Announcements Topics: -­‐  Finish sec2on 3.3, work on 4.1 and 4.2, start 4.3 * 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”) Calculus On Con2nuous Func2ons In order to start studying con2nuous-­‐2me dynamical systems, we need to develop the usual tools of calculus on con2nuous func2ons:  
 
 
 
Limits Con2nuity these are all defined Deriva2ves in terms of limits Integrals }
Rates of Change The rate of change of a func2on tells us how the dependent variable changes when there is a change in the independent variable. Geometrically, the rate of change of a func2on corresponds to the slope of it’s graph. Secant Lines and Tangent Lines A secant line is a line that intersects two points on a curve. A tangent line is a line that just touches a curve at a point and most closely resembles the curve at that point. y
tangent
P
Q
secant
y = f(x)
x
Average Rate of Change = Slope of Secant Line The average rate of change of f(t) from t=t1 to t=t2 corresponds to the slope of the secant line PQ. y
Q
f(t2)
f( t1)
P
t1
t2
secant
y = f(t)
t
Average Rate of Change = Slope of Secant Line Alterna(ve Nota(on: The average rate of change y
of f(t) from the base point f(t0 +Δt )
t=t0 to t=t0+Δt is f(t0)
Q
secant
P
t0 +Δt
t0
Δt
y = f(t)
t
Average Rate of Change = Slope of Secant Line Example #2 (modified): Find the average rate of change of the func2on h(t) = t 2 + 1
star2ng from 2me t=1 and las2ng 1, 0.1, and 0.01 units of 2me. € t0 . Δt t0+Δt 1 1 2 1 0.1 1.1 1 0.01 1.01 f(t0+Δt) -­‐ f(t0) Δt Es2ma2ng the Slope of the Tangent Steps: 1. Approximate the tangent at P using secants intersec2ng P and a nearby point Q. 2. Obtain a beber approxima2on to the tangent at P by moving Q closer to P, but Q ≠ P. 3. Define the slope of the tangent at P to be the limit of the slopes € PQ as Q approaches P of secants (if the limit exists). y
P
Q3
Q2
Q1
y = f(t)
t
€
Instantaneous Rate of Change = Slope of Tangent Line The instantaneous rate of change of f(t) at t=t0 corresponds to the slope of the tangent line at t=t0. Δf
f '(t 0 ) = lim
Δt →0 Δt
Note: The slope of the curve y=f(t) at P is the slope of its tangent line at P. y
f(t0 +Δt )
f(t0)
tangent
Q
P
t0
t0 +Δt
y = f(t)
t
Instantaneous Rate of Change = Slope of Tangent Line This special limit is called the deriva*ve of f at t0 and is denoted by f ’(t0) (read “f prime of t0”). Alterna2ve nota2on: y
f(t0 +Δt )
f(t0)
tangent
Q
P
t0
t0 +Δt
y = f(t)
t
Instantaneous Rate of Change = Slope of Tangent Line Example #10 (modified): 2
h(t) = t + 1
(a) Guess the limit of the slopes of the secants as the second point approaches the base point. t0 Δt t0+Δt f(t0+Δt) -­‐ f(t0) Δt 1 0.1 1.1 2.1 1 0.01 1.01 2.01 1 0.001 1.001 2.001 y
€
(b) Use this to find the equa2on of the tangent line to h(t) at t=1. . x
The Limit of a Func2on y
Nota(ons: y=f(x)
f (2) = 5
means that the y-­‐value of the func2on AT x=2 is 5 5
2
€
2
⎧ x 2 − 4
⎪
if x ≠ 2 €
f (x) = ⎨ x − 2
⎪⎩5
if x = 2
x
lim f (x) = 4
x →2
means that the y-­‐values of the func2on APPROACH 4 as x APPROACHES 2 €
The Limit of a Func2on Defini(on: lim f (x) = L
x →a
“the limit of f(x), as x approaches a, equals L” means that the values of f(x) (y-­‐values) approach the number L more and more closely as x approaches a more and more closely (from either side of a), but x≠ a. €
Limit of a Func2on Some examples: y
y
y
x=3
y=g(x)
y=h(x)
y=f(x)
x
x
Note: f may or may not be defined at x=a. Limits are only asking how f is defined NEAR a. x
Lee-­‐Hand and Right-­‐Hand Limits lim− f (x) = L
x →a
means f (x)
→
L as x →
a from the lee (x < a).
lim+ f (x) = L
x →a
€
€
€ means f (x)
→
L as x →
a from the right (x
> a).
** The full limit exists if and only if €
the lee and right €
€ limits both exist (equal a real number) and are the same value. Lee-­‐Hand and Right-­‐Hand Limits For each func2on below, determine the value of the limit or state that it does not exist. y
y
y
y=f(x)
x
y=g(x)
x
y=h(x)
x
Evalua2ng Limits We can evaluate the limit of a func2on in 3 ways: 1.  Graphically 2.  Numerically 3.  Algebraically Evalua2ng Limits Example: Use a table of values to es2mate the value of x 2 −16
lim
x →4 x − 4
x f(x) 3.5 3.9 3.99 4 undefined 4.01 4.1 4.5 €
€
€
Evalua2ng Limits Algebraically BASIC LIMITS Limit of a Constant Func2on Limit of the Iden2ty Func2on lim x = a
lim c = c, where c ∈ R
x →a
x →a
Example: lim 2 = 2
Example: y
x=3
€ lim
x →3
x →3
y=2
x
€
y
y=x
x
LIMIT LAWS [used to evaluate limits algebraically] Suppose that c is a constant and the limits exist. Then lim f (x) and lim g(x)
x →a
x →a
[ f (x) + g(x)] = lim f (x) + lim g(x)
1. lim
x →a
x →a
x →a
€
[ f (x) − g(x)] = lim f (x) − lim g(x)
2. lim
x →a
x →a
x →a
€
€
€
c f (x)] = c lim f (x)
[
3. lim
x →a
x →a
LIMIT LAWS [used to evaluate limits algebraically] Con2nued… [ f (x) ⋅ g(x)] = lim f (x) ⋅ lim g(x)
4. lim
x →a
x →a
x →a
€
€
[ f (x) ÷ g(x)] = lim f (x) ÷ lim g(x),
5. lim
x →a
x →a
x →a
if lim g(x) ≠ 0
x →a
Evalua2ng Limits Algebraically Example: Evaluate the limit and jus2fy each step by indica2ng the appropriate Limit Laws. lim (x 2 − 5x + 6)
x →1
= lim x 2 − lim 5x + lim 6
x →1
x →1
x →1
= lim x ⋅ lim x − 5lim x + lim 6
x →1
x →1
= (1)(1) − 5(1) + 6
=2
x →1
x →1
Direct Subs2tu2on Property From the previous slide, we have lim (x 2 − 5x + 6) = 2
x →1
f (1)
f (x)
No2ce that we could have s€imply found the €
€
value of the limit by plugging in x=1 into the func2on. Direct Subs2tu2on Property Direct SubsFtuFon Property: If f(x) is an algebraic, exponenFal, logarithmic, trigonometric, or inverse trigonometric func2on, and a is in the domain of f(x), then lim f (x) = f (a)
x →a
€
€
Equal Limits Property Consider the func2ons: x 2 − 4 f (x) =
g(x) = x + 2.
x −2
y
y
y=g(x)
y=f(x)
€
€
4
4
2
2
2
x
2
* Note: f(x)=g(x) everywhere except at x=2 x
€
Equal Limits Property Example: x2 − 4
Calculate lim
.
x →2
Note: direct subs2tu2on does not work x −2
€
FACT: If f (x)
= g(x)
when x ≠ a , then lim
f (x) = lim g(x)
x →a
x →a
provided the limits exist. €
€
Strategy for Evalua2ng Limits #
= ±∞ THENTHELIMIT$.%
0
MAYWANTTODOLEFTANDRIGHTCASES
4RY$IRECT3UBSTITUTION
)FYOUGET
€
€
3IMPLIFY
€
real # THENTHELIMITEXISTSANDEQUALS
THATREALNUMBER
0
0
INDETERMINATE
Evalua2ng Limits Algebraically Evaluate each limit or state that it does not exist. 1
1
−
x+2
2
x
lim
lim
(a) x →−1
(b) x →2
x +1
x −2
€
€
1− x
x −2
(c) lim
€ (d) lim
x →1 x −1
x →4 4 − x
€
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 verFcal 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 bobom) 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 FuncFon 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 bobom) 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 FuncFon 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
€€