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

53
Calculus Workbook
Section 3.7 Derivatives of Logarithmic Functions
Preliminary Example. Given to the right is a graph of
f (x) = ln x. First, sketch a rough graph of f ! (x), and then use
implicit dierentiation to derive a formula for f ! (x).
y
1
1
2

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

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

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 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)

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 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

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

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

Some Notes
f (x + h) f (x)
.
h0
h
Finding Derivatives by the definition. That is, f 0 (x) = lim
Experienced calculus students often bemoan the limit definition of the derivative, often referring to it as the long way
to take the derivative. They know ther

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

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

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)]

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

Calculus 1 (Math-UA-121)
Fall 2015
Homework 4
Due: Friday, October 2
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. Evalua

Calculus 1 (Math-UA-121)
Fall 2015
Homework 5
Due: Friday, October 9
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. Use th

Calculus 1 (Math-UA-121)
Fall 2015
Homework 6
Due: Friday, October 23
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. (a) I

AP Calculus
Position, Velocity, Acceleration Practice
Name_
Date_ Period_
r o2H081D3B EKYultzat vSWo7fVt5wVa1rwey bLqLBCU.q z kARlClJ 4rhiPgShjtHsY mrdelsJesruvueyd7.S
A particle moves along a horizontal line. Its position function is s(t) for t 0. For ea

Calculus Maximus
WS 5.6: Optimization
Name_ Date_ Period_
Worksheet 5.6Optimization
Show all work. Calculator permitted. Show all set-ups and analysis. Report all answers to 3 decimals and
avoid intermediate rounding error.
Multiple Choice
1. An advertise

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

Math 121 Indeterminate Forms Worksheet Solutions
1.
2.
lim
ln x
x 1
lim
1 cos x
sin x
x
x
0
3.
sin 2 x
lim
x 0
3x 2
4.
lim xe
5.
6.
7.
8.
9.
1
(No L'hopital's Rule. Don't break your pledge!)
1
(Is L'hopital's Rule the best way to do this?)
3
x
x
x
1
lim e