unit 1 limits. slide 2 1.1 limits limit – assume that a function f(x) is defined for all x near c...
TRANSCRIPT
Slide 2
1.1 LimitsLimit – Assume that a function f(x) is defined for all x near c (in some open interval containing c) but not necessarily at c. We say that, “the limit of f(x) as x approaches c is equal to L”
* c and L are
limx cf x L
Slide 3
1.1 LimitsExistence of a Limit
– The functions value must the same value
from
– If a function is not defined on one side of c, the limit
Slide 4
1.1 ExamplesFind f(-2),
1) 2)
22 2
lim , lim , lim .xx x
f x f x f x
4
2
-2
-4
-5
4
2
-2
-4
-5
Slide 5
1.1 ExamplesFind f(-2),
3) 4)
22 2
lim , lim , lim .xx x
f x f x f x
4
2
-2
-4
-5
4
2
-2
-4
-5
Slide 6
1.1 Examples
4
2
-2
-4
-5 5
4
3
3
1
1
3
5
5
5) lim
6) lim
7) lim
8) lim
9) lim
10) lim
11) lim
12) lim
x
x
x
x
x
x
x
x
f x
f x
f x
f x
f x
f x
f x
f x
The graph of f(x) is shown below. Use the graph to evaluate each limit.
Slide 7
1.1 Examples
Piecewise Function – A function obtained by together
multiple functions on .
Sketch the graph and then evaluate each limit.
2
2
3, for 1 3 , for 1 2
5 , for 2
x xf x x x
x x x
1
1
1
2
13) lim
14) lim
15) lim
16) lim
x
x
x
x
f x
f x
f x
f x
6
4
2
-2
-4
-6
-5 5
Slide 8
1.1 LimitsEvaluating limits without graphing
1) Substitute the value you are into x.
*If you get a definite answer, then the limit equals
.
*If you get , we have more work to do.
2) the expression.
*
*Multiply by
3)
00
Slide 9
1.1 ExamplesDetermine the value of each limit without graphing.
2
22
2 317) lim
5 14x
x xx x
518) lim
xe
Slide 10
1.1 ExamplesDetermine the value of each limit without graphing.
22
2 419) lim
12 3t
tt
3
2
420) lim
2x
x xx
Slide 11
1.1 ExamplesDetermine the value of each limit without graphing.
2
4
2 921) lim
4h
h hh
3
1
122) lim
1x
xx
Slide 12
1.1 ExamplesDetermine the value of each limit without graphing.
16
423) lim
16x
xx
0
4 224) lim
h
hh
Slide 13
1.2 ExamplesDetermine the value of each limit without graphing.
23
31
1) lim9x
xx
2
22) lim
4x
xx x
Slide 15
1.3 More Evaluating LimitsEvaluating limits
If you substitute and get , then the limit equals
either or .
non-zero #0
Evaluate each limit.
5
5
5
41)
5lim
lim
lim
x
x
x
f xx
f x
f x
f x
Slide 16
1.3 ExamplesEvaluate each limit.
2
2
2
2
32)
2
lim
lim
lim
x
x
x
xf x
x
f x
f x
f x
2
2
1
1
1
23)
1lim
lim
lim
x
x
x
xf x
xf x
f x
f x
Slide 17
1.3 ExamplesInvestigate left and right hand limits, then evaluate.
3
21
14) lim
1x
xx
2
22
25) lim
6y
y yy y
Slide 18
1.3 Examples6) Sketch a possible graph of f(x).
2
2
4
0 1 2 3 4 3
lim 1
lim 3
lim 2
x
x
x
ff ff f
f x
f x
f x
6
4
2
-2
-4
-6
-5 5
Slide 19
1.4 Limits to Infinity
Limit of f as x approaches infinity – means the limit of f as x moves
increasingly far to the .
Limit of f as x approaches negative infinity – means the limit of f as
x moves increasingly far to the .
1limx x
1lim
x x
2
1limx x
2
1lim
x x
Slide 20
1.4 ExamplesEvaluate each limit without graphing.
2
5 2
20 31) lim
3 4 5x
x xx x
27 92) lim
4 3x
xx
Slide 22
1.5 Vertical and Horizontal AsymptotesVertical Asymptote
If , then f(x) has a vertical
asymptote at .
If , then f(x) has a vertical
asymptote at .Horizontal Asymptote
If , then f(x) has a horizontal asymptote
at .
If , then f(x) has a horizontal asymptote
at .
limx a
f x
limx a
f x
limx
f x
limx
f x
Slide 24
1.5 ExamplesGive the equations of all asymptotes.
3 2
2
2 33)
2 6 8x x x
f xx x
2
2
2 154)
9x x
f xx
Slide 26
1.6 Continuity
A function f is continuous at x = c if and only if the following three
conditions are met.
1)
2)
3)
Slide 27
1
-1
0.5
1.6 ContinuityFour types of discontinuities.1) 2)
3) 4)
4
2
-2
-4
-5
4
2
-2
-4
-5
4
2
-2
-4
-5
Slide 28
1.6 ExamplesFind each point of discontinuity and identify the type.
2
11)
4 3x
f xx x
2
22)
3 2x
f xx x
Slide 29
1.6 ExamplesFind each point of discontinuity and identify the type.
2 5, for 3
3) 2, for 3 1, for 3
x xf x x
x x
2 3, for 2
4)7 , for 2x x
f xx x
Slide 30
1.6 Examples5) Explain why f is not continuous at x = 2.
2 3 10
, for 22
4, for 2
x xx
f x xx
Slide 31
1.6 Examples6) Find the value of k that will make f continuous at x = –3.
2 5 24
, for 33
k, for 3
x xx
f x xx
Slide 32
1.6 Examples7) Find the value of c that will make f continuous at x = –1.
2 5, for 1
3, for 1cx x
f xcx x
Slide 33
1.6 ContinuityIntermediate Value Theorem
If f is continuous on [a, b] and w is between f(a) and f(b),
then there exists at least one c on [a, b] such that .
Slide 34
1.7 Rate of Change
Average Rate of Change
Amount of change divided by the time it takes.
*
1) Find the average rate of change of on [5, 21]. 4f x x
Slide 35
1.7 Rate of ChangeSketch the graph of f(x) = 2x² on [0, 5].
Calculate the average rate ofchange for each interval.2) [2, 4]
3) [2, 3]
4) [2, 2.5]
5) [2, 2.01]
50
40
30
20
10
2 4
Slide 36
1.7 Rate of Change
What were we really finding?
*
*To calculate: find the limit of the slope of the secant line as the
points get closer and closer together.Write this limit using the points (a, f(a)) and (a + h, f(a + h))
Slide 39
1.7 Rate of ChangeNormal Line
*8) Write equations of the tangent and normal lines at x = -1.
2 5 6f x x x
Slide 40
1.8 Definition of the DerivativeWe use the derivative to help us find the slope of the tangent line.
Notation:
Prime:
: dydx
1) If f(6) = -5 and f’(6) = 2, write equations of the tangent and normal lines at x = 6.
Slide 41
1.8 Definition of the Derivative
Definition of the Derivative:The derivative of a function f is another function f’ whosevalue at x is
provided this limit exists.Alternate Form:
0
limh
f a h f ah
' lim
x a
f x f af a
x a
Slide 42
1.8 Examples f(x) = x² + 3x - 1
2) Use the definition of the derivative to find f’(-3).
3) Use the alternate form of the derivation to find f’(2).