Lecture 10:Infinite Limits and Limits at Infinity
Limits of polynomial and exponential functions
Exponential functions as |x| ! 1:
We use:
•
•
•
•
For base b > 1, the exponential function bx grows without bound as x increases.
Write all exponential functions in the form eax with base e: for any b > 0, b = eln b , so
⇢
b > 1 =) a = ln b > 0,
x
bx = eln b = ex ln b = e(ln b)x , where
b < 1 =) a = ln b < 0.
To understand eax as x ! 1, equivalently consider e ay as y ! +1 (let y = x).
8
<1 if a > 0,
o
é
his
ax
if a = 0,
lim e = 1
x!1
:
o
x
ya o
0
if a < 0.
fig
ly
IF
x
ex
ex
ex
e
Y
ty
ly
lin
xt
x
SFU MATH 154, Fall 2023
8/11
Lecture 10:Infinite Limits and Limits at Infinity
Limits of polynomial and exponential functions
e
Behaviour of a few more functions. . . :
Examples 10.5:
•
P (x) =
1
1+e
(recall Lecture 6)
Is
4(x 2)
Pix
fig
•
Pixie
0
f
o
x
a
as
as x
a
Other rational functions
x4
x
f (x) =
,
g(x)
=
1 + x4
1 + x3
It
s
Ea
o
as x so
•
Trigonometric functions sin x and cos x
GI
II
•
tan x =
ONE
sin x
cos x
vertical
asymptote
whenever
sx o
IE
SFU MATH 154, Fall 2023
9/11
Lecture 10:Infinite Limits and Limits at Infinity
Self-Study Question
Self-Study Question
To model the time dependence of populations in limited-resource environments, one
b
frequently uses a logistic function of the form P (t) =
(t 0) (here a, b > 0 and c
1 + ceat
are constants).
(a) What is the limiting population as t ! 1 if a > 0?
aso
eat
fig
longPst
0
so
HTylarge
0
(b) What is the limiting population lim P (t) if a < 0?
so
find
eat
Example: While developing the competitive
exclusion principle, G.F. Gause modelled a
population of the protozoan Paramecium
with the formula
P (t) = 64/ 1 + 31e 0.7944t .
t!1
0
so
Pit
b
g
fig
(c) Assume that P (t) models a population which is sustainable in the long run. Based on
your answers to (a) and (b), should a be positive or negative?
Need
a co
(d) What is the meaning of the parameter b?
carrying capacity
The carrying capacity of an environment is the maximum population size that it can sustain, given the available resources
of food, water, habitat etc.
SFU MATH 154, Fall 2023
10/11
Lecture 10:Infinite Limits and Limits at Infinity
Biological Example 1
Biological Example 1: Monod Growth Model
In the 1930s and 1940s, the French biologist Jacques Monod studied the growth rates of E. coli bacteria in the presence of a
single limiting nutrient, such as glucose. Based on his laboratory experiments, he proposed the following equation for the
per capita reproduction rate R(N ) as a function of N , the concentration of nutrients:
For positive constants S and c, the growth rate is
R(N ) =
SN
.
c+N
(a) Plot the Monod growth function R(N ) for (b) What is the growth rate as the nutrient
S = 6 and c = 3.
concentration goes to 0?
rinses
his
(c) Find the limit of R(N ) as N ! 1.
Intuition
1
hint
fig
Rini
large N
lis
RINI W
SE
o
S
E
S
(d) What is the interpretation of the horizontal asymptote? Propose meanings for the
constants S and c.
rate as nutrient concentration
Q
R kl
SFU MATH 154, Fall 2023
bitinggouth
is high
nutrient level where growth rate is
c is
him
11/11
Lecture 11: Derivatives and Rates of Change
Lecture 11: Definition of a Derivative
Learning Objectives:
(a) Introduce the concept of a derivative as a slope, and a rate of change
(b) Use the formal definition to evaluate the derivative of a function
(c) Find the equation for the tangent line
(d) Approximate the derivative of a function
Recall
Y
n
by rise
o
stere
of line
gradient
ly
rise
g
run
I
SFU MATH 154, Fall 2023
1/14
Lecture 11: Derivatives and Rates of Change
Review: the slope
Recall the concept of the slope of a line; how do we extend this concept to a general curve
y = f (x)?
The tangent line to the curve y = f (x) at x = a is the line that touches the curve at a.
by
fix
slope of secant line through
saggy
t
la flat
joy Jaffa
f la
Fath
h
xn a
x
at
he o
Thx
Y
is
slope
ix flat
and
fitted
of torgentine
is
Gna
f
al
fila
derivative
jiffy
off at
a
The derivative of a function f (x) at x = a is the slope of its tangent line at a.
SFU MATH 154, Fall 2023
This function f is incasing
f
a
so
2/14
Lecture 11: Derivatives and Rates of Change
Formally, the derivative is defined as a limit:
The definition of a derivative
Recall
f la
is read as
f of
a
The derivative of a function f at a number a, denoted by f 0 (a), is defined to be
f 0 (a) = lim
x!a
f (x)
x
f (a)
f (a + h)
= lim
h!0
a
h
f (a)
,
É
if this limit exists.
IL
f prime of
readas
a
If the curve is y = f (x), we can also write this as
g
dy
= lim
x!0
dx
y
.
x
p
read as
Note
If
is
not
SFU MATH 154, Fall 2023
d f d
a
slope of the secant
x
line is
fraction Id's
cannot be
cancelled I
limit as
sip
in the
ox o become
oftgtlv.ie
3/14