suppose you drive 200 miles, and it takes you 4 hours. then your average speed is: if you look at...
DESCRIPTION
for some very small change in t where h = some very small change in t We can use the TI-84 to evaluate this expression for smaller and smaller values of h. 2.1 Rates of Change and LimitsTRANSCRIPT
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Suppose you drive 200 miles, and it takes you 4 hours.
Then your average speed is:mi200 mi 4 hr 50 hr
distanceaverage speed elapsed time
xt
If you look at your speedometer during this trip, it might read 65 mph. This is your instantaneous speed.
2.1 Rates of Change and Limits
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A rock falls from a high cliff.
The position of the rock is given by: 216y t
After 2 seconds:216 2 64y
average speed: av64 ft ft322 sec sec
V
What is the instantaneous speed at 2 seconds?
2.1 Rates of Change and Limits
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instantaneousyVt
for some very small change in t
2 216 2 16 2hh
where h = some very small change in t
We can use the TI-84 to evaluate this expression for smaller and smaller values of h.
2.1 Rates of Change and Limits
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instantaneousyVt
2 216 2 16 2hh
h yt
1 80
0.1 65.6.01 64.16.001 64.016.0001 64.0016.00001 64.0002
We can see that the velocity approaches 64 ft/sec as h becomes very small.
We say that the velocity has a limiting value of 64 as h approaches zero.
(Note that h never actually becomes zero.)
2.1 Rates of Change and Limits
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2
0
16 2 64limh
hh
The limit as h approaches zero:
2
0
16 4 4 64limh
h h
h
2
0
64 64 16 64limh
h hh
0
lim 64 16h
h
0
64
2.1 Rates of Change and Limits
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Definition: LimitLet c and L be real numbers. The function f has limit L as x approaches c if, for any given positive number ε, there is a positive number δ such that for all x,
Lxfcx
)(lim
εL||f(x)δc||x 0
2.1 Rates of Change and Limits
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a
L
f
Lxfax
)(lim DNExfax
)(lim
DNE = Does Not Exist
a
fL1
L2
2.1 Rates of Change and Limits
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Definition: One Sided Limits
Left-Hand Limit: The limit of f as x approaches a from the left equals L is denoted
Lxfax
)(lim
Right-Hand Limit: The limit of f as x approaches a from the right equals L is denoted
Lxfax
)(lim
2.1 Rates of Change and Limits
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2.1 Rates of Change and Limits
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Definition: Limit
Lxfax
)(lim if and only if
Lxfax
)(lim Lxfax
)(limand
2.1 Rates of Change and Limits
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)(lim xfax
DNExfax
)(lim
DNE = Does Not Exist
Possible Limit Situations
a
f
a
f
2.1 Rates of Change and Limits
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1 2 3 4
1
2
At x = 1: 1
lim 0x
f x
1
lim 1x
f x
1 1f
left hand limit
right hand limit
value of the function
1
limx
f x does not exist
because the left and right hand limits do not match!
2.1 Rates of Change and Limits
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At x = 2: 2
lim 1x
f x
2
lim 1x
f x
2 2f
left hand limit
right hand limit
value of the function
2
lim 1x
f x
because the left and right hand limits match.
1 2 3 4
1
2
2.1 Rates of Change and Limits
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At x =3: 3
lim 2x
f x
3
lim 2x
f x
3 2f
left hand limit
right hand limit
value of the function
3
lim 2x
f x
because the left and right hand limits match.
1 2 3 4
1
2
2.1 Rates of Change and Limits
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Use your calculator todetermine the following:
(a)
(b)
xx
x
sinlim0
xx
1coslim0
2.1 Rates of Change and Limits
1
DNE
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Suppose that c is a constant and the following limits exist )(lim)(lim xgxf
axax and
)(lim)(lim)()(lim.1 xgxfxgxfaxaxax
)(lim)(lim)()(lim.2 xgxfxgxfaxaxax
)(lim)(lim.3 xfcxcfaxax
2.1 Rates of Change and Limits
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Suppose that c is a constant and the following limits exist )(limand)(lim xgxf
axax
)(lim)(lim)()(lim.4 xgxfxgxfaxaxax
)(lim
)(lim
)()(lim.5
xg
xf
xgxf
ax
ax
ax
2.1 Rates of Change and Limits
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ccax
lim.7
axax
lim.8
nax
n
axxfxf )(lim)(lim.6
where n is a positive integer.
nn
axax
lim.9 where n is a positive integer.
nn
axax
lim.10 where n is a positive integer.
nax
nax
xfxf )(lim)(lim.11
where n is a positive integer.
2.1 Rates of Change and Limits
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Evaluate the following limits. Justify each step using the laws of limits.
523lim.1 2
3
xx
x
523lim.2
1 xx
x
3 2
22lim.3 xx
x
16
-5/4
2
nsinlim.4 x
x6
2.1 Rates of Change and Limits
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1. If f is a rational function or complex:a. Eliminate common factors.b. Perform long division.c. Simplify the function (if a complex fraction)
2. If radicals exist, rationalize the numerator or denominator.
3. If absolute values exist, use one-sided limits and the following property.
00
aifaaifa
a
2.1 Rates of Change and Limits
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11lim.1 2
3
1
xx
x
hh
h
11lim.20
22
lim.42
x
xx
111lim.3 2
0 ttt
3/2 DNE
1/2
DNE
2.1 Rates of Change and Limits
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Theorem
If f(x) g(x) when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a, then
)(lim)(lim xgxfaxax
2.1 Rates of Change and Limits
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The Squeeze (Sandwich) Theorem
If f(x) g(x) h(x) when x is near a (except possibly at a) and
Lxhxfaxax
)(lim)(lim then Lxgax
)(lim
2.1 Rates of Change and Limits
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Show that: 2
0
1lim sin 0x
xx
The maximum value of sine is 1, so 2 21sinx xx
The minimum value of sine is -1, so 2 21sinx xx
So: 2 2 21sinx x xx
2.1 Rates of Change and Limits
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2 2 2
0 0 0
1lim lim sin limx x x
x x xx
2
0
10 lim sin 0x
xx
By the sandwich theorem:
2.1 Rates of Change and Limits
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2.1 Rates of Change and Limits
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0lim )/sin(
0
x
xex
1)/sin(1 x
1)/sin(1 eee x
exexex x )/sin(
0lim0lim00
xeandex
xx
Therefore,
0lim )/sin(
0
x
xex
2.1 Rates of Change and Limits
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1θ
sinθlimProve0θ
OAT ΔOAPSector 0AP ΔArea
(1)tanθ21π(1)*
2πθ(1)sinθ
21 2
θ tan21
2θθ sin
21
simplify and divide by sin θ
θ sinθ tan
θ sinθ1 θ cos
θ θ sin1
1θ cos lim1 limSince0θ0θ
1θ
sinθlimtheorem pinching the By0θ
2.1 Rates of Change and Limits
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0θ
θcos1limProve0θ
-
θ)0θ(sin)1θ(cos0 22
2.1 Rates of Change and Limits
P(cos , sin )
Q(1,0)
PQPQ ___
0
θθsin1θcos2θ(cos0 22
θθcos220
θθcos120
2θθcos10
2θθcos10
2
2θ
θθcos10
02θ limSince
0θ
0θ
θcos1lim0θ
-
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The notationlim ( )x a
f x
means that the values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a (on either side) but not equal to a.
2.2 Limits Involving Infinity
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a
f
Vertical Asymptote
lim ( )x a
f x
2.2 Limits Involving Infinity
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Vertical AsymptoteThe line x = a is called a vertical asymptote of the curve y = f(x) if at least one of the following statements is true:
lim ( )x a
f x
lim ( )
x af x
lim ( )x a
f x
2.2 Limits Involving Infinity
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0lim lnx
x
f(x) = ln x has a vertical asymptote at x = 0 since
f(x) = tan x has a vertical asymptote at x = /2 since
/ 2lim tan
xx
2.2 Limits Involving Infinity
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35lim
( 5)
x
x
ex
2
2 5( )4 3
xf xx x
2.2 Limits Involving Infinity
-∞
x = 3x = 1
Determine the equations of thevertical asymptotes of
Find the limit
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Let f be a function defined on some interval (a, ∞). Then
lim ( )x
f x L
means that the value of f(x) can be made as close to L as we like by taking x sufficiently large.
2.2 Limits Involving Infinity
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Horizontal Asymptote
lim ( )x
f x L
L
f
2.2 Limits Involving Infinity
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2.2 Limits Involving Infinity
Definition End Behavior ModelSuppose that f is a rational function as follows:
01
1
01
1
......)(
bxbxbaxaxaxf m
mm
m
nn
nn
mnxf
x
if0)(lim1.
mnbaxf
x
if)(lim2.
mnxfx
if)(lim3.
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Horizontal AsymptoteThe line y = L is called a horizontal asymptote of the curve y = f(x) if either
lim ( )x
f x L
lim ( )x
f x L
or
2.2 Limits Involving Infinity
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lim 0x
xe
f(x) = e x has a horizontal asymptote at y = 0 since
2.2 Limits Involving Infinity
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1lim 0nx x
If n is a positive integer, then
1lim 0nx x
2.2 Limits Involving Infinity
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Find the limit
2
2
2lim3 1t
tt t
2
2lim9 1y
y
y
2.2 Limits Involving Infinity
-1/3
2/3
1/3
xxxx
x 5342
3
3
lim
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Find the limit
2.2 Limits Involving Infinity
xx
x
sinlim
Use squeeze theorem
1sin1 x
01sin1
xxx
xx
01lim,01lim
xx xx0sinlim
xx
x
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2.2 Limits Involving Infinity
xx
x
sinlim
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A function is continuous at a point if the limit is the same as the value of the function.
This function has discontinuities at x = 1 and x = 2.
It is continuous at x = 0 and x =4, because the one-sided limits match the value of the function
1 2 3 4
1
2
2.3 Continuity
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Definition: Continuity
A function is continuous at a number a if That is,1. f(a) is defined2. exists3. )()(lim afxf
ax
)(lim xfax
)()(lim afxfax
2.3 Continuity
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Definition: One Sided Continuity
A function f is continuous from the right at a number a if
and f is continuous from the left at a if
)()(lim afxfax
)()(lim afxfax
2.3 Continuity
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1. Removable discontinuity
2.3 Continuity
123)(
2
x
xxxf
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2. Infinite discontinuity
2.3 Continuity
2
1)(x
xf
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3. Jump discontinuity
2.3 Continuity
xx
xf )(
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4. Oscillating discontinuity
2.3 Continuity
xxf 1sin)(
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Definition: Continuity On An Interval
A function f is continuous on an interval if it is continuous at every number in the interval. (If f is defined on one side of an endpoint of the interval, we understand continuous at the endpoints to mean continuous from the right or continuous from the left).
2.3 Continuity
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Theorem
1. f + g 2. f – g3. cf4. fg5. f / g if g(a) 06. f(g(x))
If f and g are continuous at a and c is a constant, then the following functions are also continuous at a:
2.3 Continuity
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Theorem(a) Any polynomial is continuous
everywhere; that is, it is continuous on = (-∞, ∞).
(b) Any rational function is continuous whenever it is defined; that is, it is continuous on its domain.
2.3 Continuity
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Any of the following types of functions are continuous at every number in their domain: Polynomials; Rational Functions, Root Functions; Trigonometric Functions; Inverse Trigonometric Functions; Exponential Functions; and Logarithmic Functions.
2.3 Continuity
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If f is continuous at b and , then . In other words,
bxfax
)(lim)())((lim bfxgf
ax
))(lim())((lim xgfxgfaxax
2.3 Continuity
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If g is continuous at a and f is continuous at g(a), then the composite function f(g(x)) is continuous at a.
2.3 Continuity
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The Intermediate Value TheoremSuppose that f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b). Then there exists a number c in (a, b) such that f(c) = N.
a
f
b
f(a)
f(b)
c
f(c)=N
2.3 Continuity
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Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.
)2,1(;12 xx
2.3 Continuity
1)( 2 xxxf
0111)1( 2 f
0122)2( 2 f
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Graph Continuous at x=0?
)(xfy )0(f )(lim0
xfx
xxf )(
xxxf
2
)(
xxf 1)(
xxxf )(
||)( xxf
xxxf sin)(
xx
xf )(
xxf )(
xxxf cos1)(
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Graph Continuous at x = 0?
)(xfy )0(f )(lim0
xfx
xxf )(
xxxf
2
)(
xxf 1)(
xxxf )(
||)( xxf
xxxf sin)(
xx
xf )(
xxf )(
xxxf cos1)(
0 0 yes
undefined 0 no
undefined DNE no
undefined 1 no
0 0 yes
undefined1 no
undefined DNE no
0 DNE no
undefined 0 no
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Definition: LimitLet c and L be real numbers. The function f has limit L as x approaches c if, for any given positive number ε, there is a positive number δ such that for all x,
Lxfcx
)(lim
εL||f(x)δc||x 0
2.3 Continuity
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Showthat limx1
(5x 3)2
Solution Set c = 1 and f(x) = 5x - 3 and L = 2.For any given > 0, there exists a > 0 such that
0 < |x - 1| < whenever |f(x) - 2| <
2.3 Continuity
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•|(5x - 3) - 2| < •|5x - 5| < •5|x - 1| < •|x - 1| < /5•So if = /5
Then limx1
(5x 3)2
1- 1 1+
2+
2-
2
2.3 Continuity
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Showthat limx 2
3x-15
Solution Set c = 2 and f(x) = 3x - 1 and L = 5.For any given > 0, there exists a > 0 such that
0 < |x - 2| < whenever |f(x) - 5| <
2.3 Continuity
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•|(3x - 1) - 5| < •|3x - 6| < •3|x - 2| < •|x - 2| < /3•So if = /3 2- 2 2+
5+
5-
5
513lim2
xx
2.3 Continuity
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DefinitionAverage Rate of ChangeThe average rate of change of a quantity over a period of time is the amount of change divided by the time it takes.
2.4 Rates of Change and Tangent Lines
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Find the average rate of change of f(x) = x2 - 2xover the interval [1,3] andthe equation of the secant line.
f(1) = -1f(3) = 3
21313
1313
)()f()f(
(3,3)
(1,-1)
y = mx + b 3 = 2*3 + b
b = -3 y = 2x - 3
2.4 Rates of Change and Tangent Lines
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2.4 Rates of Change and Tangent Lines
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Slope of a Curve
Definition Slope of a Curve at a Point The slope of the curve y = f(x) at the point P(a, f(a)) is
hafhafm
h
)()(lim0
provided the limit exists
or axafxfm
ax
)()(lim
The tangent line to the curve at P is the line through P with this slope.
2.4 Rates of Change and Tangent Lines
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Find the slope of the parabola y = x2 at the point (2,4)
2.4 Rates of Change and Tangent Lines
Demonstration
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2.4 Rates of Change and Tangent Lines
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Normal to a curve
The normal line to a curve at a point is the line perpendicularto the tangent at that point.
2.4 Rates of Change and Tangent Lines
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Find an equation of the normal line to the curvey = 9 – x2 at x = 2
hxfhxfm
h
)()(lim0
hxhxm
h
)9()(9lim22
0
hxhxhxm
h
)9()2(9lim222
0
hhxhm
h
2
0
2lim
hxmh
2lim0
xm 2
2.4 Rates of Change and Tangent Lines
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At x = 2, the slope of the tangent line is -2(2) = -4,so the slope of the normal line is ¼.
y = mx + b
5= (1/4) (2) + b
5= 1/2 + bb = 9/2
y= (1/4) x + 9/2
2.4 Rates of Change and Tangent Lines