Find the open intervals where $f$ is increasing or decreasing.

$$f(x) = \ln |x^2 - 1|$$

For the 3d state of hydrogen, at what radius is the electron probability a maximum? Compare your answer with the radius of the Bohr orbit for (n=3).

Fill in the blanks with correct vocabulary or words. Nonpolar solutes in a solution are called ________.

Evaluate the integral.

$$\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x ^ { 2 } } \int _ { 0 } ^ { x + y } ( 2 x - y - z ) d z\ d y\ d x$$

Lanni Products is a start-up computer software development firm. It currently owns computer equipment worth $30,000 and has cash on hand of$20,000 contributed by Lanni’s owners. For each of the following transactions, identify the real and/or financial assets that trade hands. Are any financial assets created or destroyed in the transaction? Lanni takes out a bank loan. It receives $50,000 in cash and signs a note promising to pay back the loan over three years.

Prove that the set is a subspace of the appropriate $R^n$.

$$\begin{Bmatrix}\begin{bmatrix}u_1\\u_2\\u_3\\u_4\end{bmatrix}\in R^4:u_1+5u_3=0\text{ and }4u_2-3u_4=0\end{Bmatrix}$$

Which of the following individuals would best represent our understanding of fluid versus crystallized intelligence? a. James is 80 and has just solved a math equation that has been puzzling him for the last 40 years. b. Luis is 22 and has written several successful novels. c. Alice is 25 and has discovered a new chemical element. d. As a high school student, Alex changed the way people, thought about a local homeless man by doing research into his life e. After watching the sky for 50 years Nate finally discovered a new planet.

How might a geographer classify economic activities that take place at your school? Explain.

If A and B denote arbitrary subsets of a metric space M, prove that: $\operatorname{int}(M-A)=M-\overline{A}$.

Write each expression in the standard form a + bi. (-8 + 4i) - (2 - 2i)

Find the matrix products.

$$\left[\begin{array}{ccc}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right]\left[\begin{array}{rrr}{3} & {-4} & {0} \\ {1} & {2} & {-5} \\ {6} & {-3} & {-1}\end{array}\right]$$

What is mental age?

Use the QR algorithm to find the eigenvalues. In each matrix, all eigenvalues have equal magnitude, so Hessenberg may be needed. Compare the results of QR algorithm before and after reduction to Hessenberg form. (a)

$$\left[ \begin{array} { r r r r } { - 5 } & { - 10 } & { - 10 } & { 5 } \\ { 4 } & { 16 } & { 11 } & { - 8 } \\ { 12 } & { 13 } & { 8 } & { - 4 } \\ { 22 } & { 48 } & { 28 } & { - 19 } \end{array} \right]$$

, (b)

$$\left[ \begin{array} { r r r r } { 7 } & { 6 } & { 6 } & { - 3 } \\ { - 26 } & { - 20 } & { - 19 } & { 10 } \\ { 0 } & { - 1 } & { 0 } & { 0 } \\ { - 36 } & { - 28 } & { - 24 } & { 13 } \end{array} \right]$$

, (c)

$$\left[ \begin{array} { r r r r } { 13 } & { 10 } & { 10 } & { - 5 } \\ { - 20 } & { - 16 } & { - 15 } & { 8 } \\ { - 12 } & { - 9 } & { - 8 } & { 4 } \\ { - 30 } & { - 24 } & { - 20 } & { 11 } \end{array} \right]$$

Solve each equation by any method. $\frac { 5 } { x - 1 } - \frac { 3 x } { x + 1 } = \frac { 9 } { x ^ { 2 } - 1 }$