Derivatives
The Slope of a CurveorHow Secant Lines become Tangent Lines1
What is the slope of the line through these two points?(0, 0)(2,
4)This is called a secant line through the curve y = x2 because it
passes through two points on the curve.
Finding the slope of a line is easyWhat about finding the slope
of a curve?2
What is the slope of the line through this point?(1, 1)This is
called a tangent line through the curve y = x2 at the point (1,
1).
Since the slope of a curve like this one is always changing, we
can only talk about slope in terms of specific points or intervals
on the curve3
(0, 0)(1, 1)Lets start by drawing a secant line through the
point (1, 1) and some other point close to it.
This is clearly not the slope at (1, 1)So now what do we
do?Since the slope of a curve like this one is always changing, we
can only talk about slope in terms of specific points or intervals
on the curveTry a point even closer4
(1, 1)Lets start by drawing a secant line through the point (1,
1) and some other point close to it.
This is a lot closer to the slope at (1, 1)How much closer can
we get?Since the slope of a curve like this one is always changing,
we can only talk about slope in terms of specific points or
intervals on the curveIf we use limits, we can get as close to the
point as we want.
5
What is the slope of the line at (1, 1)?(1, 1)And the first
thing we do when taking a limit is
Since the slope of a curve like this one is always changing, we
can only talk about slope in terms of specific points or intervals
on the curve
Plug it inNow what?6
What is the slope of the line through this point?(1, 1)So the
slope of the tangent line through the curve y = x2 at the point (1,
1) is 2Since the slope of a curve like this one is always changing,
we can only talk about slope in terms of specific points or
intervals on the curve
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f(x)x
But we need to be able to write and read this formula in text
book terms (which means replacing y with f (x))8For all of the
enrichment provided by problem-based learning, some students still
find comfort in being able to wrap it all up in one or two
formulas.
f(x)x
9For all of the enrichment provided by problem-based learning,
some students still find comfort in being able to wrap it all up in
one or two formulas.
is called the derivative of f at aWe write:
The derivative of f with respect to x is There are two formulas
for the derivative of
10There are many ways to write the derivative of
The Definition of the Derivative:The Derivative at a Point:
y2y1x2x1x2x1(Also called the Numerical Derivative in your
text)11
f prime xorthe derivative of f with respect to x
y prime
or12
Heres how to find the general derivative of a polynomial
function.
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Heres how to find the general derivative of a polynomial
function.
So in this case the derivative of f (x) isAnd what this means
isThe slope of y = x2 3 atx = 1 is
x = 2 is
x = 2 is
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