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2.1 Tangent Lines and Velocity
1. Slope of a Line:
Let L be a line passing through points x 1 , y 1 and x 2 , y 2 . Then
y y
the slope of L
m x2 x1
2
1
the equation of L
y y 1 mx x 1
If these two points are on a curve y fx, then this line is also called
1.4 Continuity and Its Consequences
Graphically, a function f is continuous at x a if the graph of f has
no break up (i.e., hole, vertical asymptote, jump) at x a.
Examples: Functions that are not continuous at x 0, or x 2.
y
y 20
1.0
y
1.4
1.2
1.0
0.8
De
Review: Polynomials and Special Functions
Polynomials:
Consider Px a n x n a n1 x n1 a 1 x a 0 where a n 0. Px is an
nth degree polynomial.
1. Zeros:
The following statements are equivalent: let c be a real number,
a. x c is a zero of Px;
b. Pc 0;
c. Px x
1.2 The Concepts of Limit
What is a limit (of a sequence of real numbers)?
Example: two sequences of numbers:
2. 01, 2. 001, 2. 0001, 2. 00001, 2. 000001,
1. 99, 1. 999, 1. 9999, 1. 99999, 1. 999999
Example: a sequence of numbers:
1 , 2 , 3 , , 999 , .
Math 131 - Fall, 2009 Homework 1 - Due Friday, Aug. 28 (10pts) Name: _Solutions_
Mathematics Quote: The pleasure we obtain from music comes from counting, but counting
unconsciously. Music is nothing but unconscious arithmetic. Gottfried Wilhelm Leibniz (
1. 3 Computation of Limits
v. Exponential functions.
1. Limits of Some Special Functions:
i. For any polynomial px and any real number a,
ii. Power of functions. Let lim xa fx L. Then for n a positive
integer.
n
lim fx L n .
xa
lim
xa
if n is odd
n
L
if n
MATH 131-03, Fall, 2013
Quiz 8(a)
Name: _
No Calculator. Show your work in details.
1. (3pts) Find h x where
(1) h x 2 x cot x
(2) h x
x
sec x
tan x
2. (2pts) Let H sin . Compute H and H .
3. (3pts) Let f x x 2 cos x .
(1) Give the equation of the tangen
MATH 131-03, Fall, 2013 Quiz 7 - Take-Home Due Oct 7 in class Name: _Solutions_
No Calculator. You must work on this quiz on your own. Show your work in details.
1. (1pts) State the definitions of f a and f x .
fa h fa
provided the limit exists. provided
MATH 131-03, Fall, 2013
Quiz 8
Name: _Solutions_
No Calculator. Show your work in details.
1. (3pts) Find h x where
(1) (a) hx 2 x cotx , h x 2 x ln2 cotx 2 x csc 2 x
(b) hx 3 x cscx , h x 3 x ln3 cscx 3 x cscx cotx
x secx
1 secx tanx tanx x secx sec 2
MATH 131-03, Fall, 2013 Quiz 7 - Take-Home Due Oct 7 in class Name: _
No Calculator. You must work on this quiz on your own. Show your work in details.
1. (1pts) State the definitions of f a and f x .
2. (2pts) Let fx
x . Derive f 1 by definition.
2x 1
3
MATH 131-03, Fall, 2013
Quiz 5
Name: _Solutions_
No Calculator. Show your work in details.
1. (3pts) The graph of fx is given below. Find the following based on the graph.
(1) lim x1 fx
(2) lim x3 fx
(3) lim x fx 2
(4) lim x fx 2
(5) Give all asymptotes
MATH 131-03, Fall, 2013
Quiz 6(a)
Name: _
No Calculator. Show your work in details.
1. (5pts) Compute the following limits:
2x 6 1
(1) lim x
3x 3 2x
(2) lim x 2x 4x 2 2x
2x
2x
(3) lim x 2e 2x 5e 2x
3e 4e
(4) lim x arctane 3x
2. (2pts) Find the slope of
MATH 131-03, Fall, 2013
Quiz 5(a)
Name: _
No Calculator. Show your work in details.
1. (3pts) The graph of fx is given below. Find the following based on the graph.
(1) lim x1 fx
(2) lim x3 fx
(3) lim x fx
(4) lim x fx
(5) Give all asymptotes of y fx
3.4 - Increasing and Decreasing Functions
Theorem: Suppose that f is differentiable on an interval I.
1. Increasing and Decreasing Functions
(i) If f x 0 for all x in I, then f is increasing on I.
(ii) If f x 0 for all x in I, then f is decreasing on I.
P
4.5 - Fundamental Theorem of Calculus
More Examples: Evaluate the following definite integrals.
1. Fundamental Theorem of Calculus, Part I
a.
x x 2x 4
dx
1
x2
1 e 2x 2 e x
c.
dx
0
ex
If f is continuous on a, b and Fx is an antiderivative of f, then
b
a
1. Absolute Minimum and Absolute Maximum:
Example: Let fx 2x 3 3x 2 12x 5. Find the absolute extrema of
fx on (i) 0, 4 and (ii) 2, 1.
Let f be continuous on a, b. Then f must attain its absolute maximum
and minimum values on a, b. Furthermore, if f is als
3.1 - Application of Differentiation:
1. Linear Approximation:
Applications:
(1) What are 1. 001 , cos0. 001, ln1. 001 (without a calculator)?
(2) If we know the temperature is 91 yesterday and 87 today, can we
predict the temperature tomorrow?
Idea:
Exam
3.3 - Critical Numbers, and Maximum & Minimum Values
y
A point c in the domain of f is called a critical number of f if
(i) f c 0 or (ii) f c is not defined.
15
is given at the right.
1. Critical Numbers
Example: The graph of f x
10
(1) Find all possible
3.2 - Indeterminate Forms and LHopitals Rule:
1. Indeterminate Forms:
fx
is called indeterminate if either both fx and gx
gx
approach 0 or , that is,
fx
0 or .
0
gx
The limit: lim xa
Suppose that fa 0 and ga 0
Using linear approximations for fx and gx,
f
Example: Find a value of c satisfying the conclusion of Rolles
Theorem for fx x 3 3x 2 2x 2 on the interval 0, 1.
2.9 - Rolles Theorem and The Mean Value Theorem
1. Rolles Theorem
Theorem: Suppose that fx is continuous on the interval a, b and
is differen
Example: Write the expression in summation notation:
4.2 - Sums and Sigma Notation
1. Sums and Sigma Notation
a. 1 2 2 2 1000 2
The sum of the first 10 integers:
1 2 3 4 5 6 7 8 9 10 510 5 55
The sum of the first 100 integers:
1 2 3 4 99 100 50100 50 5050
4.3 Areas
1. Approximations of Areas
Known area formulas:
Consider to approximate the area A under the curve y fx on a, b
where f is continuous and nonnegative, fx 0, on a, b. Here are
examples that A is approximated by a sum of the areas of 10 and 15
rec
3.6 - Curve Sketch
f 3 1 4 0
9
,
f 1 1 4 0
1
2
Example Let fx x x 4 . Find
1. domain of f , zeros of f and vertical asymptotes;
2. the intervals of increase and decrease;
3. all local extrema;
4. the intervals of concavity;
5. all inflection points; and
Math 131 Fall, 2008
Review for the Final Exam
The final exam is designed as a two-hour exam with 200 points. You will have three hours to complete the
exam. The final exam consists of two parts:
Part I - without calculator: conceptual questions, limits, d
Example: The graph of f is given below. Determine graphically the
interval on which f is
3.5 - Concavity
1. Concave up and concave down
y
For a function f that is differentiable on an interval I, the graph of f is
If f is concave up on a, b, then the sec
2. Indefinite Integrals
4.1 - Antiderivative
1. Antiderivatives
Definition: The function Fx is an antiderivative of fx if
F
x fx.
Definition: Let F be an antiderivative of f. The indefinite integral of f is
defined by
fxdx Fx C
Theorem: Suppose that F an
Example: Evaluate the following integrals:
4.6 - Integration by Substitution
Integration by Substitution
a.
Let Fx be an antiderivative of fx F x fx . Then
f gxg xdx Fgx C.
2x 2 1 dx
2
b.
0
x
2x 2
1
dx
a. x 2x 2 1 dx
Derivation: u gx, du g xdx and f gxg
4.4 - Definite Integrals
Note that:
1. Definite Integrals
(1) A definite integral fxdx is a constant (an indefinite integral
a
fxdx Fx C is a function in x.
b
Recall: The Riemann Sum for the area of the region under the curve
y fx for x in a, b :
An
fc