NAME: JIUSHU HE
MATLAB ASSIGNMENT 1
the point C is (month, day) where month and day
are the month and day of your birthday (December 27th).
1. Use MATLAB to draw the vectors
. The three
vectors must be placed at the point A.
Math 280 Review Problems
1. Find the equation for the plane that contains the point P(2,1,1) and also contains the line with
parametric equations x = 1 + 3t, y = 2t , z = 2 t.
2. Find an equation for the plane that contains the points P(3,2,-1), Q(4,1,1),
For functions of one variable, the derivative gives the rate of change of y with respect to x, which
is also the slope of the tangent line at a given point. For a function of two variables, it will be
difficult to talk about rates of change o
Quiz 5 solutions
1. z f (x, y) is defined implicitly by the equation 2x 2 y y 2 z3 xz 2 .
using implicit differentiation.
Differentiate, treating x as a variable, y as a constant, and z as a function of x.
In 2, we have used two coordinate systems, rectangular (or Cartesian) coordinates and polar
coordinates, to describe the location of points and sets of points. You can review polar
coordinates, if necessary, in Appendix H in the back of your b
The other type of vector product is called the cross product or vector product. a b is a vector
orthogonal to both a and b.
From a geometric viewpoint, if a and b are non-zero vectors in V3 that form angle , 0 ,
then a b a b sin n with n a uni
Test 2 solutions
1. For rt t 2 , t 3 3 , 2t , find the following: (See 10.2-10.3 for similar problems.)
rt 2t, t 2 , 2
b) the tangent vector to the curve at t = 3
r3 6,9, 2 You could give the unit
From calculus, we have a formula for the length of a parametrically defined curve in the plane:
We can expand this to curves in space defined by parametric equations (or a vector function.)
dx dy dz
The conic sections are curves in the xy-plane, with equations that have x and y only to the first or
second power (no other functions of x or y and no other exponents) and always include at least
one x2 or y2. The name comes from the observ
Quiz 2 solutions
1. a 3,1,0 , b 2,1,1
a) Find ab.
i j k
a b 3 1
0 i 3j 5k
b) Prove that your answer from part (a) is orthogonal to both a and b.
a a b 3,1,0 1,3,5 3 3 0
b a b 2,1,1 1,3,5 2 3 5 0
2. Find both para
Maximum and minimum values of a function of two variables are similar to maximum and
minimum values of a function of a single variable. A function z = f(x, y) has a local maximum
(or relative maximum) point at (a, b) if f a, b f x, y for all
The partial derivatives fx and fy give the slopes of the function surface in the x and y directions.
If the graph of the function represents the surface of a hill, we could use fx to find how steep the
hill is facing to the east, and fy to fi
Test 3 solutions
1. f(x, y) xy
a) Find the gradient f .
f x, y f x , f y y,x
b) Evaluate the gradient at the point P(3, 1).
f 3,1 1,3
c) What is the maximum rate of change of f at P, and in which direction does the maximum rate
Quiz 6 solutions
1. Use Lagrange multipliers to maximize f(x, y) 4 x 2 y 2 subject to the constraint 2x + y = 1.
We need to solve f g .
f 2x,2y , g 2,1
2x + y = 1
Then x , y , substitute into the constraint to get 2 1
In single-variable calculus, we have functions of one independent variable. We are familiar with
graphs, derivatives, and integrals of such functions. In multivariable calculus, we look at
functions of several variables, that is, more
Test 1 solutions
(Homework problems similar to the test problems are noted.)
a 1,1,2 and b 2,1,3 (See 9.2 and 9.3, esp. 9.3 #15, 25, and quiz 1)
Find the following:
a) a b = (1)(2) + 1(1) + 2(3) = 5
b) a vector of length 2 in the opp
There are several ways to define a surface.
As the graph of a function z = f(x, y): We have already seen that the graph of a function of two
independent variables is a surface in 3. For example, the graph of f(x, y) = 6 3x 2y is a
During this course we will look at a variety of different types of functions. They differ not only
in the number of variables, but also in what the domain and range consist of.
We are very familiar with functions of a single variable, also ca
Now back to functions of several variables. We already looked briefly at functions of 2 variables
in section 9.6. In this section we will add to that, and also expand what we have done to
functions of more than two independent variables.
Previously, we have usually used functions of only one independent variable, but there are many
familiar examples of functions of two independent variables.
The temperature at a particular moment of a location on earth depends on the location,
As you know, when we have functions of a single variable that are composite functions, we need
to use the chain rule to find the derivative. If y f x and x gt then the composite function
is y f gt and the derivative of the composite can be fo
Quiz 3 solutions
A curve is defined by vector function r(t ) 2 sin t,2 cos t, t .
1. Sketch or completely describe the curve.
x 2 sin t
y 2 cos t
so x 2 y 2 4 , as t increases, z increases
a left-handed helix o
Quiz 7 solutions
Completely express each of the following integrals, but do not evaluate.
1. Express as a double integral the mass of the lamina bounded by y 1 x 2 and y 0 , with density
function x, y 2xy . Do not evaluate.
Now that we have defined vectors from two different perspectives, geometric and algebraic, and
know how to add vectors and multiply them by scalars, we will look at whether there is a
reasonable and useful way to multiply two vectors together.
MATH 280-Assignment #3: VECTOR FUNCTIONS AND SPACE CURVES
The objective of this assignment is to visualize the tangent vector of a space curve at a
specific point. The following sequence of examples illustrate the procedure.
Consider the vector
NAME: JIUSHU HE
MATLAB ASSIGNMENT 3
Exercise MATH 280-3 Due Monday November 9
Consider the vector function ()=<sin(2),cos(2),>, 02. We can use plot3 to generate the
graph of ():
Write ()=<(),(),()> where ()=in(
MA 280-Assignment #1:
VECTORS AND THE GEOMETRY
OF SPACE- Part I
The quiver command allows us to draw vectors in the plane
For instance, if we wish to draw the two vectors a <2,3>
and place them at the points
MA 280-Assignment #2:
VECTORS AND THE GEOMETRY
OF SPACE- Part II
We can use the dot product command to write an equation of the plane
through the point (2, 4, -1) with normal vector n= 2,3,4> :
Thus the equation of the desired plane is
NAME: JIUSHU HE
MATLAB ASSIGNMENT 2
Exercise MA 280-2 Due October 19
Use MATLAB to
a. Plot the planes x+ yz=2 and 2 x y+ 3 z=1 on the
b. Find two points on the line of intersection of the
planes in part(a)
c. Plot th
Worksheet 4 - Assignment (contributes 5% of your final grade)
Instructions: Some of the answers are given below, but you need to show all your work for
questions if you are instructed to do so. NO Work NO CREDIT. -Due date: December 8,
1. Which of t
Finding limits of functions of two variables is very similar to finding limits of functions of one
variable. lim f x, y L means that we can make f as close to the limit L as desired by
x,y a, b
taking (x, y) sufficiently close to (a, b). Th
Quiz 4 solutions
1. For a particle with the position function r(t ) 6ti t 3 j 3t 2k , find each of the following:
a) the velocity v(t)
vt r(t ) 6i 3t 2 j 6t k
b) the acceleration a(t)
at r(t ) 6tj 6k
c) the speed v(t)
Math 280 Review Solutions
1. To write an equation for a plane, we need a point and a normal vector. The direction vector for the line
v = 3,2,1 is in the plane. Point Q(1, 0, 2) is on the line so it is on the plane also. Then vector PQ
= 1,1,1 is in the p
Topics for Multivariable Calculus final exam:
Ch. 9: Topics from this chapter will occur on the test primarily in the context of other
problems from later material:
vectorssum, difference, scalar multiple, magnitude, unit vector
dot product, angle between
The method of Lagrange multipliers is used to find the absolute maximum or minimum value of
a function subject to a condition or restriction. We looked at using partial derivatives to find, for
example, the tops of hills or the highest point