Math 131 - Fall, 2009 Homework 2 - Due Tuesday, Sept. 1 Name: _Solutions_
Mathematics Quote: I consider that I understand an equation when I can predict the properties of its
solutions, without actually solving it. Dirac
Instructions for homework:
(1) Sho
Math 131 - Fall, 2009 Homework 2 - Due Tuesday, Sept. 1 Name: _
Mathematics Quote: I consider that I understand an equation when I can predict the properties of its
solutions, without actually solving it. Dirac
Instructions for homework:
(1) Show your wor
Basic Precalculus Toolbox
Often the diculties students run into in calculus stems from precalculus deciencies. To excel in
calculus you must be strong in algebra, functions, and mathematical models. Algebra skills and your
understanding of functions come
Precalculus Review Notes Page 1
Precalculus Review
Exponent Laws:
a s a t = a s +t
Example: 2 5 2 3 = 2 5+3 = 2 8
as
= a s t
at
Example:
(a )
Example: 2 3
st
()
= a st
(a b )s
= as bs
EXAMPLE:
26
= 2 6 3 = 2 3
23
5
= 2 35 = 215
Example: (2 3) = 2 5 35
5
(
18.01 Calculus
Jason Starr
Fall 2005
Lecture 32. December 9, 2005
Practice Problems. Course Reader: RI.
1. Using p ower series to solve calculus problems. The reason power series are useful is
because they allow us to describe functions that have no direc
18.01 Calculus
Jason Starr
Fall 2005
Lecture 31. December 8, 2005
Practice Problems. Course Reader: 7B4, 7B6, 7C1, 7C5, 7D1, 7D2.
1. Power series. Given a real number a and a sequence of real numbers (cn )n0 , there is an
associated expression, called a p
18.01 Calculus
Jason Starr
Fall 2005
Lecture 30. December 6, 2005
Practice Problems. Course Reader: 6C2.
1. Sequences By denition, a sequence of real numbers is a rule assigning to each counting number
n an associated real number an . The integer n is cal
18.01 Calculus
Jason Starr Fall 2005
Lecture 29. December 2, 2005
Homework. Problem Set 8 Part I: (c), (d) and (e); Part II: Problems 1 and 2.
Practice Problems. Course Reader: 6B7.
1. A problem with Riemann integrals. Riemann integrals are dened in ve
18.01 Calculus
Jason Starr Fall 2005
Lecture 28. December 1, 2005
Homework. Problem Set 8 Part I: (a) and (b).
Practice Problems. Course Reader: 6A1, 6A2.
1. Indeterminate forms. Expressions of the form 0/0, /, 0 , , 0 and 0 are called indeterminate fo
18.01 Calculus
Jason Starr
Fall 2005
Lecture 27. November 22, 2005
Homework. Problem Set 7 Part II: Problem 2.
Practice Problems. Course Reader: 5F2, 5F3, 5F4, 5F5.
1. Integration by parts. The dierential form of the product rule is,
d(uv ) = udv + v du.
18.01 Calculus
Jason Starr Fall 2005
Lecture 26. November 18, 2005
Homework. Problem Set 7 Part I: (f)(g); Part II: Problem 1 and Problem 2 (a), (b).
Practice Problems. Course Reader: 5E8, 5E10, and please read through Part II of Problem
Set 7.
18.01 C
18.01 Calculus
Jason Starr
Fall 2005
Lecture 25. November 17, 2005
Homework. Problem Set 7 Part I: (a)(e)
Practice Problems. Course Reader: 5D2, 5D6, 5D7, 5D10, 5D14
1. Inverse hyperbolic functions. There are a few other useful formulas for hyperbolic fun
18.01 Calculus
Jason Starr
Fall 2005
Lecture 24. November 15, 2005
Practice Problems. Course Reader: 5A1, 5A2, 5A3, 5A5, 5A6.
1. Inverse functions. Let a, b, s and t be constants. Let y = f (x) be a function dened on the
interval,
a x b,
and whose values
18.01 Calculus
Jason Starr
Fall 2005
Lecture 23. November 8, 2005
Homework. Problem Set 6 Part I: (i) and (j); Part II: Problem 2.
Practice Problems. Course Reader: 4I1, 4I4, 4I6.
1. Tangent lines to parametric curves. This short section was not explicitl
18.01 Calculus
Jason Starr
Fall 2005
Lecture 22. November 4, 2005
Homework. Problem Set 6 Part I: (f)(h); Part II: Problem 2 (a) and (c).
Practice Problems. Course Reader: 4G1, 4G4, 4G6, 4H1, 4H3.
1. Surface area of a right circular cone. Before attacking
18.01 Calculus
Jason Starr
Fall 2005
Lecture 21. November 3, 2005
Homework. Problem Set 6 Part I: (a) (e); Part II: Problem 1.
Practice Problems. Course Reader: 4E2, 4E5, 4E7, 4F1, 4F6.
1. Parametric equations. To this point in the course, plane curves we
18.01 Calculus
Jason Starr
Fall 2005
Lecture 20. November 1, 2005
Practice Problems. Course Reader: 4C2, 4C6, 4D1, 4D4, 4D8.
1. Average values. Given a function f (x) dened on some interval [a, b], what is the average
value of f (x)? A reasonable rst appr
18.01 Calculus
Jason Starr
Fall 2005
Lecture 19. October 28, 2005
Homework. Problem Set 5 Part I: (d) and (e); Part II: Problems 2 and 3.
Practice Problems. Course Reader: 4A1, 4A3, 4B1, 4B3, 4B6.
1. Dierentials revisited. In a typical applied integration
18.01 Calculus
Jason Starr
Fall 2005
Lecture 18. October 25, 2005
Homework. Problem Set 5 Part I: (c).
Practice Problems. Course Reader: 3G1, 3G2, 3G4, 3G5.
1. Approximating Riemann integrals. Often, there is no simpler expression for the antideriva
tive
18.01 Calculus
Jason Starr
Fall 2005
Lecture 17. October 21, 2005
Homework. Problem Set 5 Part I: (a) and (b); Part II: Problem 1.
Practice Problems. Course Reader: 3F1, 3F2, 3F4, 3F8.
1. Ordinary dierential equations. An ordinary dierential equation is a
18.01 Calculus
Jason Starr
Fall 2005
Lecture 16. October 20, 2005
Practice Problems. Course Reader: 3D1, 3D3, 3D7, 3E3, 3E4.
1. Dummy variables. Give a Riemann integrable function f (x) dened on an interval [a, b], the
notation,
b
f (x)dx,
a
is shorthand
18.01 Calculus
Jason Starr
Fall 2005
Lecture 15. October 18, 2005
Homework. Problem Set 4 Part I: (d) and (e); Part II: Problem 2.
Practice Problems. Course Reader: 3B6, 3C2, 3C3, 3C4, 3C6.
1. The Riemann sum for the exponential function. The problem is t
18.01 Calculus
Jason Starr
Fall 2005
Lecture 14. October 14, 2005
Homework. Problem Set 4 Part II: Problem 2.
Practice Problems. Course Reader: 3B1, 3B3, 3B4, 3B5.
1. The problem of areas. The ancient Greeks computed the areas of triangles, quadrilaterals
18.01 Calculus
Jason Starr
Fall 2005
Lecture 13. October 13, 2005
Homework. Problem Set 4 Part I: (a) and (b); Part II: Problem 3.
Practice Problems. Course Reader: 3A1, 3A2, 3A3.
1. Dierentials. An alternative notation for derivatives is dierential notat
18.01 Calculus
Lecture 12. October 6, 2005
Jason Starr
Fall 2005
,
Homework. Problem Set 3 Part I: (i) and (j).
This was a guest lecture by Sabri Kilic. Notes from the lecture will not be posted. As always,
please do the required reading in the course tex
18.01 Calculus
Jason Starr Fall 2005
Lecture 11. October 4, 2005 Homework. Problem Set 3 Part I: (g) and (h). Practice Problems. Course Reader: 2E4, 2E8, 2E9. 1. Related rates. A situation that arises often in practice is that two quantities, say x and y