NAME: JIUSHU HE
COURSE: MATH280
CRN: 23986
MATLAB ASSIGNMENT 1
Let
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 vec
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
Section 9.4
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 V
Math 280
CRN 30210
Test 2 solutions
18 pts.
1. For rt t 2 , t 3 3 , 2t , find the following: (See 10.2-10.3 for similar problems.)
a) rt
(10.2 #3-14)
rt 2t, t 2 , 2
b) the tangent vector to the curve
Section 10.3
From calculus, we have a formula for the length of a parametrically defined curve in the plane:
2
2
dx dy
dt
dt dt
b
L
a
We can expand this to curves in space defined by parametric e
Conic Sections
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 includ
Math 280
Section 30210
Quiz 2 solutions
2 pts.
1. a 3,1,0 , b 2,1,1
a) Find ab.
i j k
a b 3 1
2
1
0 i 3j 5k
1
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
Math 280
CRN 30210
Quiz 7 solutions
Completely express each of the following integrals, but do not evaluate.
2 pts.
1. Express as a double integral the mass of the lamina bounded by y 1 x 2 and y 0 ,
Section 9.3
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
reas
Section 11.2
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
Math 280
Section 30210
Quiz 4 solutions
5 pts.
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 acce
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 i
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 mult
Section 11.8
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 fin
Section 9.7
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
coordin
Math 280
CRN 30210
Quiz 5 solutions
3 pts.
3 pts.
1. z f (x, y) is defined implicitly by the equation 2x 2 y y 2 z3 xz 2 .
z
a) Find
using implicit differentiation.
x
Differentiate, treating x as a va
Math 280
CRN 30210
Quiz 6 solutions
4 pts.
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 2
2y
2x + y = 1
Preliminary remarks
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
Math 280
CRN 30210
Test 1 solutions
(Homework problems similar to the test problems are noted.)
1.
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)(
Section 10.5
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 e
Section 10.1
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 ver
Section 11.1
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
Section 9.6
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
Section 11.5
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
Math 280
Section 30210
Quiz 3 solutions
2 pts.
2 pts.
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
zt
t=/2
so x 2 y
Math 280
CRN 30210
Test 3 solutions
16 pts.
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 o
Section 11.6
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
Section 11.7
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 rela
Section 11.3
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 variable
Section 9.5
lines in 3-space:
The vector equation of a line is r r0 tv , where r0 is the
position vector of a given point P0 on the line, v is a vector
parallel to the line, and t is a parameter. For
% In Which Steve Tries To Be Neat With His MATLAB Homework
% January 25, 2018 (note the TWO spaces before this line! What happens if you
change it to one?)
% Problem 1.
% I don't understand the rref c