At the points where the graph is discontinuous or not differentiable, state why.
The graph of
is continuous on
The graph of
has a removable discontinuity at
and a jump discontinuity at
See
[link] .
The graph of is differentiable on
The graph of
is not differentiable at
because it is a point of discontinuity, at
because of a sharp corner, at
because it is a point of discontinuity, and at
because of a sharp corner. See
[link] .
Finding an equation of a line tangent to the graph of a function
The equation of a tangent line to a curve of the function
at
is derived from the point-slope form of a line,
The slope of the line is the slope of the curve at
and is therefore equal to
the derivative of
at
The coordinate pair of the point on the line at
is
If we substitute into the point-slope form, we have
The equation of the tangent line is
The equation of a line tangent to a curve of the function
f
The equation of a line tangent to the curve of a function
at a point
is
Given a function
find the equation of a line tangent to the function at
Find the derivative of
at
using
Evaluate the function at
This is
Substitute
and
into
Write the equation of the tangent line in the form
Finding the equation of a line tangent to a function at a point
Find the equation of a line tangent to the curve
at
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