Section 2.6 Differentiability. Local Linearity Local linearity is the idea that if we look at any...
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Transcript of Section 2.6 Differentiability. Local Linearity Local linearity is the idea that if we look at any...
![Page 1: Section 2.6 Differentiability. Local Linearity Local linearity is the idea that if we look at any point on a smooth curve closely enough, it will look.](https://reader037.fdocuments.us/reader037/viewer/2022103004/56649c755503460f94928108/html5/thumbnails/1.jpg)
Section 2.6Differentiability
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Local Linearity• Local linearity is the idea that if we look at any
point on a smooth curve closely enough, it will look like a straight line
• Thus the slope of the curve at that point is the same as the slope of the tangent line at that point
• Let’s take a look at this idea graphically
![Page 3: Section 2.6 Differentiability. Local Linearity Local linearity is the idea that if we look at any point on a smooth curve closely enough, it will look.](https://reader037.fdocuments.us/reader037/viewer/2022103004/56649c755503460f94928108/html5/thumbnails/3.jpg)
3 2( ) 9 6f x x x x
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Once we have zoomed in enough, the graph looks linear!
Thus we can represent the slope of the curve at that point with a tangent line!
![Page 10: Section 2.6 Differentiability. Local Linearity Local linearity is the idea that if we look at any point on a smooth curve closely enough, it will look.](https://reader037.fdocuments.us/reader037/viewer/2022103004/56649c755503460f94928108/html5/thumbnails/10.jpg)
The tangent line and the curve are almost identical!
Let’s zoom back out
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Differentiability• We need that local linearity to be able to
calculate the instantaneous rate of change– When we can, we say the function is differentiable
• Let’s take a look at places where a function is not differentiable
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• Consider the graph of f(x) = |x|
• Is it continuous at x = 0?• Is it differentiable at x = 0?
– Let’s zoom in at 0
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• No matter how close we zoom in, the graph never looks linear at x = 0– Therefore there is no tangent line there so it is not
differentiable at x = 0
• We can also demonstrate this using the difference quotient
h
xfhxfxf
h
)()(lim)('
0
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Definition
• The function f is differentiable at x if
exists
• Thus the graph of f has a non-vertical tangent line at x
• We have 3 major cases– The function is not continuous at the point– The graph has a sharp corner at the point– The graph has a vertical tangent
h
xfhxfxf
h
)()(lim)('
0
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Example
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• Note: This is a graph of • It has a vertical tangent at x = 0
– Let’s see why it is not differentiable at 0 using our power rule
Example
31
)( xxf
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• Is the following function differentiable everywhere?
• Graph• What values of a and b make the following function
continuous and differentiable everywhere?
Example
0for
0for)(
2 xx
xxxf
0for)1(
0for2)(
2 xxb
xaxxg
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13) A cable is made of an insulating material in the shape of a long, thin cylinder of radius r0. It has electric charge distributed evenly throughout it. The electric field, E, at a distance r from the center of the cable is given by
• Is E continuous at r = r0?
• Is E differentiable at r = r0?
• Sketch a graph of E as a function of r.
0
20
0
kr for r r
E rk for r rr