PAPER HOMEWORK 1
21-259: Calculus in Three Dimensions, Spring 2015
Due: Wednesday, January 21st, in lecture
1. In R3 we will encounter the following type of objects: points (0dimensional), curves (1-dimensional), surfaces (2-dimensional), and R3 itself.
F
CALCULUS IN THREE DIMENSIONS (21-259)
SPRING 2015
Instructor:
Dr. Dana Mihai, 8128 Wean Hall
dmihai@andrew.cmu.edu
Lecture:
Lecture 1: MWF, DH A302, 12:30-1:20 pm
Lecture 2: MWF, BH A51, 9:30-10:20 am
Course website:
http:/www.cmu.edu/blackboard
Office ho
Review 12.2: Vectors in R2
Geometrically, a vector can be represented by an arrow with a specied
direction and length (magnitude). Notation: !, or v
v
Algebraically, a vector in R2 can be represented by an ordered pair of
real numbers: v = ha; bi :
The le
1. You are using a jet pilot dogght simulator. Your monitor tells you that two missiles
have honed in on your plane along the directions (3,5,2) and (1,3,-2). In what direction
should you turn to have the maximum chance of avoiding both missiles?
2. Consi
(PG 13: for some reference to violence and use of weapons)
1. A superhero stands on the top of a building and wants to shoot a laser gun at his evil
nemesis, who is standing on the corner of the street unaware of the danger hes in. The
computer programmer
13.1: Vector Functions and Space Curves
Vector Valued Function: rule that assigns to each element in its
domain a vector in its range:
v : R ! Rn
v(t) = hv1 (t); :; vn (t)i
so v(t) is a vector with n components.
If n = 3, v(t) = hf (t); g(t); h(t)i, where
Cylinders
0 Let C be a planar curve (generating curve)
0 Lettr be a line not in the plane of the curve (axis of the cylinder) (note: we only
consider axes parallel to the coordinate axes)
0 Cylinder: the set of all lines parallel to I (called rulings) and
13.3: Arc Length and Curvature
Review questions:
- what is the formula for the length of the curve with vector equation
r(t), a t b
- what is the formula for the arclength s(t) of a curve with vector equation
r(t)
- what is s0 (t) equal to? [Use the answe
12.1: Three-dimensional coordinate system
R3 : f(x; y; z) j x, y, z are real numbersg
Read: "R three: the set of all ordered triples (x; y; z) such that x,
y and z are real numbers"
Geometric representation of points
Cartesian coordinate system
x-axis:
Review 12.1:
R2
R2 : f(x; y) j x and y are real numbersg
Read: "R two: the set of all ordered pairs (x; y) such that x and
y are real numbers"
In R2 it makes no sense to talk about a plane, because R2 has
no knowledge of being this is a concept that mak
14.2: Limits and Continuity
Review questions:
- What does
lim
(x;y)!(a;b)
f (x; y) = L
mean? How can you show that such a limit doesn exist? How can you show
t
that a limit exists? (methods)
- True or false: If f (x; y) ! L as (x; y) ! (a; b) along every
14.5: The Chain Rule
Review questions:
- State the Chain Rule (write formulas) for the case z = f (x; y) and x
and y are functions of one variable, t. What if x and y are functions of two
variables, u and v?
- If z is dened implicitly as a function of x a
Learning goals for section 14.2 (Limits and Continuity):
At the end of this lecture you should be able to:
- nd the limit of a function of two variables if it exists, by either plugging
in the coordinates of the point, by simplifying the function, or by u
Learning goals for section 14.3: Partial Derivatives
At the end of this section you should be able to:
- nd rst partial derivatives of given functions (evaluate them at given
points): 11, 12, 15-40, 41-44, 81-83, 90, 96
- interpret the values obtained for
Sowhowé BLUE. Bum
NAME:
SECTION:
I have not received any help during this exam from unauthorized sources and the work '
submitted in this exam is entirely my own.
Signature: '
MATH 21-259, CALCULUS IN 3D
Spring 2015, February 9th
Midterm 1
Total: 100 poi
134 "3 Arctength formula
Arclength Formula for a curve y = x) or 70) in R2:
0 Partition [a,b] into n subintervals:
a=to<t1<.<ln=b
o The arciength fort e [1}, n+1] is:
Li = IAl'il = Il'(li+1) r(t:)i,
where r(r,-) -= (x(fi)=y(lz-) = (165%)-
Ari = it-+1) '
Learning goals for section 14.6: Directional Derivative and
the Gradient Vector
At the end of this section you should be able to:
- calculate and evaluate gradients of functions of several variables
- nd the directional derivative of a given function at a
Learning goals for section 14.5: The Chain Rule
At the end of this section you should be able to:
- use the chain rule and tree diagrams to calculate and evaluate total and
partial derivatives of functions of several variables: 1-6, 7-12, 13-14, 17-20,
21
Learning goals for section 12.5 (Equations of Lines and Planes):
At the end of this lecture you should be able to:
- nd vector equations, parametric equations, symmetric equations of
lines in R3 based on various information given: 2-5, 6-12, 15-16, 57-58,
Learning goals for section 12.6 (Cylinders and Quadrics):
At the end of this lecture you should be able to:
- identify a given surface as a plane, cylinder or quadric
- distinguish between a curve in R2 and a cylinder in R3 : 1, 2
- describe and sketch a
Learning goals for section 13.2 (Vector Functions and Space
Curves):
At the end of this lecture you should be able to:
- sketch plane curves and derivatives of vector function: 3-8
- nd intersections of curves, of curves and surfaces, angles of intersecti
Learning goals for section 13.3: Arc Length and Curvature
At the end of this lecture you should be able to:
- nd lengths of curves by using the arc length formula: 1-6, 11, 12, 15
- reparameterize curves in terms of the arc length: 13, 14, 16
- nd unit ta
Learning goals for section 12.4 (The Cross Product):
At the end of this lecture you should:
- know how to calculate the cross product of two vectors given on components: 1-7, 17-18
- determine whether expressions involving the dot and the cross product
ar
Learning goals for section 12.3 (The Dot Product):
At the end of this lecture you should:
- know how to calculate the dot product of two vectors by using both the
algebraic and the geometric formula: 2-10, 11-12, 13
- know that the dot product is dened on