1.
Review for Midterm I
Math 1a
October 21, 2007
Announcements
Midterm I 10/24, Hall 7-9pm, Hall A and D
Old exams and solutions on website
problem sessions every night, extra MQC hours
2.
Outline
The Intermediate Value
Limits
Theorem
Concept
Computation Derivatives
Limits involving inﬁnity Concept
Continuity Intepretations
Concept Implications
Examples Computation
3.
Outline
The Intermediate Value
Limits
Theorem
Concept
Computation Derivatives
Limits involving inﬁnity Concept
Continuity Intepretations
Concept Implications
Examples Computation
4.
The concept of Limit
Learning Objectives
state the informal deﬁnition of a limit (two- and one-sided)
observe limits on a graph
guess limits by algebraic manipulation
guess limits by numerical information
5.
Heuristic Deﬁnition of a Limit
Deﬁnition
We write
lim f (x) = L
x→a
and say
“the limit of f (x), as x approaches a, equals L”
if we can make the values of f (x) arbitrarily close to L (as close to
L as we like) by taking x to be suﬃciently close to a (on either side
of a) but not equal to a.
15.
Outline
The Intermediate Value
Limits
Theorem
Concept
Computation Derivatives
Limits involving inﬁnity Concept
Continuity Intepretations
Concept Implications
Examples Computation
16.
Computation of Limits
Learning Objectives
know basic limits like limx→a x = a and limx→a c = c
use the limit laws to compute elementary limits
use algebra to simplify limits
use the Squeeze Theorem to show a limit
17.
Limit Laws
Suppose that c is a constant and the limits
lim f (x) and lim g (x)
x→a x→a
exist. Then
1. lim [f (x) + g (x)] = lim f (x) + lim g (x)
x→a x→a x→a
2. lim [f (x) − g (x)] = lim f (x) − lim g (x)
x→a x→a x→a
3. lim [cf (x)] = c lim f (x)
x→a x→a
4. lim [f (x)g (x)] = lim f (x) · lim g (x)
x→a x→a x→a
18.
Limit Laws, continued
lim f (x)
f (x)
= x→a
5. lim , if lim g (x) = 0.
x→a g (x) lim g (x) x→a
x→a
n
n
6. lim [f (x)] = lim f (x) (follows from 3 repeatedly)
x→a x→a
7. lim c = c
x→a
8. lim x = a
x→a
9. lim x n = an (follows from 6 and 8)
x→a
√ √
10. lim n x = n a
x→a
n
11. lim f (x) = lim f (x) (If n is even, we must additionally
n
x→a x→a
assume that lim f (x) > 0)
x→a
19.
Direct Substitution Property
Theorem (The Direct Substitution Property)
If f is a polynomial or a rational function and a is in the domain of
f , then
lim f (x) = f (a)
x→a
20.
Theorem (The Squeeze/Sandwich/Pinching Theorem)
If f (x) ≤ g (x) ≤ h(x) when x is near a (as usual, except possibly
at a), and
lim f (x) = lim h(x) = L,
x→a x→a
then
lim g (x) = L.
x→a
21.
Outline
The Intermediate Value
Limits
Theorem
Concept
Computation Derivatives
Limits involving inﬁnity Concept
Continuity Intepretations
Concept Implications
Examples Computation
22.
Limits involving inﬁnity
Learning Objectives
know vertical asymptotes and limits at the discontinuities of
”famous” functions
intuit limits at inﬁnity by eyeballing the expression
show limits at inﬁnity by algebraic manipulation
23.
Deﬁnition
Let f be a function deﬁned on some interval (a, ∞). Then
lim f (x) = L
x→∞
means that the values of f (x) can be made as close to L as we
like, by taking x suﬃciently large.
Deﬁnition
The line y = L is a called a horizontal asymptote of the curve
y = f (x) if either
lim f (x) = L or lim f (x) = L.
x→∞ x→−∞
y = L is a horizontal line!
24.
Theorem
Let n be a positive integer. Then
1
limx→∞ =0
xn
limx→−∞ x1n =0
25.
Using the limit laws to compute limits at ∞
Example
Find
2x 3 + 3x + 1
lim
x→∞ 4x 3 + 5x 2 + 7
if it exists.
A does not exist
B 1/2
C0
D∞
26.
Using the limit laws to compute limits at ∞
Example
Find
2x 3 + 3x + 1
lim
x→∞ 4x 3 + 5x 2 + 7
if it exists.
A does not exist
B 1/2
C0
D∞
27.
Solution
Factor out the largest power of x from the numerator and
denominator. We have
2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 )
=3
4x 3 + 5x 2 + 7 x (4 + 5/x + 7/x 3 )
2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3
lim = lim
x→∞ 4x 3 + 5x 2 + 7 x→∞ 4 + 5/x + 7/x 3
2+0+0 1
= =
4+0+0 2
Upshot
When ﬁnding limits of algebraic expressions at inﬁnitely, look at
the highest degree terms.
28.
Solution
Factor out the largest power of x from the numerator and
denominator. We have
2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 )
=3
4x 3 + 5x 2 + 7 x (4 + 5/x + 7/x 3 )
2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3
lim = lim
x→∞ 4x 3 + 5x 2 + 7 x→∞ 4 + 5/x + 7/x 3
2+0+0 1
= =
4+0+0 2
Upshot
When ﬁnding limits of algebraic expressions at inﬁnitely, look at
the highest degree terms.
29.
Inﬁnite Limits
Deﬁnition
The notation
lim f (x) = ∞
x→a
means that the values of f (x) can be made arbitrarily large (as
large as we please) by taking x suﬃciently close to a but not equal
to a.
Deﬁnition
The notation
lim f (x) = −∞
x→a
means that the values of f (x) can be made arbitrarily large
negative (as large as we please) by taking x suﬃciently close to a
but not equal to a.
Of course we have deﬁnitions for left- and right-hand inﬁnite limits.
30.
Vertical Asymptotes
Deﬁnition
The line x = a is called a vertical asymptote of the curve
y = f (x) if at least one of the following is true:
limx→a f (x) = ∞ limx→a f (x) = −∞
limx→a+ f (x) = ∞ limx→a+ f (x) = −∞
limx→a− f (x) = ∞ limx→a− f (x) = −∞
31.
Finding limits at trouble spots
Example
Let
t2 + 2
f (t) =
t 2 − 3t + 2
Find limt→a− f (t) and limt→a+ f (t) for each a at which f is not
continuous.
32.
Finding limits at trouble spots
Example
Let
t2 + 2
f (t) =
t 2 − 3t + 2
Find limt→a− f (t) and limt→a+ f (t) for each a at which f is not
continuous.
Solution
The denominator factors as (t − 1)(t − 2). We can record the
signs of the factors on the number line.
42.
Outline
The Intermediate Value
Limits
Theorem
Concept
Computation Derivatives
Limits involving inﬁnity Concept
Continuity Intepretations
Concept Implications
Examples Computation
43.
Outline
The Intermediate Value
Limits
Theorem
Concept
Computation Derivatives
Limits involving inﬁnity Concept
Continuity Intepretations
Concept Implications
Examples Computation
44.
Continuity
Learning Objectives
intuitive notion of continuity
deﬁnition of continuity at a point and on an interval
ways a function can fail to be continuous at a point
45.
Deﬁnition of Continuity
Deﬁnition
Let f be a function deﬁned near a. We say that f is continuous at
a if
lim f (x) = f (a).
x→a
46.
Free Theorems
Theorem
(a) Any polynomial is continuous everywhere; that is, it is
continuous on R = (−∞, ∞).
(b) Any rational function is continuous wherever it is deﬁned; that
is, it is continuous on its domain.
47.
Outline
The Intermediate Value
Limits
Theorem
Concept
Computation Derivatives
Limits involving inﬁnity Concept
Continuity Intepretations
Concept Implications
Examples Computation
48.
The Limit Laws give Continuity Laws
Theorem
If f and g are continuous at a and c is a constant, then the
following functions are also continuous at a:
1. f + g
2. f − g
3. cf
4. fg
f
5. (if g (a) = 0)
g
49.
Transcendental functions are continuous, too
Theorem
The following functions are continuous wherever they are deﬁned:
1. sin, cos, tan, cot sec, csc
2. x → ax , loga , ln
3. sin−1 , tan−1 , sec−1
50.
Outline
The Intermediate Value
Limits
Theorem
Concept
Computation Derivatives
Limits involving inﬁnity Concept
Continuity Intepretations
Concept Implications
Examples Computation
51.
The Intermediate Value Theorem
Learning Objectives
state IVT
use IVT to show that a function takes a certain value
use IVT to show that a certain equation has a solution
reason with IVT
52.
A Big Time Theorem
Theorem (The Intermediate Value Theorem)
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f (a) and f (b), where f (a) = f (b). Then
there exists a number c in (a, b) such that f (c) = N.
54.
Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b]
f (x)
x
55.
Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b]
f (x)
f (b)
f (a)
x
a b
56.
Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f (a) and f (b), where f (a) = f (b).
f (x)
f (b)
N
f (a)
x
a b
57.
Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f (a) and f (b), where f (a) = f (b). Then
there exists a number c in (a, b) such that f (c) = N.
f (x)
f (b)
N
f (a)
x
a c b
58.
Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f (a) and f (b), where f (a) = f (b). Then
there exists a number c in (a, b) such that f (c) = N.
f (x)
f (b)
N
f (a)
x
a b
59.
Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f (a) and f (b), where f (a) = f (b). Then
there exists a number c in (a, b) such that f (c) = N.
f (x)
f (b)
N
f (a)
x
a c1 c2 c3 b
60.
Using the IVT
Example
Prove that the square root of two exists.
Proof.
Let f (x) = x 2 , a continuous function on [1, 2]. Note f (1) = 1 and
f (2) = 4. Since 2 is between 1 and 4, there exists a point c in
(1, 2) such that
f (c) = c 2 = 2.
61.
True or False
At one point in your life your height in inches equaled your weight
in pounds.
62.
Outline
The Intermediate Value
Limits
Theorem
Concept
Computation Derivatives
Limits involving inﬁnity Concept
Continuity Intepretations
Concept Implications
Examples Computation
63.
Outline
The Intermediate Value
Limits
Theorem
Concept
Computation Derivatives
Limits involving inﬁnity Concept
Continuity Intepretations
Concept Implications
Examples Computation
64.
Concept
Learning Objectives
state the deﬁnition of the derivative
Given the formula for a function, ﬁnd its derivative at a point
“from scratch,” i.e., using the deﬁnition
Given numerical data for a function, estimate its derivative at
a point.
given the formula for a function and a point on the graph of
the function, ﬁnd the (slope of, equation for) the tangent line
65.
The deﬁnition
Deﬁnition
Let f be a function and a a point in the domain of f . If the limit
f (a + h) − f (a)
f (a) = lim
h
h→0
exists, the function is said to be diﬀerentiable at a and f (a) is
the derivative of f at a.
66.
Outline
The Intermediate Value
Limits
Theorem
Concept
Computation Derivatives
Limits involving inﬁnity Concept
Continuity Intepretations
Concept Implications
Examples Computation
67.
The Derivative as a function
Learning Objectives
given a function, ﬁnd the derivative of that function from
scratch and give the domain of f’
given a function, ﬁnd its second derivative
given the graph of a function, sketch the graph of its
derivative
68.
Derivatives
Theorem
If f is diﬀerentiable at a, then f is continuous at a.
70.
The second derivative
If f is a function, so is f , and we can seek its derivative.
f = (f )
It measures the rate of change of the rate of change!
71.
Outline
The Intermediate Value
Limits
Theorem
Concept
Computation Derivatives
Limits involving inﬁnity Concept
Continuity Intepretations
Concept Implications
Examples Computation
72.
Implications of the derivative
Learning objectives
Given the graph of the derivative of a function...
determine where the function is increasing and decreasing
determine where the function is concave up and concave down
sketch the graph of the original function
ﬁnd and interpret inﬂection points
73.
Fact
If f is increasing on (a, b), then f (x) ≥ 0 for all x in (a, b)
If f is decreasing on (a, b), then f (x) ≤ 0 for all x in (a, b).
Fact
If f (x) > 0 for all x in (a, b), then f is increasing on (a, b).
If f (x) < 0 for all x in (a, b), then f is decreasing on (a, b).
74.
Deﬁnition
A function is called concave up on an interval if f is
increasing on that interval.
A function is called concave down on an interval if f is
decreasing on that interval.
75.
Fact
If f is concave up on (a, b), then f (x) ≥ 0 for all x in (a, b)
If f is concave down on (a, b), then f (x) ≤ 0 for all x in
(a, b).
Fact
If f (x) > 0 for all x in (a, b), then f is concave up on (a, b).
If f (x) < 0 for all x in (a, b), then f is concave down on
(a, b).
76.
Outline
The Intermediate Value
Limits
Theorem
Concept
Computation Derivatives
Limits involving inﬁnity Concept
Continuity Intepretations
Concept Implications
Examples Computation
77.
Computing Derivatives
Learning Objectives
the power rule
the constant multiple rule
the sum rule
the diﬀerence rule
derivative of x → e x is e x (by deﬁnition of e)
78.
Theorem (The Power Rule)
Let r be a real number. Then
dr
x = rx r −1
dx
79.
Rules for Diﬀerentiation
Theorem
Let f and g be diﬀerentiable functions at a, and c a constant.
Then
(f + g ) (a) = f (a) + g (a)
(cf ) (a) = cf (a)
80.
Rules for Diﬀerentiation
Theorem
Let f and g be diﬀerentiable functions at a, and c a constant.
Then
(f + g ) (a) = f (a) + g (a)
(cf ) (a) = cf (a)
It follows that we can diﬀerentiate all polynomials.
81.
Derivatives of exponential functions
Fact
dx
= ex
dx e
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