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Section 2.6 Differentiability

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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

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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!

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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

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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

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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

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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

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13) A cable is made of an insulating material in the shape of a long, thin cylinder of radius r 0. 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 = r 0 ? Is E differentiable at r = r 0 ? Sketch a graph of E as a function of r.

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