Announcements Topics: -­‐  finish sec2on 4.2; work on sec2ons 4.3 and 4.4; start sec2on 4.5 * Read these sec2ons and study solved examples in your textbook! Work On: -­‐  Prac2ce problems from the textbook and assignments from the coursepack as assigned on the course web page (under the link “SCHEDULE + HOMEWORK”) Infinite Limits Example: Use a table of values to es2mate the value of 1
lim
x →0 x
x f(x) 0.1 0.01 0.001 0 undefined -­‐0.001 -­‐0.01 -­‐0.1 €
€
Infinite Limits Defini&on: lim f (x) = ∞
x →a
“the limit of f(x), as x approaches a, is infinity” means that the values o€f f(x) (y-­‐values) increase without bound as x becomes closer and closer to a (from either side of a), but x ≠
a. €
Defini&on: lim f (x) = −∞
x →a
“the limit of f(x), as x approaches a, is nega2ve infinity” means that the values of f(x) (y-­‐values) decrease without bound as x becomes closer and closer to a (from either side of a), but x ≠ a. €
Infinite Limits Example: Determine the infinite limit. (a) lim x + 2
x →−1
€
€
x +1
(b) lim csc x
x →π +
Note: Since the values of these func2ons do not approach a real number L, these limits do not exist. Ver2cal Asymptotes Defini&on: The line x=a is called a ver9cal asymptote of the curve y=f(x) if either lim f (x) = ±∞ or lim f (x) = ±∞
x →a−
€ Example: x →a +
Basic func2ons we know that have VAs: Limits at Infinity The behaviour of func2ons “at” infinity is also known as the end behaviour or long-­‐term behaviour of the func2on. What happens to the y-­‐values of a func2on f(x) as the x-­‐values increase or decrease without bounds? lim f (x) = ?
x →−∞
lim f (x) = ?
x →∞
Limits at Infinity Possibility: y-­‐values also approach infinity or -­‐ infinity Examples: f (x) = e
f (x) = −0.01x 3
x
20
25
15
€
€
10
-25
. -5
0
5
-25
0
5
10
15
20
25
30
25
50
75
100
Limits at Infinity Possibility: y-­‐values approach a unique real number L Examples: 3x + 1
f (x) =
x −2
sin x
f (x) =
x
16
€
1
8
0.5
-10
. -5
0
5
10
15
20
25
-25
0
-8
-16
€
-0.5
25
50
75
100
125
150
Limits at Infinity Possibility: y-­‐values oscillate and do not approach a single value Example: 1.5
1
0.5
f (x) = sin x
-30
-25
-20
-15
-10
-5
0
-0.5
. -1
-1.5
5
10
15
20
25
30
€
Limits at Infinity Defini&on: Defini&on: lim f (x) = L
lim f (x) = L
x →−∞
x →∞
“the limit of f(x), as x approaches ∞
, equals L” “the limit of f(x), as x approaches −∞
, equals L” means that the values of f(x) (y-­‐values) can be made as € as we’d like to L by close taking x sufficiently large. means that the values of f(x) (y-­‐values) can be made as € as we’d like to L by close taking x sufficiently small. €
Calcula2ng Limits at Infinity *The Limit Laws listed previously are s2ll valid if “ x →
a ” is replaced by “ x →
∞ ” Limit Laws for Infinite Limits (abbreviated): €
€
∞+∞=∞
∞⋅ ∞ = ∞
c⋅∞=∞
where c is any non-­‐zero constant €
Calcula2ng Limits at Infinity Theorem: 1
If r>0 is a ra2onal number, then lim r = 0.
x →∞
x
If r>0 is a ra2onal number such that x is defined for all x, then lim 1 = 0.
r
x →−∞
x
r
€
€
€
Calcula2ng Limits at Infinity Examples: Find the limit or show that it does not exist. 2x 2 + 1
(a) xlim
2
→−∞
4x − x
€
€
arctan(3x
+ 5) €
(b) lim
[
] x →∞
Horizontal Asymptotes Defini&on: The line y=L is called a horizontal asymptote of the curve y=f(x) if either lim f (x) = L or lim f (x) = L
x →∞
x →−∞
Example: €
€
Basic func2ons we know that have HAs: . Limits at Infinity What about the limits at infinity of these func2ons? 2x
ln x
e
(a) f (x)
= (b) g(x) =
x
10x
Which part (top or bogom) goes to infinity faster? €
€
Limits at Infinity y = e 2x
y = 10x
20
15
15
10
y= x
y = ln x
10
5
€
-5
-4
€
-3
-2
-1
5
0
0
1
2
3
4
5
10
15
20
5
€
€
25
30
35
Comparing Func2ons That Approach ∞
at ∞
Suppose lim
f (x)
= ∞
and lim g(x) = ∞
x →∞
x →∞
f (x)
= ∞.
1.  f(x) approaches infinity faster than g(x) if lim
x →∞ g(x)
€
€
f (x)
= 0.
2.  f(x) approaches infinity slower than g(x) if lim
x →∞ g(x)
€
f
(x)
3.  f(x) and g(x) approach infinity at the same rate if lim
= L.
x →∞ g(x)
€
where L is any finite number other than 0. € €
€
Comparing Func2ons That Approach ∞
at ∞
The Basic Func2ons in Increasing Order of Speed Func9on Comments Goes to infinity slowly aln x
n with ax
β x with ae
€
€
€
n>0
β >0
€ €
Approaches infinity faster for larger n
Approaches infinity faster for larger β
€
Note: €
The constant a
can be any posi2ve number and does not €
change the order of the func2ons. €
€
€
Comparing Func2ons That Approach ∞
at ∞
y = ex
10
y=x
y = x2
7.5
€ €
€
€
y= x
5
y = ln x
2.5
0
2.5
5
7.5
10
12.5
€
€
15
17.5
Limits at Infinity What about the limits at infinity of these func2ons? −0.5
x
e
(a) f (x)
= −2 (b) g(x) =
−2
10x
5x
−x
Which part (top or bogom) goes to 0 faster? €
€
Limits at Infinity Semilog Graphs 10
0
5
10
15
y = ln5 − 2ln x
20
25
30
35
5
-5
0
5
10
-10
€
y = −x
-10
€
€
20
y = −0.5ln x
25
30
35
y = ln10 − 2ln x
-5
-15
15
€
Comparing Func2ons That Approach 0 at ∞
Suppose lim
f (x)
= 0 and lim g(x) = 0.
x →∞
x →∞
f (x)
1.  f(x) approaches 0 faster than g(x) if lim
= 0.
x →∞ g(x)
€
€
€
€
f (x)
= ∞.
2.  f(x) approaches 0 slower than g(x) if lim
x →∞ g(x)
€
f (x)
3.  f(x) and g(x) approach 0 at the same rate if lim
= L.
x →∞ g(x)
€
where L is any finite number other than 0. €
Comparing Func2ons That Approach 0 at ∞
The Basic Func2ons in Increasing Order of Speed Func9on Comments −n
with n>0
Approaches 0 faster for larger n
β >0
ae
− β x 2 with β > 0
ae
€
Approaches 0 faster for larger β
ax
− β x with €
€
€
€
€
Approaches 0 really €fast Note: Again, a can be any posi2ve constant and this €will not affect the ordering. €
€
Comparing Func2ons That Approach 0 at ∞
y =e
y = x −2
−x
4
€
€
3
€
2
1
-2
€
y =e
-1
−x 2
0
1
2
3
4
5
6
7
Comparing Func2ons That Approach 0 at ∞
y = e−x
0
1
2
3
-1
4
5
6
7
8
9
€Semilog Graphs €
-2
-3
-4
-5
€
y =e
−x 2
y = −2ln
y =ln−2ln
x x
y = −x 2
y = −x
€€
Limits of Sequences Recall: The solu2on of the discrete-­‐2me dynamical system m
t +1 = f (m
t ) is a sequence of values of m
t for t = 0, 1, 2, … €
€
Solu9on: €
{m0, m1, m2, m3, ...}
€
Limits of Sequences To determine the limit of the solu2on to a discrete-­‐2me dynamical system, we define an associated func9on, m(t)
, which is defined for all t ∈
R . If this func2on has a limit at infinity, then the €
sequence shares this limit (although the converse is not necessarily true). €
Limits of Sequences Example: 1
The solu2on of M
t +1 = 2 M
t +1
is given by Mt = M
()
1
0 2
t
+2
€
If M
0 = 10,
what will eventually happen to the concentra2on of methadone in the pa2ent’s € blood? Con2nuity Intui&ve idea: A process is con&nuous if it takes place without interrup2ons or an abrupt change. Geometrically, a func2on is con2nuous if it’s graph has no break in it. y
y=f(x)
f(a)
a
x
Con2nuity Defini&on: A func2on f is con9nuous at the point x=a if f(x) approaches f(a) as x approaches a, i.e. y
y=f(x)
f(a)
lim f (x) = f (a)
x →a
€
a
x
Con2nuity Implicitly requires 3 things: 1. lim
f (x)
exists x →a
2. f (a)
is defined 3. lim f (x) = f (a)
x →a
y
y=f(x)
f(a)
€
€
€
If f is not con2nuous at a (i.e. f fails to meet at least one of the three condi2ons above), then we say that f is discon9nuous at x=a. a
x
Con2nuity Example: Find the discon2nui2es of the func2on and explain why it is discon2nuous there. x +1
h(x) = 2
x − 2x − 3
€
Start by looking at x-­‐values where f(x) is not defined and then check the 3 condi2ons of con2nuity. . y
x
Con2nuity Example: Find the discon2nui2es of the func2on and explain why it is discon2nuous there. ⎧2 − x
g(x) = ⎨
⎩ x −1
€
x ≤1
x >1
Start by looking at x-­‐values where f(x) is changes . from one ‘piece’ to another and then check the 3 condi2ons of con2nuity. y
x
Which Func2ons Are Con2nuous? Defini0on: A func2on is said to be con2nuous if it is con2nuous at every point in its domain. Basic Con&nuous Func&ons:   polynomials h(x) = −x 7 + 2x 4 −1
ex: f (x) = 4
  ra2onal func2ons €
€
3x − 4
f
(x)
=
, x ≠1
ex: 1− x
  root func2ons f (x) = x, x ≥ 0
€ ex: €
h(x) =
5
, x∈R
2
1+ x
g(x) = 3 x , x ∈ R
Which Func2ons Are Con2nuous? Basic Con&nuous Func&ons:  algebraic func2ons ex: f (x) =
x + 16
x2 +1
 absolute value func2on €
f (x) = x
 exponen2al and logarithmic func2ons g(x) = log 5 x, x > 0
€ ex: f (x) = e x
 trigonometric and inverse trigonometric func2ons €x
€ ex: f (x) = sin
g(x) = arctan x
Which Func2ons Are Con2nuous? Combining Con&nuous Func&ons: The sum, difference, product, quo2ent, and composi2on of con2nuous func2ons is con2nuous where defined. Example: arctan(e x + x 2 )
Determine where h(x)
= is con2nuous. x +1
€
. Limits of Con2nuous Func2ons Example: arctan(e x + x 2 )
.
Evaluate lim
x →0
x +1
€
Note: By the defini2on of con2nuity, if a func2on is con2nuous at x=a, then we can evaluate the limit simply by direct subs2tu2on. The Deriva2ve Recall: The instantaneous rate of change of the func2on f(x) at x=a is f (a + h) − f (a)
f '(a) = lim
h →0
h
y
tangent
f(a+h)
f(a)
(provided this limit exists). P
a
Geometrically, this number represents the slope of the tangent to the curve at (a, f(a)). Q
a+h
mP = lim mPQ
Q →P
€
y = f(x)
x
The Deriva2ve Defini&on: Given a func2on f(x), the deriva9ve of f with respect to x is the func&on f ’(x) defined by df
f (x + h) − f (x)
= f '(x) = lim
h →0
dx
h
The domain of this func2on is the set of all x-­‐
values for which the limit exists. €
The Deriva2ve Interpreta0ons of f ’: 1.  The func2on f ’(x) tells us the instantaneous rate of change of f(x) with respect to x for all x-­‐values in the domain of f ’(x). 2. The func2on f ’(x) tells us the slope of the tangent to the graph of f(x) at every point (x, f(x)), provided x is in the domain of f ’(x). The Deriva2ve Example: Find the deriva2ve of y
f (x) = x + 3
€
and use it to calculate the instantaneous rate of change of f(x) at x=1. Sketch the curve f(x) and the tangent to the curve at (1,2). x
The Deriva2ve y
Example: Find the deriva2ve of x
f (x) = x 2 + 2x.
€
Sketch the graph of f(x) and the graph of f ’(x). y
x
Rela2onship between f ’ and f If f is increasing on an interval (c,d): The deriva2ve f ’ is posi2ve on (c,d). The rate of change of f is posi2ve for all x in (c,d). The slope of the tangent is posi2ve for all x in (c,d). If f is decreasing on an interval (c,d): The deriva2ve f ’ is nega2ve on (c,d). The rate of change of f is nega2ve for all x in (c,d). The slope of the tangent is nega2ve for all x in (c,d). Cri2cal Numbers Defini2on: c is a cri9cal number of f if c is in the domain of f and either f ’(c)=0 or f ’(c) D.N.E. Differen2able Func2ons A func2on f(x) is said to be differen9able at x=a if we are able to calculate the deriva2ve of the func2on at that point, i.e., f(x) is differen2able at x=a if f (a + h) − f (a)
f '(a) = lim
h →0
h
exists. €
Differen2able Func2ons Geometrically, a func2on is differen2able at a point if its graph has a unique tangent line with a well-­‐
defined slope at that point. 3 Ways a Func2on Can Fail to be Differen2able: y
y
x
y
x
x
Graphs y
Example: 2
f
(x)
=
x
− 6x .
(a)  Sketch the graph of (b) By looking at the graph of f, sketch the graph of f ’(x). €
x
y
x
Rela2onship Between Differen2ability and Con2nuity If f is differen2able at a, then f is con2nuous at a. Continuous Functions
Differentiable
Functions