3.2: continuity
DESCRIPTION
3.2: Continuity. Objectives: To determine whether a function is continuous Determine points of discontinuity Determine types of discontinuity Apply the Intermediate Value Theorem. CONTINUITY AT A POINT—No holes, jumps or gaps!!. A function is continuous at a point c if: - PowerPoint PPT PresentationTRANSCRIPT
3.2: DifferentiabilityObjective: Students will be able to…• Find points where the derivative
fails to exist.
There are 4 instances where the derivative fails to exist…
Derivative
Where f’(x) Fails to Exist
O Is the function f(x) = |x-2| differentiable at x=2?
O Find the left and right hand derivative:
O Look at the graph…what does it look like at x=2?
DNExfxf
x
2)2()(lim
2
1. CORNER:O One sided derivatives exist, but
differ
Given: f(x)=x2/3. Find f ’(0 ).
0)0()(lim
0 xfxf
x
• What is the domain of f(x)? What is the domain of f’(x)?
• Graph the function. What do you see at x=0?
2. A CUSP: (rational exponent—numerator even, denominator odd)
O an extreme cornerO The slopes of the secant lines
approach ∞ from one side and -∞ from the other side
Find the derivative of f(x)=x1/3 at x =0.
Look closer ….
Look at graph of function.
00lim
31
0 xx
x
3. VERTICAL TANGENT(rational exponent—numerator and denominator are odd)
O Slopes of the secant lines approach either ∞ or -∞ from both sides
Find f’(0) if
Graph f(x). What do you see?
0,20,2
)(xx
xf
4.DISCONTINUITYO Will cause one or both of the one
sided derivatives to be ±∞
Find all the points in the domain where f is not differentiable. Justify your answer.
1. f(x)= |x +3| -1
2. 3 62)( xxf
Find all points where f is not differentiable. Justify your answer.
1. 2.
1,
1,)(
2 xx
xxxf 3
23)( xxg
Most functions are differentiable wherever they are defined. Look for corners, cusps, PODs, and vertical tangents.
O Polynomials→ differentiableO Rational, Trig, Exponential, Log
functions→ differentiable in domains
Given:
O Where is y not differentiable?
Think of the graph….check endpoint of interval
1 xy
Differentiability Implies Continuity
O If f has a derivative at x=a, then f is continuous at x=a
O Is the converse true?
Differentiability Implies Local Linearity
O A function that is differentiable at ‘a’ closely resembles its own tangent line very close to ‘a’
O p.107: Exploration
Is f(x)=sin x differentiable at x=л? If so, what is f’(л)?
Intermediate Value Theorem for Derivatives
O If a and b are any two points in an interval on which f is differentiable, then f’ takes on every value between f’(a) and f’(b).