Exam 12: Mathematical Problem Set

Use synthetic division to show that the number c=-2 is a lower bound for the zeros of $f(x)=x^{4}-3 x^{3}-4 x^{2}+8 x-2 .$ Be sure to reference any properties, theorems or tests used.

Find a polynomial of degree 3 with real number coefficients that has 3+i and -7 as zeros.

Sketch the region of feasible points and solve the linear programming problem. Maximize C=4 x-5 y , subject to: x\geq2 y\geq0 x+y\leq x+2y\geq8

For the function $f(x)=\frac{2\left(x^{2}-16\right)}{x^{2}-9}$ what are the intercepts and what are the asymptotes?

Which transformation was not performed on y=cos x to obtain $y=-4 \cos (2 x+\pi / 3) ?$

The graph of the polar equation $r=\frac{4}{3-4 \sin \theta}$ is a(n)

A Solve the system graphically. Round the answer to the nearest hundredth. $\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$ $y=x^{2}$

Solve the system graphically. Round the answer to the nearest hundredth. $\frac{x^{2}}{2}+\frac{y^{2}}{5}=1$ $y=\frac{1}{3} x$

What is the smallest interval for $\theta(0 \leq \theta \leq k)$ that gives a complete graph of the polar equation $r=4 \cos 5 \theta$ ? Sketch the graph to show which petals are generated by positive values of $r$ and which by negative values of $r$ .

Simplify: (csc x-tan x) sin x cos x=

The table shows average salaries for employees of Fairto Middlin Corporation. Find a linear regression equation for the data and graph the equation with a scatter plot of the data. (Use x=0 for 1975 , and express values to the nearest hundredth.) Use this data to estimate the average salary in 2012 . $\begin{array}{cc}\text { Year } & \text { Average Salary (\$) } \\\hline 1975 & 18,632 \\1980 & 19,563 \\1985 & 21,324 \\1990 & 22,817 \\1995 & 25,099 \\2000 & 26,103\end{array}$

The graph below shows the regression line for the data. The correlation coefficient is $\mathrm{r}=0.920$ . Is it correct to say this data show strong positive correlation?

The following three transformations are applied (in the given order) to the graph of $y=x^{2}:$ I. A vertical stretch by a factor of 3 II. A horizontal shift right 5 units III. A vertical shift down 6 units Write an equation for the graph produced as a result of applying these transformations.

Let $f(x)=\sqrt{x-3}$ and $g(x)=x^{2}+1$ Compute $g \circ f$ and state its domain in interval notation.

Determine the sum, if it exists, of the infinite geometric series $4+\frac{4}{3}+\frac{4}{9}+\frac{4}{27}+\ldots$

The graph of the function $x^{2}+y^{2}+20 x-16 y+80=0$ is a circle. What are the coordinates of the center and the length of the radius?

Write a proof by mathematical induction of the statement: 1³ + 2³ + 3³ + ... + $n^{3}=\frac{n^{2}(n+1)^{2}}{4}$

Show that the points A(1,-3), B(-8,1), C(-4,10) , and D(5,6) are the vertices of a square.

Solve the double inequality $-12<\frac{5 x-6}{2} \leq-7$ Express your answer in interval notation.

The graph below shows the regression line for the data. The correlation coefficient is $\mathrm{r}=0.947$ . Is it correct to say this data show strong positive correlation?