Calculus 1 (Math-UA-121)
Fall 2015
Homework 11
Due: Friday, December 11
at the start of class
Give complete, well-written solutions to the following exercises.
Make sure to write your name and discussion section number, and to staple your solution.
1. Rec

Calculus 1 (Math-UA-121)
Fall 2015
Homework 2
Due: Friday, September 18
at the start of class
Give complete, well-written solutions to the following exercises.
Make sure to write your name and discussion section number, and to staple your solution.
1. Let

Section 6.1 Integration by Parts
2010 Kiryl Tsishchanka
Integration by Parts
FORMULA FOR INTEGRATION BY PARTS:
Z
Z
udv = uv vdu
EXAMPLE 1: Find
Z
ln xdx.
Solution: We have
Z
ln x = u
ln xdx = d(ln x) = du
1
dx = du
x
EXAMPLE 2: Find
Z
dx = dv
Z
Z
1
x=v

Section 5.4 The Fundamental Theorem of Calculus
2010 Kiryl Tsishchanka
The Fundamental Theorem of Calculus
EXAMPLE: If f is a function whose graph is shown below and g(x) =
Zx
f (t)dt, find the values
0
of g(0), g(1), g(2), g(3), g(4), and g(5). Then sket

Section 5.1 Areas and Distances
2010 Kiryl Tsishchanka
Areas and Distances
We can easily find areas of certain geometric figures using well-known formulas:
However, it isnt easy to find the area of a region with curved sides:
METHOD: To evaluate the area

Section 1.4 Calculating Limits
2010 Kiryl Tsishchanka
Calculating Limits
LIMIT LAWS: Suppose that c is a constant and the limits
lim f (x) and
xa
lim g(x)
xa
exist. Then
1. lim [f (x) + g(x)] = lim f (x) + lim g(x)
xa
xa
xa
2. lim [f (x) g(x)] = lim f (x)

19
Calculus Workbook
Sections 2.1 & 2.6 Tangents, Velocities, and Other Rates of Change
y
f(x)
x
Let t = tangent line to f (x) at x = a
Slope of secant line P Q =
As h 0, the slope of the secant line P Q approaches
. Therefore:
The slope of the tangent li

Section 3.1 Exponential Functions
2010 Kiryl Tsishchanka
Exponential Functions
DEFINITION: An exponential function is a function of the form
f (x) = ax
where a is a positive constant.
1 x
x
10
1 3
-3
= 103 = 1000
10
1 2
-2
= 102 = 100
10
1
1
-1
= 101 =

Section 4.3 Derivatives and the Shapes of Graphs
2010 Kiryl Tsishchanka
Derivatives and the Shapes of Graphs
INCREASING/DECREASING TEST:
(a) If f (x) > 0 on an (open) interval I, then f is increasing on I.
(b) If f (x) < 0 on an (open) interval I, then f

AP Calculus AB Worksheet
Find the general antiderivatives.
1.
2.
3.
4.
5.
6.
7.
8.
9.
Use the Fundamental Theorem to calculate the definite integrals.
10.
11.

NYUStudentActivitiesBoard
2016-2017
2015-2016 New Club in Development (NCD) Petition
Applicant Club Name: _
Submitted by: _
We the undersigned are NYU Students who support the development of said club.
Please submit this form to the Student Activities Boa

ANTIGONE quotes
Women:
o Ismene: [] Remember we are women, / were not born to contend with men. Then too, / were
underlings, ruled by much stronger hands, / so we must submit to this, and things still worse. I, for
one, Ill beg the gods forgive me - / Im

Antigone
Sophocles
Translated by E.F. Watling
Characters:
Ismene daughter of Oedipus
Antigone daughter of Oedipus
Creon King of Thebes
Haemon Son of Creon
Teiresias a blind prophet
A Sentry
A Messenger
Eurydice wife of Creon
Chorus of Theban Elders
The fo

Section 5.5 The Substitution Rule
2010 Kiryl Tsishchanka
The Substitution Rule
THEOREM (The Fundamental Theorem Of Calculus, Part II): If f is continuous on [a, b], then
Zb
f (x)dx = F (b) F (a) = F (x)
a
ib
a
where F is any antiderivative of f, that is F

Section 4.5 Optimization Problems
2010 Kiryl Tsishchanka
Optimization Problems
EXAMPLE 1: A farmer has 2400 ft of fencing and wants to fence off a rectangular field that
borders a straight river. He needs no fence along the river. What are the dimensions

5
Calculus Workbook
Section 2.2 The Limit of a Function
Definition. We say that lim f (x) = L, which is read the limit as x approaches a of f (x) equals L, if
xa
we can make f (x) arbitrarily close to L by taking x values suciently close to a but not equa

75
Calculus Workbook
Section 4.5 Indeterminate Forms and LHospitals Rule
Preliminary Example. Use a table of values to estimate the value of lim
x0
exact value of the limit?
The 5 basic indeterminate forms:
0
,
0
,
,
,
sin(2x)
. Is it possible to find the

Review of Inverses
(See text material in 1.5 and 1.6)
Functions 0 and 1 are called inverses to each other if each one undoes the effect of the other, that
is for + in the domain of 0 and , in the domain of 1
" if 0 + , then 1, + so that 10 + +, and
2) if

105
Calculus Workbook
Section 5.4 Fundamental Theorem of Calculus
Preliminary Example. Let f be the function shown in the diagram
to the right, and define
g(x) =
!
x
y
8
f (t) dt.
0
4
f
1
!4
Example. Calculate
d
dx
!
1
x
t2 dt
and
d
dx
!
2
x
cos t dt.
2
3

25
Calculus Workbook
Section 2.8 The Derivative as a Function
Preliminary Example. Below and to the right, you are given the graph of a function f (x). With a
straightedge, draw in tangent lines to f (x) and estimate their slopes from the grid to fill in

33
Calculus Workbook
Section 3.1 Information
Alternate Notations for the derivative of y = f (x)
1. f ! (x) or f !
2. y !
dy
3. dx
the derivative of y with respect to x
d
4. dx [f (x)]
List of Shortcut Formulas
1.
d
dx [c]
2.
d
n
dx [x ]
3.
d
dx [cf (x)]

93
Calculus Workbook
Section 5.1 Areas and Distances
Example 1. The following data is gathered as a small plane travels down the runway toward takeo. How far
did the plane travel in the 10 second period? (Give a range of values.)
time (sec)
velocity (ft/s

12
Sonoma State University
Section 2.4 Continuity
y
y
f
f
a
x
a
x
Definition. We say that a function f is continuous at a if lim f (x) = f (a). (Graphically, continuity means
xa
that f has no breaks or jumps at x = a.) In particular, if f is continuous at

Section 8.7 Taylor and Maclaurin Series
Taylor and Maclaurin Series
In the preceding section we were able to find power series representations for a certain restricted
class of functions. Here we investigate more general problems: Which functions have pow

Section 3.7 Indeterminate Forms and LHospitals Rule
2010 Kiryl Tsishchanka
Indeterminate Forms and LHospitals Rule
THEOREM (LHospitals Rule): Suppose f and g are differentiable and g (x) 6= 0 near a
(except possibly at a). Suppose that
lim f (x) = 0 and
x

12
Sonoma State University
Section 2.4 Continuity
y
y
f
f
a
x
a
x
Definition. We say that a function f is continuous at a if lim f (x) = f (a). (Graphically, continuity means
xa
that f has no breaks or jumps at x = a.) In particular, if f is continuous at

109
Calculus Workbook
Section 5.5 The Substitution Rule
Preliminary Example. Evaluate
Example 1. Evaluate
Z
sin x
dx.
cos2 x
Z
2x
p
1 + x2 dx.
110
Sonoma State University
Example 2. Evaluate
Z
2
Z
e
2
x ex dx.
0
Example 3. Evaluate
1
ln x
dx.
2x
The Subst

Calculus 1 (Math-UA-121)
Fall 2015
Homework 8
Due: Friday, November 6
at the start of class
Give complete, well-written solutions to the following exercises.
Make sure to write your name and discussion section number, and to staple your solution.
1. Moore

Calculus 1 (Math-UA-121)
Fall 2015
Homework 9
Due: Friday, November 13
at the start of class
Give complete, well-written solutions to the following exercises.
Make sure to write your name and discussion section number, and to staple your solution.
1. Find

NYU - Tandon School of Engineering
125 Pre-Calculus Problems
This is a set of practice PreCalc questions that reflect material you are expected (1) to
have mastered in high school and (2) to have at your fingertips while taking Math at the
NYU SoE. If you

NYU-Tandon School of Engineering
MA1024/MA1324
Review Problems for Final Exam
(1) Find the domain and range of the given function.
1
(a) f (x) =
4 x2
(b) f (x) = |x + 2| 1
16
(c) y = 2
x 16
(d) g(x) = 4p tan2 (x)
(e) h(x) = 2 sin2 (x)
(2) (a) Let f (t) =

NYU-Tandon School of Engineering
MA1024/MA1324
Review Problems for Exam 1
(1) A space station has a population of 400 people at time t = 0. The population
increases by 4.67% a year. Find a formula for P (t), the population at time t.
(t > 0, measured in y

NYU-Tandon School of Engineering
MA1024/MA1324
Review Problems for Exam 2
(1) Let f (x) =
x
.
x+1
(a) Use the limit definition of the derivative to find f 0 (x).
(b) Find the equation of the tangent line to the curve y = f (x) at x = 2.
In Problems 2-16 ,

NYU-Tandon School of Engineering
MA1024/MA1324
Review Problems for Exam 3
(1) Estimate f (2.8) given that f (3) = 2 and f 0 (x) = (x3 + 5)
1/5
.
(2) Find the linear approximation of f (x) = ln(2x + 1) at a = 0 and use it to approximate the value of ln(1.1