Section 7.1 Areas Between Curves
2010 Kiryl Tsishchanka
Areas Between Curves
We dene the area A of S as the limiting value of the sum of the areas of the above approximating
rectangles:
n
A = lim
n
i=1
[f (x ) g (x )] x
i
i
From this it follows that the a
Section 2.6 Implicit Dierentiation
2010 Kiryl Tsishchanka
Implicit Dierentiation
Some functions can be described by expressing one variable explicitly in terms of another
variable for example,
1x
y = x2 , y =
, y = tan 2x
1 + x3
or, in general, y = f (x).
MATHUA 122: Calculus 11 - Fall 2016 Section:
Name:
dx
1' A; (x+2)(x +5)
: J 4* \l. . 1 7L \3
\ i
U3?
f
a
9M
ay
Worksheet 6: Improper Integrals
Date: 2. Determine all p > 0 such that f 24 dx converges.
e x(lnx)P
L'" if bk: 1 7:; CvJ :' g? (31311
J9 '" a ~
Math 122 - Summer 2016 - Final Exam - Version A
You have 110 minutes to complete this final exam. Books, notes and electronic devices are not permitted. Read
and follow directions carefully. Show and check all work. Label graphs and include units where ap
Math 122 - Summer 2016 - Midterm Exam 2 - Version A
You have 110 minutes to complete this midterm exam. Books, notes and electronic devices are not permitted.
Read and follow directions carefully. Show and check all work. Label graphs and include units wh
MATH-UA 122: Calculus II - Fall 2016 Instructor: Fanny Shum
Practice Exam 2
No calculators, notes, or other outside materials are permitted. Show all work to receive full credit.
1. Which of the following sequences is both bounded and monotonic?
n2
(a)
(b
MATH-UA 122: Calculus II - Fall 2016 Section 004 Quiz 1: Oct 6, 2016
Name:_ Grade: / 10
No calculators, notes, or other outside materials are permitted. Show all work to receive full credit.
Multiple Choice (1 point each)
1. What form is the partial dec
MATH-UA 122: Calculus II - Fall 2016 Section 004 Quiz 1: Sep 22, 2016
Name: Grade:
No calculators, notes, 01' other outside materials are permitted. Show all work to receive full credit.
/10
Multiple Choice (1 pez'nt each)
1. Determine the best ehoice
MATH-UA 122: Calculus II - Fall 2016
Section:
Worksheet 7: Area and Volumes
Name:
Date:
ONLY set up the integrals for each question.
1. Find the area of the region bounded by y = cos x, y = 2 cos x, for 0 x 2.
2. Find the area of the region bounded by x =
Notes
Welcome, Review and the
Substitution Rule 5.5
Vindya Bhat
1/48
Class Plan
Notes
Announcements
Welcome
I
I
I
I
Syllabus review
Course calendar review
NYU Classes demo
WebAssign demo
5.4 The Fundamental Theorem of Calculus
(review)
5.5 The Substitut
Math 122 - Summer 2016 - Midterm Exam 1 - Version A
You have 110 minutes to complete this midterm exam. Books, notes and electronic devices are not permitted.
Read and follow directions carefully. Show and check all work. Label graphs and include units wh
MATH-LIA 122: Calculus 1i 3E1]! 2016 Sect.ion:_. Worksheet 1: Integration
I.
Name:_f_m_.m_ Date:
1. Which of the following re equai to
X [15 1133; dx? U '- \rn &U. r 9335'
Please cireie all of the aforrect answers. You do not need to justify your solution
Math 122 - Fall 2016 - Homework - Chapter 8
Give complete, well written solutions to the following exercises:
1. The Fibonacci sequence is defined by the equations
f1 = 1,
f2 = 1,
fn = fn1 + fn2 ,
n 3.
Show that the following statement is true.
X
n=2
1
=1
Section 1.3 The Limit of a Function
2010 Kiryl Tsishchanka
The Limit of a Function
DEFINITION: We write
lim f (x) = L
xa
and say
the limit of f (x), as x approaches a, equals L
if we can make the values of f (x) arbitrary close to L (as close to L as we l
Section 2.1 Derivatives and Rates of Change
2010 Kiryl Tsishchanka
Derivatives and Rates of Change
The Tangent Problem
EXAMPLE: Graph the parabola y = x2 and the tangent line at the point P (1, 1).
Solution: We have:
DEFINITION: The tangent line to the cu
Section 1.5 Continuity
2010 Kiryl Tsishchanka
Continuity
DEFINITION 1: A function f is continuous at a number a if
lim f (x) = f (a)
xa
REMARK: It follows from the denition that f is continuous at a if and only if
1. f (a) is dened.
2. lim f (x) and lim+
Section 1.6 Limits Involving Innity
2010 Kiryl Tsishchanka
Limits Involving Innity
I. Innite Limits
DEFINITION: The notation
lim f (x) =
xa
means that the values of f (x) can be made arbitrary large (as large as we like) by taking x
suciently close to a
Section 2.4 The Product and Quotient Rules
2010 Kiryl Tsishchanka
The Product and Quotient Rules
THE PRODUCT RULE: If f and g are both dierentiable functions, then
d
d
d
[f (x)g(x)] = g(x) [f (x)] + f (x) [g(x)]
dx
dx
dx
or
[f (x)g(x)] = f (x)g(x) + f (x)
MATH-UA 122.007 Calculus II
Name:
Worksheet for Section 5.15.5: Review of the Integral, Integration by Substitution
Spring 2014
Please get into groups of three.
The Denite Integral
1. The rate at which the worlds oil is being consumed is increasing. Suppo
Solutions
MATH-UA 122 Calculus II
Name:
Worksheet for Section 5.15.5: Review of the Integral, Integration by Substitution
Spring 2014
Please get into groups of three.
The Denite Integral
1. The rate at which the worlds oil is being consumed is increasing.
Solutions
MATH-UA 122 Calculus II
Name:
Worksheet for Section 6.2: Integration by Parts
Spring 2014
Evaluate the integrals, integrating by parts. In the case of a denite integral, your answer should
be a number. For indenite integrals, your answer should
MATH-UA 122.007 Calculus II
Name:
Worksheet for Section 6.2: Integration by Parts
Spring 2014
Evaluate the integrals, integrating by parts. In the case of a denite integral, your answer should
be a number. For indenite integrals, your answer should be the
Math 122 - Fall 2016 - Homework - Chapter 6
Give complete, well written solutions to the following exercises:
1. Evaluate the following integrals. Indicate the technique(s) you are using.
Z
Z
sin x cos x
2
(a)
dx
(d) (2x2 + 1)ex dx
4
4
Z sin x + cos x
Z
1
Notes
Integration by Parts 6.1
Vindya Bhat
1/20
Class Plan
Notes
Announcements
5.5 The Substitution Rule (Review)
6.1 Integration by Parts
Summary
2/20
Recall
Notes
Theorem (Substitution Rule)
If u = g(x) is a differentiable function whose range is
an i
Section 8.3 The Integral and Comparison Tests
2010 Kiryl Tsishchanka
The Integral and Comparison Tests
The Integral Test
THE INTEGRAL TEST: Suppose f is a continuous, positive, decreasing function on [1, )
and let an = f (n). Then the series
an is converg
Section 8.7 Taylor and Maclaurin Series
Taylor and Maclaurin Series
In the preceding section we were able to nd power series representations for a certain restricted
class of functions. Here we investigate more general problems: Which functions have power
Section 8.2 Series
2010 Kiryl Tsishchanka
Series
DEFINITION: An innite series is an expression that can be written in the form
a1 + a2 + a3 + . . . + ak + . . .
The numbers a1 , a2 , a3 , . . . are called the terms of the series.
Consider
s1 = a1
s2 = a1