Find the equation of the normal line to the curve y = −3x$^2$+x at the point (−1, −4).

Rod $AB$ moves over a small wheel at $C$ while end $A$ moves to the right with a constant velocity $\mathbf{v}_A$. Using the method of Sec. 15.9, derive expressions for the horizontal and vertical components of the velocity of point $B$.

Find the area under the standard normal distribution curve.
Between $z=-0.96$ and $z=-0.36$

Find a normal vector to the level curve f(x, y) = c at P. f(x, y) = x² + y² c = 25, P(3, 4)

Formulate Points $G(7,1)$ and $H(-1,4)$ lie on the coordinate plane.
Think of the $x$-coordinates 7 and $-1$ as points on a number line. What is the midpoint of the segment connecting them? What is the midpoint of the segment connecting $y$-coordinates 1 and 4 ? What is the midpoint of $\overline{G H}$ ?

What occurs when carbonic acid reacts with calciterich rocks such as limestone?

Provide an economic use for each mineral listed below:

$$\begin{matrix} \text{Galena:} & \text{\_\_\_\_\_}\\ \text{Hematite:} & \text{\_\_\_\_\_}\\ \text{Graphite:} & \text{ \_\_\_\_\_}\\ \text{Sphalerite:} & \text{\_\_\_\_\_}\\ \text{Gypsum:} & \text{\_\_\_\_\_}\\ \text{Calcite:} & \text{\_\_\_\_\_}\\ \end{matrix}$$

Show the conjecture is false by finding a counterexample. If the product of two numbers is positive, then the two numbers must both be positive.

Tell whether the difference between the two positive integers is always, sometimes, or never positive. Explain your reasoning.

The ocean basin (a) is covered by undersea mountains. (b) is mostly flat but features mountains and trenches. $(c)$ has not been mapped sufficiently to be understood. (d) is an area of low pressure.

A particle moves along the curve of intersection of the surfaces $y = x^2$ and $z = 2x^3 / 3$ with constant speed $v = 6$. It is moving in the direction of increasing $x$. Find its velocity and acceleration when it is at the point $(1, 1, 2/ 3)$.

Let y = e˟+eˉ˟/2. Find an equation of the line normal to the curve at x = 1.

Let a be a positive constant. Without solving the equation, explain why the solution x of the equation a x= 4 a+3 must be positive.

An airplane at a constant altitude a flies a horizontal distance d toward you at velocity v. You observe for time t and measure Its angles of elevation $\angle E_{1}$ and $\angle E_{2}$ at the start and end of your observation. Find the missing information. $a=2 \mathrm{mi}, v=\square \mathrm{mi} / \mathrm{min}, t=15 \mathrm{s}, m \angle E_{1}=40, m \angle E_{2}=50$