MATH 1431

Chap. 2 Chap. 3
MATH 1431 - 23974
Annalisa Quaini
quaini@math.uh.edu
Office : PGH 662
Lecture : TuTh 5:30PM-7:00PM
Office hours : W 8AM-10AM
Daily quiz 2 is due on Thursday before class.
You are responsible for weekly quizzes.
http://www.math.uh.edu/∼quaini
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Chap. 2 Chap. 3
Sect. 2.6
Solving Inequalities
Solve the inequality for
x(3x − 12)(4x − 36) ≥ 0
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Chap. 2 Chap. 3
Sect. 2.6
Solve the inequality for
4
1
+
>0
x −1 x −6
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Chap. 2 Chap. 3
Sect. 2.6
The Intermediate Value Theorem
Intermediate value theorem
If f is continuous on [a, b] and K is any number between f (a) and
f (b), then there is at least one number c in the interval (a, b) such
that
f (c) = K .
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Chap. 2 Chap. 3
Sect. 2.6
Definition
A function is bounded on an interval I if there are numbers k and
K such that
k ≤ f (x) ≤ K , for all x in I
Examples:
f (x) = sin(x),
f (x) = x 2 ,
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−∞ < x < ∞
0≤x ≤1
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Chap. 2 Chap. 3
Sect. 2.6
Extreme Value Theorem
Extreme value theorem
A function f continuous on a closed interval I = [a, b] takes on
both a maximum value M and a minimum value m.
M and m are called the extreme values of the function f on I .
f is bounded on I : m ≤ f (x) ≤ M, for all x in I .
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
DIFFERENTATION
3.1 The Derivative
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Chap. 2 Chap. 3
Secant Lines
vs.
Sect. 3.1 Sect. 3.2
Tangent Lines
2
1
-1
0
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1
2
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
2
1
-1
0
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1
2
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
2
1
-1
0
Secant slope =
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f (c+h)−f (c)
h
1
=
2
∆y
∆x
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
2
1
-1
0
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1
2
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
Tangent Line Slope
2
1
-1
0
1
2
f (c + h) − f (c)
= f 0 (c)
h→0
h
Slope = lim
f 0 (c) is the derivative of f at x = c.
f 0 (c) is the rate of change of f at x = c.
The derivative measures how f (x) changes when x changes.
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
Example
Find the slope of the tangent line to f (x) = x 2 at x = 1.
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
THE DERIVATIVE
Definition
A function is said to be differentiable at c if
f (c + h) − f (c)
h→0
h
lim
exists.
If the limit exists, it is called the derivative of f at c and is
denoted by f 0 (c).
Other notation:
d
f (x) = f 0 (x)
dx
Geometrical interpretation: f 0 (c) is the slope of the tangent line
at x = c.
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
Theorem
If f is differentiable at x, then f is continuous at x.
The converse is not true!
How can the graph of a function be used to determine
if/where a function is not differentiable?
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
f (c + h) − f (c)
h→0
h
slope of the tangent: f 0 (c) = lim
Equation of the tangent line through the point (c, f (c))
y = f (c) + f 0 (c)(x − c)
Need: POINT + SLOPE
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
Tangent Lines
2
1
-2
-1
0
1
2
Identify the values of x where the slope of the tangent line is
positive
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
Tangent Lines
2
1
-2
-1
0
1
2
Identify the values of x where the slope of the tangent line is
negative
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
Tangent Lines
2
1
-2
-1
0
1
2
Identify the values of x where the slope of the tangent line is ZERO
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
Examples
1
2
Find the derivative of f (x) = x 2 at x = 1, and give the
equation of the tangent line to the graph of y = x 2 at x = 1.
Find an equation for the tangent line of f (x) =
x = 2.
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1
at
x −1
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
The line passing through a point on the graph and perpendicular
to the tangent is called the normal line.
How can we use the derivative to find the slope of the
normal line to the graph of f (x) at x = c?
Remember: for perpendicular lines, neither of which vertical,
m1 m2 = −1
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
The line passing through a point on the graph and perpendicular
to the tangent is called the normal line.
How can we use the derivative to find the slope of the
normal line to the graph of f (x) at x = c?
Remember: for perpendicular lines, neither of which vertical,
m1 m2 = −1
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Chap. 2 Chap. 3
Slope of the normal line: −
Sect. 3.1 Sect. 3.2
1
f 0 (c)
Equation of the normal line
y = f (c) −
1
(x − c)
f 0 (c)
provided f 0 (c) 6= 0.
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
Find an equation for the normal line of f (x) =
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1
at x = 2.
x −1
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
The Derivative as a Function
You can compute f 0 (x) at x = 2, −3, etc.
You can also write the derivative for each and every x.
Definition
The derivative of a function f is the function f 0 with value at x
given by
f (x + h) − f (x)
,
f 0 (x) = lim
h→0
h
provided the limit exists.
To differentiate the function f is to find its derivative.
Ex. f (x) = x 2
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
3.2 SOME DIFFERENTATION
FORMULAS
Goal: Find the derivatives of polynomial and rational functions.
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
If a is a real number,
f (x) = a,
f 0 (x) = 0.
f (x) = x,
f 0 (x) = 1.
Powers
d n
x = nx n−1 ,
dx
Notice that it works also for n = 0.
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n ∈ Q.
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Chap. 2 Chap. 3
Sect. 3.1 Sect. 3.2
Examples
Find the derivative of:
1
f (x) = x 2 .
2
f (x) = x 3 .
3
f (x) = x 73 .
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