Section 1.1 HW
Please give complete, well written solutions to the following exercises.
1. Sketch the region in the plane that solves the inequality
|x| + |y | 1
Hint: Start by assuming both x and y are positive.
2. Consider the curve that satises the equ
Practice problems from old Midterms
1. False. Limit laws apply only if the individual limits exist.
2. True. Use theorem 1.5.7; If f is continuous at a, and lim g (x) = b, then lim f (g (x) = f (b); here g (t) = 4t2 12
x a
xa
3. False. Counter example: f
Multiple choice
(15 points)
For each of the following problems circle the correct statement(s):
(A) Given the functions f (x) = x2 and g (x) = x + 1, the composition f (g (x) corresponds
to which of the following?
1. f (g (x) = x2 + 1
2. f (g (x) = x2 + x
Practice problems from old Midterms
These problems are collected from several dierent midterms and represent a sample of the kinds
of questions in the midterms. This is by no means to be considered as exhaustive and for more
practice, you should look at s
Section 1.2 HW
Please give complete, well written solutions to the following exercises.
1. Let f (x) = m1 x + b1 and g (x) = m2 x + b2 . Show that f g is also a
linear function. What is the slope of f g ?
2. Let
1
.
2x
Dene fn+1 (x) = (f0 fn )(x) where n
Section 1.3 HW
Please give complete, well written solutions to the following exercises.
1. 22 on page 34.
2. In your own words, explain why
lim H (x)
x 0
does not exist according to denition 1 on page 25.
3. Find two functions f and g so that neither func
Section 1.4 HW
Please give complete, well written solutions to the following exercises.
1. If
lim (f + g ) = 2 and
xa
lim (f g ) = 1
xa
Find
lim f g
xa
2. Find all values of a so that
lim
x 0
ax + 4 2
= 1.
x
3. Compute
|2x 1| |2x + 1|
.
x0
x
Hint: As x 0,
A BRIEF INTRODUCTION TO CALCULUS
STEVEN HEILMAN
1. Applications of Calculus
It seemed a limbo of painless patient consciousness through which souls of
mathematicians might wander, projecting long slender fabrics from plane to
plane of ever rarer and paler
HW 3
Please give complete, well written solutions to the following exercises.
Section 1.5
1. Find all the values for a and b so that
ax b
x 1
2
f (x) = 2x + 3ax + b 1 < x 1
4
x>1
is continuous.
2. For what values of x is the function g (x) = (sin(3x5 + 10
Multiple choice
(15 points)
For each of the following problems circle the correct statement(s):
(A) Given the functions f (x) = x2 and g (x) = x + 1, the composition f (g (x) corresponds
to which of the following?
1. f (g (x) = x2 + 1
2. f (g (x) = x2 + x
Notes and Solved Examples
Limits Involving
Basic Ideas:
1) Vertical Asymptote:
Vertical Asymptotes (VA) occur when at some points the value of the function or the limit of the function
1
approaches as x > a. Easiest examples to see this are f (x) = 2 , w
Notes and Solved Example
2.2: Derivative a s a function
Basic Ideas:
If you compute derivatives at all points on a curve (if it exists) then for every x there is a rule that assigns
a value f (x), slope of the tangent to the curve at that point. This coll
Notes and Solved Examples
2.3: Rules for Dierentiation
Basic Ideas:
Computing derivatives using rules. Here on (for the next few sections), we are following the same pattern
as we did when we combined functions in various ways. If you have two functions,