Jason Starr
Fall 2005
18.01 Calculus
Due by 2:00pm sharp
Friday, Oct. 21, 2005
Problem Set 4
Late homework policy. Late work will be accepted only with a medical note or for another
Instituteapproved reason.
Cooperation policy. You are encouraged to work
18.01 Calculus
Due by 2:00pm sharp
Friday, Dec. 9, 2005
Jason Starr
Fall 2005
Solutions to Problem Set 8
Late homework policy. Late work will be accepted only with a medical note or for another
Instituteapproved reason.
Cooperation policy. You are encoura
18.01 UNIT 2 REVIEW; Fall 2007
The central theme of Unit 2 is that knowledge of f (and sometimes f ) tells us something about
f itself. This is even true of our rst topic, approximation. For instance, knowing that f (x) = e x
satises f (0) = 1 and f (0) =
STUDY GUIDE TO CALCULUS
This Student Study Guide accompanies the textbook Calculus by Gilbert Strang. It is correlated
section by section with the essential points of the text. The Guide contains four components which
experience has shown are most helpful
3.1 Linear Approximation
CHAPTER 3
3.1
(page 95)
APPLICATIONS OF DERIVATIVES
Linear Approximation
(page 95)
This section is built on one idea and one formula. The idea is t o use the tangent line as an approximation t o
the curve. The formula is written i
4.1 The Chain Rule
DERIVATIVES BY THE CHAIN RULE
CHAPTER 4
4.1
(page 158)
The Chain Rule
(page 158)
+
+
The function sin(3x 2) is 'composed' out of two functions. The inner function is u(x) = 32 2. The outer
function is sin u. I don't write sin x because
Jason Starr
Fall 2005
18.01 Calculus
Due by 2:00pm sharp
Friday, Sept. 16, 2005
Solutions to Problem Set 1
Part I/Part II
Part I(20 points)
(a) (2 points)
(b) (2 points)
(c) (2 points)
(d) (2 points)
(e) (2 points)
(f ) (2 points)
(g) (2 points)
(h) (2 po
Jason Starr
Fall 2005
18.01 Calculus
Due by 2:00pm sharp
Friday, Sept. 30, 2005
Solutions to Problem Set 2
Part I/Part II
Part I(20 points)
(a) (2 points)
(b) (2 points)
(c) (2 points)
(d) (2 points)
(e) (2 points)
(f ) (2 points)
(g) (2 points)
(h) (2 po
2.1 The Derivative of a Function
CHAPTER 2
2.1
(page 49)
DERIVATIVES
The Derivative of a Function
(page 49)
In this section you are mainly concerned with learning the meaning of the derivative, and also the notation.
The list of functions with known deriv
18.01 Practice Exam 2
Problem 1. (20)
Find the local maxima and minima and points of inection of
2x3 + 3x2 12x + 1 .
Then use this data to sketch its graph on the given axes, showing also where it is convex (concave up) or
concave (down). (Note that diere
Problem 1. (10 pts.) Find the tangent line to y =
1 2
x at x = 1
3
Problem 2. Find the derivative of the following functions:
a. (7 pts.)
x
1 x
b. (8 pts).
cos (2 x)
x
c. (5 pts). e 2 f ( x ) = g(x)
d. (5 pts.) ln(sin x)
Problem 3. (15 pts.) Find
by
dy
fo
18.01 Calculus
Due by 2:00pm sharp
Friday, Dec. 2, 2005
Jason Starr
Fall 2005
Solutions to Problem Set 7
Late homework policy. Late work will be accepted only with a medical note or for another
Institute-approved reason.
Cooperation policy. You are encour
18.01 Practice Questions for Exam 1
Solutions will be posted on the 18.01 website.
No books, notes, or calculators will be allowed at the exam.
1.
Evaluate each of the following, simplifying where possible; for (b) indicate reasoning.
The letters a and k
18.01 Exam 2
Tuesday, Oct. 17, 2006
Problem 1. (15 pts.) Estimate the following to two decimal places (show work)
a. (8 pts.)
b. (7 pts.)
sin( + 1/100)
101
Problem 2. (20 pts.) Sketch the graph of
y=
4
+ x + 1 on < x <
x
and label all critical points and
18.01 Practice Questions for Exam 1
Solutions will be posted on the 18.01 website
Problem 1. Evaluate each of the following:
a)
b)
d
x
dx 1 + 2x
x=1
d
(u ln 2u)
du
Problem 2.
(simplify your answer)
a) Evaluate
d
dt
1 k cos2 t, where k is constant.
b) Chec
18.01 Calculus
Due by 2:00pm sharp
Friday, Nov. 18, 2005
Jason Starr
Fall 2005
Solutions to Problem Set 6
Late homework policy. Late work will be accepted only with a medical note or for another
Instituteapproved reason.
Cooperation policy. You are encour
18.01 Practice Questions for Exam 2
Solutions will be posted on the 18.01 website. No books, notes, calculators. Show work.
1. For the function 3x5 5x3 + 1, sketch the graph over a suitable interval showing
all the local maximum and minimum points on the
18.01 Fall 2006
Exam 1 Review
General Di erentiation Formulas
(u + v)0
=
u0 + v 0
(cu)0
=
cu0
(uv)0
u 0
v
=
u0 v + uv 0 (product rule)
u0 v uv 0
(quotient rule)
v2
d
f (u(x)
dx
=
=
f 0 (u(x) u0 (x) (chain rule)
You can remember the quotient rule by rewri
18.01 Calculus
Due by 2:00pm sharp
Friday, Nov. 4, 2005
Jason Starr
Fall 2005
Solutions to Problem Set 5
Late homework policy. Late work will be accepted only with a medical note or for another
Instituteapproved reason.
Cooperation policy. You are encoura
Jason Starr
Fall 2005
18.01 Calculus
Due by 2:00pm sharp
Friday, Oct. 14, 2005
Solutions to Problem Set 3
Part I/Part II
Part I(20 points)
(a)(2 points)
(b)(2 points)
(c)(2 points)
(d)(2 points)
(e)(2 points)
(f )(2 points)
(g)(2 points)
(h)(2 points)
(i)