9.1 - Review / Preview

[9.1] - Review / Preview

We started in on section 9.1 in Schlicker's "Active Calculus". With what we discussed about the distance formula, you should be able to answer questions 1 and 2 below.

Now Read section 9.1.4 and 9.1.5 on Traces and Contour Maps/Level curves. Then complete question 3.

Consider the point $P=(x,y,z)$ [I'm using an older diagram here. Just translate these parameters in the diagram to this notation: $a\to x; b\to y; c\to z$.]
1. What are the coordinates of the closest point to $P$ on the $\color{red}y$-axis? And what is the distance between these two points?
  Closest point, let's call it $W$, has the same $y$ coordinate. But to be on the $y$ axis $x=0$ and $z=0$, so $W=(0,y,0)$
  
  Distance $PW$ is $\sqrt{(x-0)^2 + (y-y)^2 +(z-0)^2}=$ $\sqrt{x^2+z^2}$ .
2. What are the coordinates of the closest point to $P$ on the $\color{red}xz$ plane? And what is the distance between these two points?
  The $xz$ plane means $y=0$. Closest point is $S=(x,0,z)$ .
  
  The distance $PS$ is $\sqrt{(x-x)^2+(y-0)^2+(z-z)^2}=$ $|y|$ (absolute value of $y$).
The work above should help you answer Schlicker's question 8 [in section 9.1]. Record your answers to question 8 below.
With what we did above, we can answer part b) first: The points whose distance from the $y$-axis equals the distance from the $xz$ plane means... $$\begineq \sqrt{x^2+z^2} = |y|\\ x^2+z^2 = y^2\endeq$$ This is the first equation in b.)

a.) The term on the left represents a circle in the $xz$ plane centered on $(x,0,z)=(0,0)$ (which is where the $y$ axis is...) and the term on the right means that the radius is $y$. So as the $y$ coordinate grows, the radius of the circle grows too. So this is A cone opening along the $y$-axis.
Answer Schlicker's question 13 about $z=6x^2-2y^2+5$, and record your answers below. [Hint: You can use Geogebra to graph an equation and a plane together.]
1. Upward parabola? $y=$constant. e.g. $y=0$.
2. Downward parabola: E.g.$x=$0.
3. Intersecting straight lines: $z=5$.