APMA1650: Homework 1 Solutions
Elvis Nunez (elvis [email protected])
February 8, 2017
Problem 1. Suppose you flip a coin 20 times.
a) What is the probability of flipping exactly 5, 10, 12 Heads?
Let A5 , A10 , A12 be the events in which flip 5, 10, and 12 h
APMA 1650 : Homework 2 Solutions
[Wednesday, February 15]
1. A medical condition is carried by 1% of the population. A diagnostic test for the condition has the following
accuracy. If an individual has the condition, the test correctly detects this 95% of
APMA 1650 Homework 5 Solutions
Exercise 1 Let the CDF of a random variable X
0
x
F (x) = x82
16
1
be
x 0,
0 < x < 2,
2 x < 4,
x 4.
a) Find the density function (PDF) of X.
Solution. The PDF of a function F (x) is f (x) =
d
(0) = 0
dx
d
dx F (x),
so we ca
APMA 1650 Homework 3 Solutions
Exercise 1 A single cell will die with probability p or split into two with
probability 1 p, producing a second generation of cells. Each cell in the second
generation (if there are any) will die or split into two with the s
APMA 1650 Homework 4
Due Wednesday March 1st.
Maximum mark: 10 points. All exercises 2 points. JUSTIFY all answers.
Exercise 1 Suppose a building has 10 floors. m people get into the elevator
at level 0 and each one independently and uniformly at random c
APMA 1650 Homework 2
Due Wednesday February 15th.
Maximum mark: 10 points. All exercises 2 points. JUSTIFY all answers.
Exercise 1 A medical condition is carried by 1% of the population. A diagnostic test for the condition has the following accuracy. If a
APMA 1650 Homework 5
Due Wednesday March 15th.
Maximum mark: 10 points. All exercises 2 points. JUSTIFY all answers.
Exercise 1 The response times on an online computer terminal have a Gammatype distribution with mean 4 seconds and variance 8 seconds. Wri
APMA 1650 Homework 1
Due Wednesday February 8th.
Maximum mark: 10 points. All exercises 2 points. Complete and submit all
unstarred questions. Do NOT submit the starred question 1 d) it is a harder
question that you may try for fun. JUSTIFY all answers.
E
APMA 1650 Homework 4
Due Wednesday March 1st.
Maximum mark: 10 points. All exercises 2 points. JUSTIFY all answers.
Exercise 1 Suppose a building has 10 floors. m people get into the elevator
at level 0 and each one independently and uniformly at random c
APMA 1650 Homework 5
Due Wednesday March 8th.
Maximum mark: 10 points. All exercises 2 points. JUSTIFY all answers.
Exercise 1 Let the CDF of a random variable X be
0
x 0,
x
0
< x < 2,
F (x) = x82
2 x < 4,
16
1
x 4.
a) Find the density function (PDF) of
APMA 1650 Homework 3
Due Wednesday February 22th.
Maximum mark: 10 points. All exercises 2 points. JUSTIFY all answers.
Exercise 1 A single cell will die with probability p or split into two with
probability 1 p, producing a second generation of cells. Ea
CLPS0900
Quantitative Methods
Sample Questions for Exam 1
KEY
Below is a sample of questions from past first exams. If any revisions to these study questions are
needed, they will be posted on the course website. A key will be posted for these questions;
CLPS0900Exam1
AdditionalStudyQuestions
Below are additional study questions you may find helpful as you prepare for the first exam. These are representative questions taken from an exam bank,
and therefore emphasize text coverage. A key is attached, but s
CLSP0900
Study questions for Exam 1
Below are additional study questions you may find helpful as you prepare for the first exam. These are questions taken from an exam bank, and therefore
emphasize text coverage. A key is attached. Formulas to be provided
APMA 1650/1655 Homework 5
March 11, 2016
Due before class on Friday, March 18th. It can be dropped off in the APMA 1650 homework box
on the first floor of the APMA department, 182 George St OR at class (before it starts) on Friday.
APMA 1650: Complete all
Applied Math 33
Second Exam, 6 November, 2001.
Points are as indicated. Please identify in a clear manner work that you want considered for
credit. You may use your text book and one (1) calculus book for reference. Calculators that
are capable of solving
AM 33, Fall 2001, Third Practice Exam
1. Let
2 ; 0<t<2
.
t ; t2
f (t) =
Compute the Laplace transform of f . Solve the IVP
y (t) + 2y (t) = f (t),
y (0) = 1.
2. Solve the following integro-dierential equation by taking Laplace transform
t
y (t) = 2y (t) +
AM 33, Fall 2001, Second Practice Exam
1. Find the general solution to the dierential equation
y + y 6y = e2x 2.
2. One solution of
x2 y 2xy + (2 9x2 )y = 0
is y1 (x) = xe3x .
Find a second linearly independent solution y2 .
Calculate the Wronskian W (y
AM 33, Fall 2001, Practice Exam
1. Find all critical points of the dierential equation
dy
= 4y y 3 .
dt
Classify each critical point as stable, unstable or semistable.
Let y (t) be the solution to this dierential equation with y (0) = 1. Without solving
t
dy/dt
2
2
y
Figure 1: graph of f (y ) in the rst question
Fall 2001, AM33 Solution to the practice exam
1. Find all critical points of the diferential equation
dy
= 4y y 3
dt
Classify each critical point as stable, unstable or semistable.
Solution The cri
Fall 2001, AM33 Solution to hw 10
1. Section 6.3, problem 9
t , if t 2
0,
elsewhere
f (t) =
f is nonzero only between and 2 , so we only need to integrate in that region:
2
Lcfw_f =
2
(t )est dt
=
=
es
s2
test dt
2
est dt
e2s
(1 + s)
s2
Where the rst in
Solution to AM 33 HW 8
1. 6.1.2.
2
t
(t 1)1
f (t) =
1
0t1
1<t2
2<t3
Solution: f (t) is continous at [0,1) and (1,3].
2. 6.1.5. Find the Laplace transform of each of the following functions:
(a) t
(b) t2
(c) tn , where n is a positive integer.
Solution:
(
Fall 2001, AM33 Solution to hw7
1. Section 3.4, problem 41 We are solving the ODE
t2 y + 3ty + 1.25y = 0
By problem 38 x = log t turns this DE into a constant coecient DE.
x = log t t = ex
dt
= ex = t
dx
By the chain rule
dy dt
dy
dy
=
=
t
dx
dt dx
dt
We
Solutions to Homework No. 6
SECTION 3.3
Problem 4:
Using Theorem 3.3.1, all we have to do is check whether the Wronskian of f (x) = exp(3x) and g (x) =
exp(3(x 1) is identically zero or not. Since W (f, g ) := f g f g = 3(x 1) exp(3x) exp(3(x 1)
3 exp(3x
Solution to AM 33 HW 5
1. 3.1.15. Solve y + 8y 9y = 0,
y (1) = 1,
y (1) = 0
Solution:
the characteristic equation is:
x2 + 8x 9 = 0
x1,2 = 1, 9
let
y = aex1 + be9(x1)
then
y = aex1 9be9(x1)
plug the initial value in:
a+b=1
a 9b = 0
solve it, get
b=
1
,
10
Fall 2001, AM33 Solution to hw4
1. Problem 3, page 84
Sketch the graph of f (y ) versus y , determine the critical points, and classify each one as
asymptotically stable or unstable.
dy
= y (y 1)(y 2)
dt
Solution
The required graph is:
2.5
x*(x-1)*(x-2)
2
Solutions to Homework No. 3
SECTION 2.4
Problem 10:
Let f (t, y ) = (t2 + y 2 )3/2 . Clearly both f and f /y are continuos on all of the ty plane so that the hypotheses of Theorem 2.4.2 are satised everywhere. Note that this does not mean that we have exi
AM33 Homework assignment # 2
1. (2.2.11)
xdx + yex dy = 0
y (0) = 1
Solution:
xex dx = y dy
therefore
xex dx =
y 2
+c
2
=
y dy
xdex
= xex
ex dx
= xex ex
so
y2
= c + ex xex
2
1
since y (0) = 1, so c = 2 , i.e,
y 2 = 1 + 2ex 2xex
y=
2
2ex 2xex 1
the inter
Fall 2001, AM33 Solution to hw1
1. Find the general solution to the following problems:
(a) Problem 1, p38
y + 3y = t + e2t
Solution
This is a linear equation with constant coecients and solved by nding an integrating
factor (t). (t) has to satisfy:
(t)y