Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why. $f(x)=2 x^{2}$; [-1, 1]

A student mixes 1.0 L of water at $40^{\circ} \mathrm{C}$ with 1.0 L of ethyl alcohol at $20^{\circ} \mathrm{C}.$ Assuming that no heat is lost to the container or the surroundings, what is the final temperature of the mixture?

A length of blood vessel is measured as 2.7 cm, with the radius measured as 0.7 cm. If each of these measurements could be off by 0.1 cm, estimate the maximum possible error in the volume of the vessel.

Determine whether the matrix is symmetric, skew-symmetric, or neither. $A$ square matrix is skew-symmetric when $A^T = −A$.

$$A=\begin{bmatrix} 0 & 2 & -1 \\ -2 & 0 & -3 \\ 1 & 3 & 0 \end{bmatrix}$$

When a 0.80-kg crown is submerged in water, its apparent weight is measured to be 7.3 N. Is the crown pure gold?

Cerebrosides are found in the brain and in the myelin sheath of nerve tissue. The structure of the cerebroside phrenosine. (a) What hexose is formed on hydrolysis of the glycoside bond of phrenosine? Is phrenosine an $\alpha$- or $\beta$-glycoside? (b) Hydrolysis of phrenosine gives, in addition to the hexose in part (a), a fatty acid called cerebronic acid, along with a third substance called sphingosine. Write structural formulas for both cerebronic acid and sphingosine.

A lamp filament radiates energy at a rate of 100 W when the temperature of the surroundings is $20^{\circ} \mathrm{C},$ and only 99.5 W when the surroundings are at $30^{\circ} \mathrm{C}.$ If the temperature of the filament is the same in each case, what is its temperature in Celsius?

Why should you indent the statements in the body of a loop?

Verify that each solution satisfies the differential equation.

$$\begin{matrix} \text{Differential Equation} & \text{Solutions}\\ \ {y^{\prime \prime}+9 y=0} & \ {\{\sin 3 x, \cos 3 x\}}\\ \end{matrix}$$

Find (a) the image of v and (b) the preimage of w for the linear transformation. $T: R^{2} \rightarrow R, T\left(v_{1}, v_{2}\right)=\left(2 v_{1}-v_{2}\right), v = (2, -3), w = 4$

Prove that if A is an $n \times n$ matrix that is idemp otent and invertible, then $A=I_{n}$.

Find the nth root of the matrix B. An nth root of a matrix B is a matrix A such that $A^n = B$.

$$B=\left[\begin{array}{rrr} {8} & {0} & {0} \\ {0} & {-1} & {0} \\ {0} & {0} & {27} \end{array}\right], \ n=3$$

Let $B=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \dots, \mathbf{v}_{n}\right\}$ be a basis for the vector space V, let B' be the standard basis, and consider the identity transformation $I: V \rightarrow V$. What can you say about the matrix for I relative to B? relative to B'? when the domain has the basis Band the range has the basis B'?

Discuss heat, work, and the change in internal energy of your body when you shovel snow.