1
Minyi Yuan
STAT 250-DL1
12/13/2015
Data Analysis Assignment 5
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1 Type your name on your paper
2 Under your name, put STAT 250 with your correct section number (e.g. STAT 250-00x)
3 Type Data Analysis Assignment 5 centered
Data Analysis Assignment 3
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1 Type your name on your paper (please read the formatting file to see how)
2 Under your name, put STAT250 with your correct section number (e.g. STAT 250-0xx)
3 Type Data Analysis Assignment 3 ce
Features of Graphs
The Graph Features mathlet allows you to choose the coe cients of a degree
three polynomial and then illustrates where the graph of that polynomial is
rising (increasing), falling (decreasing), concave and convex.
Find coe cient values
Can Design
There are many factors to consider in food packaging, including marketing,
durability, cost and materials. In this problem we minimize the cost of materials
for a can.
Find the height and radius that minimizes the surface area of a can whose
vo
Derivative of the Square Root Function
a) Use implicit dierentiation to nd the derivative of the inverse of f (x) = x2
for x > 0.
b) Check your work by nding the inverse explicitly and then taking its derivative.
Solution
If youre having trouble with this
Cab Company Scheduling
let Di = # of drivers who start their 8 hour shift in period I (I = 1,2,3,4,5,6)
period 1
12:00:00 AM-4:00am
period 4
12 noon - 4:00pm
period 2
4:00am - 8:00am
period 5
4:00pm - 8:00pm
period 3
8:00am - 12 noon
period 6
8:00pm - mid
Do We Need the Quotient Rule?
The quotient rule can be di cult to memorize, and some students are more
comfortable with negative exponents than they are with fractions. In this exercise we learn how we can use the chain and product rules together in place
Exponential Growth and Inhibited Growth
dy
a) The dierential equation
= ry describes a situation in which a population
dx
size y increases at a rate proportional to its size. Use separation of variables
to nd a solution to this equation.
dy
b) The dierent
Quotient Rule Practice
Find the derivatives of the following rational functions.
a)
x2
x+1
b)
x4 + 1
x2
c)
sin(x)
x
Solution
a)
x2
x+1
The quotient rule tells us that if u(x) and v(x) are dierentiable functions,
and v(x) is non-zero, then:
0
u(x)
u0 (x) v
n1
xbar 1
s1
df
t test
test stat
p value
13 n2
0.53 xbar 2
0.52 s2
12
0.12
0.0208
0.170777
0.191577
0.437695
0.274164
0.393258
13
0.41
1.49
12
22
n1
xbar 1
s1
df
t test
test stat
p value
35 n2
75.29 xbar 2
11.39 s2
34
-9.38
3.706631
10.66478
14.37141
3.79
Solving an Optimization Problem using Implicit Dierentiation
Suppose you wish to build a grain silo with volume V made up of a steel cylinder
and a hemispherical roof. The steel sheets covering the surface of the silo are
quite expensive, so you wish to m
SOLUTIONS TO 18.01 EXERCISES
Unit 3. Integration
3A. Dierentials, indenite integration
3A-1 a) 7x6 dx. (d(sin 1) = 0 because sin 1 is a constant.)
b) (1/2)x
1/2
c) (10x9
8)dx
dx
d) (3e3x sin x + e3x cos x)dx
p
p
e) (1/2 x)dx + (1/2 y)dy = 0 implies
p
p
p
SOLUTIONS TO 18.01 EXERCISES
Unit 4. Applications of integration
4A. Areas between curves.
4A-1 a)
Z
1
(3x
2x2 )dx = (3/2)x2
1
x
(2/3)x3
1/2
1
1/2
= 1/24
b) x3 = ax =) x = a or x = 0. There are two enclosed pieces ( a < x < 0
and 0 < x < a) with the same
Using Di erentials to Study Population Dynamics
We have seen that dierentials give a convenient way for expressing linear approximations. In
this example, we explore population dynamics in the language of dierentials.
A simple generational model of popula
Antiderivative of tan x sec2 x
Compute
Z
tan x sec2 x dx in two dierent ways:
a) By substituting u = tan x.
b) By substituting v = sec x.
c) Compare the two results.
Solution
a) Compute
Z
tan x sec2 x dx by substituting u = tan x.
If u = tan x then du = s
Generalizing the Mean Value Theorem Taylor s theorem
We explore generalizations of the Mean Value Theorem, which lead to error estimates for Taylor
polynomials. Then we test this generalization on polynomial functions.
Recall that the mean value theorem s
n
t
df
0.002374
12
3.526
11
27
3.741
26
0.000458
n
t
df
22
-2.061
21
2.061
7
3.517
6
0.012564
0.051902
null mean
t
p
mean
Sx
n
test statistic
p value
5.4
-1.690026
0.048552
5.27
0.56
53
-0.13
7.28011
0.076922
-1.690026
0.048552
0.05
210
ho
h1
n
mean
>210