For each vector function r=r(t), find the unit tangent vector T and the principal unit normal vector N at t. $\mathbf{r}(t)=e^{t} \mathbf{i}+e^{-t} \mathbf{j} \quad t=0$

Given that $\mathbf{u}$ is a unit vector, prove that each vector a can be written as the sum of a vector parallel to $\mathbf{u}$ and a vector perpendicular to $\mathbf{u}$ :

$$\mathbf{a}=\underbrace{(\mathbf{a} \cdot \mathbf{u}) \mathbf{u}}_{\text {parallel to } \mathbf{u}}+\underbrace{(\mathbf{u} \times \mathbf{a}) \times \mathbf{u}}_{\text {perpendicular to } \mathbf{u}}$$

HINT: Use Exercise 42(b).

A manufacturer sells watches for $\$ 50$ per unit. The fixed cost related to this product is $\$ 10,000$ per month, and the variable cost per unit is $\$ 30$.

(a) Write the equations of the revenue and cost functions.

(b) How many watches must be sold to break even?

Mark the following true or false.

Every subgroup of an abelian group G is a normal subgroup of G.

A firm determines that x units of its product can be sold daily at p dollars per unit, where

$$x=1000-p$$

The cost of producing $x$ units per day is

$$C(x)=3000+20 x$$

What price per unit must be charged to obtain the maximum profit?

Ahard-thrown tennis ball is moving horizontally when it bangs into a vertical concrete wall at perpendicular incidence. The ball rebounds straight back off the wall. Neglect any effects due to gravity for the small time interval described here. Assume that toward the wall is the +x direction. What are the directions of its velocity and acceleration

$(c)$ just after leaving the wall?

The demand function for a brand of blank digital camcorder tapes is given by $p=-0.01 x^{2}-0.2 x+23$ where p is the unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Determine the consumers’ surplus if the unit price is $8/tape.

A meteorologist is assembling a weather balloon. The balloon's envelope has a volume of $60000 \mathrm{~L}$ and a mass of $4.0 \mathrm{~kg}$. The balloon must lift $20.0 \mathrm{~kg}$ of instruments in addition to its own mass. When properly inflated, the balloon must have a pressure of $90 \mathrm{kPa}$ at a temperature of $15^{\circ} \mathrm{C}$. The mass that the balloon can lift is the mass of the air it displaces minus the mass of the gas inside its envelope. The meteorologist wants to use methane, $\mathrm{CH}_4(\mathrm{~g})$, to fill the balloon. His colleague, however, believes that helium, despite its greater cost, is the only gas that will work. Assuming that the average molar mass of air is $28.90 \mathrm{~g} / \mathrm{mol}$, which gas is better: methane or helium? Show calculations to support your answer.

An air-track glider attached to a spring oscillates between the 10 cm mark and the 60 cm mark on the track. The glider completes 10 oscillations in 33 s. What are the (a) period, (b) frequency, (c) angular frequency, (d) amplitude, and (e) maximum speed of the glider?

A manufacturer has determined that the total cost $C$ of operating a factory is $C=0.5 x^2+15 x+5000$, where $x$ is the number of units produced. At what level of production will the average cost per unit be minimized? (The average cost per unit is $C / x$.)

determine whether u and v are orthogonal, parallel, or neither. u = 2i - 3j + k , v = -i - j - k

Carter plans to wallpaper the longest wall in his living room. The wall is twice as long as it is high and has an area of 162 square feet. Find the height of the wall.

A 8 feet

B 9 feet

C 12 feet

D 18 feet

A firm determines that x units of its product can be sold daily at p dollars per unit, where x=1000-p. The cost of producing x units per day is C(x)=3000+20x. What price per unit must be charged to obtain the maximum profit?

Samuelson and Solow reasoned that when aggregate demand was low, unemployment was

a. high, so there was upward pressure on wages and prices.

b. high, so there was downward pressure on wages and prices.

c. low, so there was upward pressure on wages and prices.

d. low, so there was downward pressure on wages and prices