part 1
1. Which story did you read?
I read "The Sniper" by Liam O'Flaherty.
2. Describe the physical setting of the story. Which words or descriptions contribute to the
emotional setting or mood? What is the mood that results from the authors use of
descr
"freedom form fear"
As i read and digested President Roosevelt 1941 speech about the Four Freedoms
immediatley President Obama came to mind. Why you may ask they both have a similar trend of
thought Roosevelt was trying to impart that the american people
1.
2.
3.
Critic: Roger Ebert
Title of Critique: the sixth sense
Provide a basic outline of the critics article. A boy can talk to the dead and the want him to do
thing for them and his psychologist is along for the ride.
4.
Summarize his or her evaluation
Part A
According to edger Allen Poe story the premature burial, Poe expressing his fears about being
bared alive do to his medical condition "catalepsy". He goes into further detail about how it has felt from
his past experiences of being bared without ha
Question 1
(Worth 10 points)
In35sentences,explainonecharacteristicofagothicsettingpresentinTheStrangeCaseofDr.
JekyllandMr.Hyde.
Essay Submission
During the dark days of winter with over cast skies, underneath, events of
the most shady kind are taking pl
Eye contact critique
I rate the film Eye Contact 2.5 stars out of a possible 5 stars. On the first glance of the short
film Eye Contact directed by Mike Foley, my thoughts were here goes another short film that tries to
scare me. I was surprised that my t
Introduction to Rates of Change
Last class, we defined the derivative as the slope of a tangent line. Today we'll see how to interpret the derivative as a rate of change, clarify the idea of a limit, and use this notion of limit to describe continuity a p
18.01 Final Answers
1. (1a) By the product rule,
(x3 ex ) = 3x2 ex + x3 ex = ex (3x2 + x3 ).
(1b) If f (x) = sin(2x), then
f (7) (x) = 128 cos(2x)
since:
f (1) (x) = 2 cos(2x)
f (2) (x) = 4 sin(2x)
f (3) (x) = 8 cos(2x)
f (4) (x) = 16 sin(2x)
f (5) (x) =
Notations
Calculus, rather like English or any other language, was developed by several
people. As a result, just as there are many ways to express the same thing,
there are many notations for the derivative.
Since y = f (x), its natural to write
y = f =
1. Compute the following derivatives:
(a) f (x) where f (x) = x3 ex
(b) f (7) (x), the seventh derivative of f , where f (x) = sin(2x)
2. (a) Find the tangent line to y = 3x2 5x + 2 at x = 2. Express your answer in
the form y = mx + b with slope m and y
1. Compute the following integral:
4
t ln t dt
1
We apply integration by parts:
v = 2 /2
t3
3
dv = t dt
u = ln t
du = 1 dt
t
Then:
4
4 4
2 3/2
2 3/2 1
t ln t
t t dt
3
13
1
t1/2
4
2 3/2
4
=
t ln t t3/2
3
9
1
16
4 3/2
=
ln 4 (4 1)
3
9
16
28
=
ln 4
3
9
t l
1. Compute the following derivatives. (Simplify your answers when possible.)
(a) f (x) where f (x) =
f (x) =
x
1 x2
1(1 x2 ) (x)(2x)
1 x2 + 2x2
1 + x2
=
=
(1 x2 )2
(1 x2 )2
(1 x2 )2
(b) f (x) where f (x) = ln(cos x)
12
sin (x)
2
1
1
( sin x) 2 sin x cos
1. Compute the following integral:
4
t ln t dt
1
2. Compute the following integral:
0 /4
tan4 sec6 d
3. Compute the following integral: 10
dx (x - 1)(x2 + 9)
4. Compute the following integral: 1
dx (5 - 4x - x2 )5/2
5. (a) Set up (but do not solve)
Do We Need the Quotient Rule?
The quotient rule can be difficult to memorize, and some students are more comfortable with negative exponents than they are with fractions. In this exer cise we learn how we can use the chain and product rules together in pl
1. Compute the following derivatives. (Simplify your answers when possible.)
(a) f (x) where f (x) =
x
1 x2
(b) f (x) where f (x) = ln(cos x)
1
2
sin (x)
2
(c) f (5) (x), the fth derivative of f , where f (x) = xex
2. Find the equation of the tangent li
Massachusetts Institute of Technology
Department of Economics
14.01 Principles of Microeconomics
Midterm Exam #1
Tuesday March 3, 2009
Last Name (Please print): _
First Name: _
MIT Email Address: _
MIT ID Number: _
Instructions. Please read carefully.
Thi
Repeated Dierentiation of Sine and Cosine
a) For which of the functions whose graphs are displayed by the Creating the
Derivative mathlet is it true that f (x) = f (x)?
b) Can you think of any other functions for which f (x) = f (x)?
Solution
a) For which
Repeated Dierentiation of Sine and Cosine
a) For which of the functions whose graphs are displayed by the Creating the
Derivative mathlet is it true that f (x) = f (x)?
b) Can you think of any other functions for which f (x) = f (x)?
1
MIT OpenCourseWare
Do We Need the Quotient Rule?
The quotient rule can be difficult to memorize, and some students are more comfortable with negative exponents than they are with fractions. In this exer cise we learn how we can use the chain and product rules together in pl
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:
u(x)
v (x)
u
Smoothing a Piecewise Polynomial
For each of the following, nd all values of a and b for which f (x) is dierentiable.
a) f (x) =
ax2 + bx + 6,
2x5 + 3x4 + 4x2 + 5x + 6,
x 0;
x > 0.
b) f (x) =
ax2 + bx + 6,
2x5 + 3x4 + 4x2 + 5x + 6,
x 1;
x > 1.
ax2 + bx +
Quotient Rule Practice
Find the derivatives of the following rational functions. a) b) c) x2 x+1 x4 + 1 x2
sin(x)
x
1
MIT OpenCourseWare http:/ocw.mit.edu
18.01SC Single Variable Calculus
Fall 2010
For information about citing these materials or our Te
The Function sinc(x)
The unnormalized sinc function is dened to be:
sinc(x) =
sin x
.
x
This function is used in signal processing, a eld which includes sound recording
and radio transmission.
1
Use your understanding of the graphs of sin(x) and x togethe
Smoothing a Piecewise Polynomial
For each of the following, find all values of a and b for which f (x) is differentiable. x 0; ax2 + bx + 6, a) f (x) = 2x5 + 3x4 + 4x2 + 5x + 6, x > 0. ax2 + bx + 6, x 1; b) f (x) = 2x5 + 3x4 + 4x2 + 5x + 6, x > 1.
1
MIT O
Limits and Discontinuity
For which of the following should one use a one-sided limit? In each case,
evaluate the one- or two-sided limit.
1. lim x
x 0
2. lim
x1
1
x+1
1
x1 (x 1)4
3. lim
4. lim | sin x|
x0
|x|
x 0 x
5. lim
Solutions
1. lim x
x 0
The functi
The Function sinc(x)
The unnormalized sinc function is dened to be:
sinc(x) =
sin x
.
x
This function is used in signal processing, a eld which includes sound recording
and radio transmission.
1
Use your understanding of the graphs of sin(x) and x togethe