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Chapter 12: Vectors and the Geometry of Space

Mon, 01 Jan 0001 00:00:00 +0000

12.1: Three-Dimensional Coordinate Systems #

Problem 3: Which of the points $A(-4, 0, -1)$, $B(3, 1, -5)$, and $C(2, 4, 6)$ is closest to the $yz$-plane? Which point lies in the $xz$-plane?

To find the distance between a point and a plane, we find the distance between that point and it’s projection to the plane in question. For example, $A$’s projection to the $yz$-plane is $(0, 0, -1)$ as the projection removes the $x$ term. Note now that the distance between these two is very simple, it is

Chapter 14: Partial Derivatives

Mon, 01 Jan 0001 00:00:00 +0000

14.1: Functions of Several Variables #

Problem 13: Find and sketch the domain of the function,

$$ f(x,y) = \sqrt{x - 2} + \sqrt{y - 1} $$

Each sqrt root cannot go negative, so $x - 2 \geq 0$ and $y - 1 \geq 0$, so the domain in set-builder notation is,

$$ \{ (x, y) \in \R^2 : x \geq 2 \ \text{and,} \ y \geq 1 \} $$

Chapter 15: Multiple Integrals

Mon, 01 Jan 0001 00:00:00 +0000

15.1: Double Integrals over Rectangles #

Problem 1: Define the solid that lies below the surface $f(x, y) = xy$ and above the rectangle

$$ R = {(x, y) : 0 \leq x \leq 6, 0 \leq y \leq 4 } $$

$(a)$ Estimate the volume by using a Riemann sum with $m = 3$, $n = 2$, where the sample point is the upper right corner of each square.