MTH 234 Groupwork 12 - 4.4, 4.5
(1) Differentiate the following:
(a) y = 6e2x
(b) y = ex
6
(c) y = e2x
3
(d) y = 7x2 e3x
(e) y = ln(5x + 3)
(f) y = ln(2 + x2 )
(g) y = y =
ln(3x)
x3
xex
(h) y = y =
ln(x2 1)
1
2
(i) y =
4e8x
5x 2
e8x + e8x
(j) y =
x
2
(k)
Math 285
(Mock) Exam 2
Name:
No calculators or other electronic devices may be used. Show
all work, have fun, and good luck! (50 points possible)
Problem 1. (10 pts) Solve the following ODE
(D4 4D3 + 7D2 6D + 2)y = 0
Hint: make use of the following factor
Math 285 Midterm 2 practice solutions
Problem 1: We know the Fourier Series of a constant is just the same
constant, so the F.S. of 2 is just 2. We now calculate the F.S. of t, which is
an odd function, so we only need to calculate the Fourier series coef
NAME:
Math 285 Midterm 2 practice
Total points: 100. Please explain all answers. Calculators, computers,
books and notes are not allowed. Suggestion: even if you cannot complete
a problem, write out the part of the solution you know. You can get partial
c
Math 285
Mock Final
Name:
No calculators or other electronic devices may be used. Show
all work, have fun, and good luck! (100 points possible)
Problem 1. (10 pts) Find the Fourier series of the function:
f (t) = |t| + t2 for t [L, L]
Solution: Observe th
Math 285
Mock Final
Name:
No calculators or other electronic devices may be used. Show
all work, have fun, and good luck! (100 points possible)
Problem 1. (10 pts) Find the Fourier series of the function:
f (t) = |t| + t2 for t [L, L]
Q2
2
Problem 2. (15)
MATH 234 Groupwork 13 - 5.3
(1) For f (x) = x(x + 5)2 , find the open intervals where f (x) is concave up or
concave down, and find all inflection points. (Hint: Sometimes doing a bit of
algebra before you start taking multiple derivatives makes life easi
Math 234 Groupwork 2
Understanding Parabolas as Translations
Standard Form: f (x) = ax2 + bx + c
Vertex Form: f (x) = a(x h)2 + k
1. For each of the functions below, write the equation of the new function after performing the following
translations and re
MATH 234 Groupwork 11 - 5.1, 5.2
(1) For the following functions, find all intervals where the function is increasing/decreasing, and locate any/all relative max/mins. Use the method of
f 0 (x) number line and make sure the min/max you are reporting is no
MATH 234 Groupwork 13 - 5.4
Point of Diminishing Returns
Suppose you are looking at a graph representing profit for a stock you are investing
in. Of the four graphs below,
which 2 would you most like to have represent your
stock profit?
Graph A
Graph B
Gr
MTH 234 Groupwork 1
Sections 1.2, 2.1
Definitions
A function from set A to set B is a rule that assigns each element of
set A to exactly 1 element of set B.
An even function is one where f (x) = f (x) and an odd function
is one where f (x) = f (x)
Marg
MATH 234 Groupwork 81
In class we explored taking derivatives of various types of functions, and came up with
the following set of rules:
Constant Rule
Constant Multiple Rule
Power Rule
Sum/Difference Rule
Product Rule
Quotient Rule
f (x) = k, Then f 0 (x
Algebra Review
Each layer of mathematics uses the layer below it to solve its problems.
First you learn numbers, the fundamental building blocks of mathematics.
Then you learn arithmetic, in which you use numbers to solve arithmetic
problems. Then you lea
MATH 234 Groupwork 8
SOLUTIONS
In class we explored taking derivatives of various types of functions, and came up with
the following set of rules:
Constant Rule
Constant Multiple Rule
Power Rule
Sum/Difference Rule
Product Rule
Quotient Rule
f (x) = k, Th
Math 234 Spring 2016 Exam 1 Version 1
Wednesday, February 17th, 2015
UIN:
Name:
Circle the section you are registered for:
ADA (Vanessa 9AM)
ADB (Vanessa 10AM) ADC (Chris 11AM)
ADD (Chris 12PM)
ADE (Mingyu 1PM)
ADF (Mingyu 2PM)
ADG (Derek 3PM)
ADH (Derek
MTH 234 Groupwork 2
Solutions
Understanding Parabolas as Translations
Standard Form: f (x) = ax2 + bx + c
Vertex Form: f (x) = a(x h)2 + k
1. For each of the functions below, write the equation of the new function
after performing the following translatio
Tuesday, February 4 Solutions Visualizing quadric surfaces
Elliptic paraboloid: z = Ax 2 + B y 2 (A, B have same sign)
If A = 0 or B = 0 our surface becomes a parabola extended out parallel to a coordinate axis. If
A = B = 0 our surface becomes the plane
In cylindrical coordinates E is the solid region
within the cylinder r = 1 bounded
above and
below by the sphere r2 +z 2 = 9 , so E = (r, , z) | 0 2, 0 r 1, 9 r2 z 9 r2 .
Thus
is R R
R 2
R1
RRR the volumeR 2
R 2 R 1
1
9r 2
dV
=
r dz dr d = 0 0 2r 9 r
The paraboloid z = 16x2 y 2 intersects thexy -plane in the circle x2 +y 2 = r2 = 16 or r =
4, so in cylindrical coordinates, E is given by (r, , z) 0 2 , 0 r 4, 0 z 16 r2 .
Thus
R /2 R 4 R 16r2 3
R /2 R 4 R 16r2 4
RRR
4(x3 + xy 2 ) dV = 4 0
(r cos3 + r3
RRR
E
xyz dV
R /3 R 2 R 5
2
4 ( sin() cos()( sin() sin()( cos() sin() d d d
0
0
R5 5
R 2
R /3 3
= 0 sin () cos() d 0 sin() cos() d 4 d
/3 1
2 1 6 5
2
= 41 sin4 () 0
sin
()
2
6 4 = 0
0
=
Assume a = 8, b = 24.
The region of integration is given in cylindrical coordinates by
E = cfw_(r, , z) | 0 2, 0 r 8, 3r z 24 . This represents the
solid region bounded below by the cone z = 3r and above by the horizontal
plane z = 24 .
R 8 R 2
R 4 R 2
R
y
1
= 1 and the point
=
x
1
(1, 1) is in the fourth quadrant of the xy-plane, so = 7
4 + 2n; z = 4. Thus,
7
2, 4 , 4 .
one set of cylindrical coordinates is
2
= 16 so r = 4; tan = 22 3 = 3 and the point
(b) r2 = (2)2 + 2 3
2, 2 3 is in the third quad
1
Part I Solutions
SOLUTIONS FOR PART I
1. NUMBERS, SETS, AND FUNCTIONS
1.1. We have at least four times as many chairs as tables. The number of
chairs (c) is at least () four times the number of tables (t ). Hence c 4t .
1.2. Fill in the blanks. The equa
Tuesday, November 1 Solutions Changing coordinates
1. Consider the region R in R2 shown below at right. In this problem, you will do a change of
coordinates to evaluate:
x 2y dA
R
y
v
(3, 4)
T
(1, 3)
R
(1, 1)
(2, 1)
S
x
u
(a) Find a linear transformation
Math 415 - Lecture 4
Powers, Transpose, Elementary Matrices, LU Factorization
Wednesday January 25 2017
Textbook: Chapter 1.4, 1.5
Suggested Practice Exercise: Chapter 1.4 Exercise 22, 27, Chapter 1.5: 4,
5, 11, 23, 29
Powers of A
Powers of A
We write: Ak
Math 415 - Lecture 5
LU decomposition
Friday, January 27th, 2016
Textbook: Chapter 1.5
Suggested Practice Exercise: Chapter 1.5 Exercise 4, 5, 11, 23, 29
1
The Piazza Problem
The Piazza Problem
Piazza Problem
Is it possible, if A is a (non zero) matrix an
MATH 415 Lecture 13
Review for Miderm 1
Wednesday February 15 2016
1
Linear systems
1.1
Systems of equations are linear combination problems, can be written as
Ax = b.
x1 2x2
x1 + 3x2
x1
1
2
+ x2
1
3
= 1
=
3
1
=
3
A system has no solution (inconsist