(a) Let $f :\mathbb{R}^3 \to \mathbb{R}, \mathbf{x}_0 \in \mathbb{R}^3$. If $\mathbf{v}$ is a unit vector in $\mathbb{R}^3$, show that the maximum value of the directional derivative of $f$ at $\mathbf{x}_0$ along $\mathbf{v}$ is $||\nabla f (\mathbf{x}_0)||$.

Suggest an explanation for why aspirin has a sour taste.

a. For what values of a and b will

$$g(x)=\left\{\begin{array}{ll}{a x+b,} & {x \leq-1} \\ {a x^{3}+x+2 b,} & {x>-1}\end{array}\right.$$

be differentiable for all values of x? b. Discuss the geometry of the resulting graph of g.

Investigate the services and fees of two banks. Bank 1 has a branch in your neighborhood or town, and Bank 2 is an online bank. Make a multimedia presentation outlining their services and fees and present it to the class.

How did Stalin gain and maintain power in the USSR?

Evaluate the following limits or state that they do not exist.

$$\lim _ { x \rightarrow 0 } \frac { 3 \sec ^ { 5 } x } { x ^ { 2 } + 4 }$$

Sharks find their prey through a keen sense of smell and an ability to detect small electrical impulses. If f(x, y, z) indicates the electrical charge in the water at position (x, y, z) and a shark senses that $\nabla f=\langle 12,-20,5\rangle$, in which direction should the shark swim to find its prey?

William Tell is said to have shot an apple off his son 's head with an arrow. If the arrow was shot with an initial speed of $55\ \mathrm{m/ s}$ and the boy was $15\ \mathrm{m}$ away , at what launch angle did Bill aim the arrow? (Assume that the arrow and apple are initially at the same height above the ground.)

Rank the following oxides in order of increasing aqueous acidity, Ga2O3, Al2O3, ln2O3.

Name three groups of birds, and describe some of their characteristics.

Find both real solutions to the equation $x^4 = 81$.

In a study of speed dating conducted at Columbia University, female subjects were asked to rate the attractiveness of their male dates, and a sample of the results is listed below (1 = not attractive; 10 = extremely attractive). Use the sign test to test the claim that the sample is from a population with a median equal to 5.

$$\begin{matrix} \text{5} & \text{8} & \text{3} & \text{8} & \text{6} & \text{10} & \text{3} & \text{7} & \text{9} & \text{8} & \text{5} & \text{5} & \text{6} & \text{8}& \text{8} & \text{7} & \text{3} & \text{5} & \text{5} & \text{6} & \text{8} & \text{7} & \text{8} & \text{8} & \text{8} & \text{7}\\ \end{matrix}$$

Find the velocity vector of a speed boat moving parallel to: 2i + j with a speed of 50 km $\mathrm{h}^{-1}$.

Simplify.

$$z + \frac { 1 } { z }$$