Download Properties of the Derivative — Lecture 9. Recall that the average

Properties of the Derivative —
Lecture 9.
Recall that the average rate of change of a function y =
f (x) over the interval from a to a + h, with h 6= 0, is the slope
∆y f (a + h) − f (a)
=
,
∆x
h
of the line between
(a, f (a)) and (a + h, f (a + h)),
and the the instant rate of change or derivative of
f (x) at a is the limit
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df f (a + h) − f (a)
0
f (a) = = lim
,
dx x=a h→0
h
the slope of the line
tangent to y = f (x) at (a, f (a)).
slope = f 0(a)
(a + h, f (a + h))
∆y
(a, f (a))
∆x = h
More generally, given a function y = f (x), the derivative of
f is the function y = f 0(x) whose domain is the set of all x at which
f 0(x) exists and is then given by the rule
f 0(x) = the derivative of f at the point x.
For many functions f we can with relative ease can calculate approximately their derivatives at particular points using a calculator
or other technical aids. How to do this, and more, is discussed at
some length in the text for the TI-83 and Excel. We urge you to
spend some time seeing how this works and becoming familiar with
it. We will bypass that here in class, but you should not take that
as a reason to skip that valuable knowledge!
Now, though, let’s calculate the actual derivatives of some simple functions, check out some standard rules of differentiation, and
then put most of our effort into learning some of the significance of
derivatives.
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Example 1. First, a constant function f (x) = a. This is a linear
function with a horizontal line for a graph, so its slope, and hence its
derivative is zero!! That is, using some of our new notation
d
a = 0.
dx
Example 2. A linear function f (x) = mx + b has for a graph
a line with slope m. So it’s derivative is simply the constant function y = m. Let’s just check that using the appropriate limit of a
difference quotient:
f (x + h) − f (x)
h→0
h
m(x + h) + b − (mx + b)
= lim
h→0
h
mx + mh − mx
= lim
= lim m = m
h→0
h→0
h
as we saw from the geometry!
f 0(x) = lim
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Example 3. Now let’s try the quadratic function f (x) = ax2.
Calculating the difference quotient for this function and then evaluating its limit, we find that for all x
f (x + h) − f (x)
h→0
h
a(x + h)2 − ax2
lim
h→0
h
2
ax + 2ahx + ah2 − ax2
lim
h→0
h
2ahx + ah2
lim
h→0
h
lim 2ax + ah = 2ax.
f 0(x) = lim
=
=
=
=
h→0
So we have neat formula for the derivative
d
(ax2) = 2ax.
dx
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A Couple Rules of Differentiation.
Later we shall assemble several rules that will greatly help us calculate the derivatives of a great many important functions including
all of the ones we’ve studied in this course! But for now here are a
couple of dandies:
Addition. If f and g are differentiable at x, then so is their
sum and
d
d
d
(f (x) + g(x)) = f (x) + g(x).
dx
dx
dx
Constant Multiple. If f is differentiable at x and if a is a
constant, then af is differentiable at x, and
d
d
(af )(x) = a f (x).
dx
dx
Example 4. What we have so far makes it easy to find the
derivative of any quadratic. Let’s try it with f (x) = 3x2 − 5x + 7:
d
d
f (x) =
(3x2 − 5x + 7)
dx
dx
=
d
d
d
(3x2) + (−5x) + (7)
dx
dx
dx
= 3
d 2
d
(x ) − 5 (x) + 0
dx
dx
= 3 · 2x − 5 = 6x − 5.
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Next, let’s consider some of the behavior of the graphs of functions.
So suppose we have a function f . Then we say that
the function f is increasing on the interval a ≤ x ≤ b if for
all a ≤ x1 < x2 ≤ b
f (x1) < f (x2).
the function f is decreasing on the interval a ≤ x ≤ b if for
all a ≤ x1 < x2 ≤ b
f (x1) > f (x2).
The function f has a
the derivative
horizontal tangent at some point a if
f 0(a) = 0.
Let’s check out these concepts in a simple example:
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Example 5. Find the intervals on which the function is
Positive (f
(x) > 0),
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Negative (f (x) < 0),
Zero (f (x) = 0).
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y = f (x)
2.5
5
10
15
20
-2.5
-5
-7.5
-10
Positive:
Negative:
Zero
Next find all intervals on which the function y = f (x) is/has
Increasing:
Decreasing:
Horizontal Tangent:
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If the function f is differentiable, then this behavior can be detected from the derivative! For example, to say that the function is
increasing on an interval a ≤ x ≤ b, simply means that for each pair
a ≤ x1 < x2 ≤ b on that interval the average rate of change from x1
to x2 is positive, and thus, the derivative at any point on the interval is non-negative! A similar argument shows that the function is
decreasing on the interval if and only if its derivative is non-positive
on that interval! Let’s summarize all of this in the table:
f (x)
increasing
decreasing
horizontal tangent
f 0(x)
f 0(x) > 0
f 0(x) < 0
f 0(x) = 0
Okay. With that information in hand, let’s try a few examples:
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Example
6. Sketch the graph of the derivative of the function:
y = f (x)
Example
7. Sketch the graph of the derivative of the function:
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y = f (x)
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Example 8. Sketch the graph of the derivative of the function:
a
b
c
y = f (x)
x-intercepts of f 0(x):
f 0(x)
positive:
negative:
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Example 9. Sketch the graph of the derivative of the function:
y = f (x)
x-intercepts of f 0(x):
f 0(x)
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positive:
negative:
Example 10. Sketch the graph of the function y = f (x) whose
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derivative
is:
y = f 0(x)
Example 11. Sketch the graph of the function y = f (x) whose
derivative
is:
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y = f 0(x)
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1
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Example 12. Sketch the graph of the function y = f (x) whose
derivative is:
a
b
c
y = f 0(x)
Horizontal tangents of f (x):
f (x)
increasing:
decreasing:
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Example 13. Sketch the graph of the function y = f (x) whose
derivative is:
y = f 0(x)
Horizontal tangents of f (x):
f (x)
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increasing:
decreasing: