration of the ball at any time t? (e) How long is the ball in the air? (f) What is the velocity of the ball upon impact with the ground? What is its speed? (g) What is the total distance traveled by the ball?

Find all functions M(x, y) such that the vector field $\mathbf{F}=M(x, y) \mathbf{i}+(x \sin y-y \cos x) \mathbf{j}$ is conservative.