Calculus 3.2

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3.2 Differentiability
Arches National Park
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Arches National Park
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To be differentiable, a function must be
continuous and smooth.
Derivatives will fail to exist at
corner
cusp
discontinuity
vertical tangent
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Most of the functions we study in calculus will
be differentiable.
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Derivatives on the TI-89
You must be able to calculate derivatives with
the calculator and without.
Today you will be using your calculator, but be
sure to do them by hand when called for.
Remember that half the test is no calculator.
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Find at x 2.
Example
returns
It is not a lower case letter d.
returns
or use
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Warning
The calculator may return an incorrect value if
you evaluate a derivative at a point where the
function is not differentiable.
Examples
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Graphing Derivatives
Graph
What does the graph look like?
This looks like
Use your calculator to evaluate
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Two theorems
Since a function must be continuous to have a
derivative, if it has a derivative then it is
continuous.
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Intermediate Value Theorem for Derivatives
p