Applications of Derivatives

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Chapter 4

• Applications of Derivatives

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4.1 Extreme Values of Functions
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(0, 2)
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The Extreme Value Theorem
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Local Extreme Values
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Find the extreme values of
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Find the extreme values of
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p.193 (1-29, 35-41)odd
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4.2 Mean Value Theorem
Rolle's Theorem Suppose that y f(x) is
continuous at every point of a, b and
differentiable at every point of (a, b).
If f(a) f(b) 0, then there is at least
one number c in (a, b) at which f '(c)
0. Proof Being continuous, f assumes absolute
max and min values on a, b. These can occur
only 1. at interior points where f ' is zero 2.
at interior points where f ' does not exist 3.
at the endpoints of the function's domain, in
this case, a and b. By the hypothesis f has a
derivative at every integer point of a, b.
That rules option 2. If either the max or min
occurs at a point c inside the interval, then f
'(c) 0 a previous theorem we have found a point
for Rolle's Theorem. If both max and min are
min are at a or b then the max and min values of
f are both 0. Thus, f has the constant value 0
so f ' 0, throughout (a, b) and c can be taken
anywhere in the interval.
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Mean Value Theorem for Derivatives
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Proof We picture the graph of f as a curve in
the plane and draw a line through the points A(a,
f(a)) and B(b, f(b)). The line is the graph of
the function The vertical difference between
the graphs of f and g at x is The function h
satisfies the hypotheses of Rolle's Theorem on
a, b. It is continuous on a, b and
differentiable on (a, b) because f and g are.
Also, h(a) h(b) 0 because the graphs of f
and g both pass through A and B. Therefore h'
0 at some point c in (a, b). This is the point
we want. To verify we differentiate both sides
of our previous equation with respect to x and
then set x c
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• Ex The function y x2 is
• Decreasing?
• Increasing?

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p.202 (1-9, 15-37)odd
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4.3 Connecting f and f with the graph of f
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Same directions as above for
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Concavity Test
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Second Derivative Test for Local Extrema
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Example A particle is moving along a horizontal
line with position function
t 0 Find the velocity and
acceleration and describe the motion of the
particle.
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Procedure for Graphing y f(x) by Hand. 1. Find
y' and y". 2. Find the rise and fall of the
graph. 3. Determine the concavity of the
curve. 4. Make a summary and show the curve's
general shape. 5. Plot specific points and
sketch the curve. Note Use zeros if you know
them. Example Sketch the graph of the function
f(x) x4 - 4x3 10 using the following
steps. (a) Identify where the extrema of f
occur. (b) Find the intervals on which f is
increasing or decreasing. (c) Find where the
graph of f is concave up or down. (d) Sketch a
possible graph for f.
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Lets examine page 209 Example 4 to discuss how
to graph a function based on its
derivative. Lets examine page 214 Example 9
to discuss how to graph a function based on its
derivative.
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p.215 (1-51)odd
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4.4 Modeling and Optimization
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Example Find two numbers whose sum is 20 and
whose product is as large as possible.
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Example A rectangle is to be inscribed in a
semicircle of radius 2. What is the largest area
the rectangle can have, and what dimensions give
that area?
Example An open-top box is to be made by
cutting congruent squares of side length x from
the corners of a 20 by 25 inch sheet of tin and
bending up the sides. How large should the
squares be to make the box hold as much as
possible? What is the resulting maximum volume?
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Example You have been asked to design a one
liter oil can shaped like a right circular
cylinder. What dimensions will use the least
material?
A drilling rig 12 mi offshore is to be connected
by pipe to a refinery onshore, 20 mi straight
down the coast from the rig. If underwater pipe
costs 50,000 per mile and land-based pipe costs
30,000 per mile, what combination of the two
will give the least expensive connection?
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Ex Suppose c(x) x3 6x2 15x, where x
represents thousands of units. Is there a
production level that minimizes average cost? If
so, what is it?
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p.226 (1-41) odd
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4.6 Related Rates
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How rapidly will the fluid level inside a
vertical cylindrical tank drop if we pump the
fluid out at the rate of 3000 L/min?
Example A hot air balloon rising straight up
from a level field is tracked by a range finder
500 ft from the liftoff point. At the moment the
range finder's elevation angle is p/4, the angle
is increasing at the rate of 0.14 rad/min. How
fast is the balloon rising at that moment?
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Example Water runs into a conical tank at the
rate of 9 ft3/min. The tank stands point down
and has a height of 10 ft and a base of 5 ft.
How fast is the water level rising when the water
is 6 ft deep?
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p.251 (1-35) odd