Transcript BCC.01.8 – What Derivatives Tell us About Functions

B.3.5 - Graphical
Differentiation
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Lesson Objectives
• 1. Given the equation of a function, graph it and then make
conjectures about the relationship between the derivative function
and the original function
• 2. From a function, sketch its derivative and from a derivative,
graph an original function
• 3. Given a mathematical statement about a function and its
derivative, give a contextual interpretation of the mathematical
statement
• 4. Determine graphically where a function is and isn’t
differentiable
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Fast Five
• 1. Find f’(x) if f(x) = -x2 + 2x
• 2. Sketch a graph whose first
derivative is always negative
• 3. The fuel consumption
(measured in litres per hour) of
a car traveling at a speed of
v km/hr is c = f(v). What is the
meaning of f `(v)? What are its
units? Write a sentence that
explains the meaning of the
equation f `(30) = -0.05.
• 4. Graph the derivative of the
function shown in the next
graph
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(A) Important Terms
• Recall the following terms as they were presented in
previous lessons:
• turning point: points where the direction of the function
changes
• maximum: the highest point on a function
• minimum: the lowest point on a function
• local vs absolute: a max can be a highest point in the
entire domain (absolute) or only over a specified region
within the domain (local). Likewise for a minimum.
• "end behaviour": describing the function values (or
appearance of the graph) as x values getting infinitely large
positively or infinitely large negatively or approaching an
asymptote
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(A) Important Terms
• increase: the part of the domain (the interval) where the function
values are getting larger as the independent variable gets higher; if
f(x1) < f(x2) when x1 < x2; the graph of the function is going up to the
right (or down to the left)
• decrease: the part of the domain (the interval) where the function
values are getting smaller as the independent variable gets higher; if
f(x1) > f(x2) when x1 < x2; the graph of the function is going up to the
left (or down to the right)
• “concave up” means in simple terms that the “direction of opening” is
upward or the curve is “cupped upward”
• An alternative way to describe it is to visualize where you would draw
the tangent lines => you would have to draw the tangent lines
“underneath” the curve
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(A) Important Terms
• Concavity is best “defined”
with graphs
• (ii) “concave down”
means in simple terms that the
“direction of opening” is
downward or the curve is
“cupped downward”
• An alternative way to describe
it is to visualize where you
would draw the tangent lines 
you would have to draw the
tangent lines “above” the curve
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(B) New Understanding of Concavity
• In keeping with the idea of concavity and the
drawn tangent lines, if a curve is concave up and
we were to draw a number of tangent lines and
determine their slopes, we would see that the
values of the tangent slopes increases (become
more positive) as our x-value at which we drew
the tangent slopes increase
• This idea of the “increase of the tangent slope is
illustrated on the next slides:
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(B) New Term – Concave Up
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(B) New Term – Concave Down
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(C) Functions and Their Derivatives
• In order to “see” the connection between a graph
of a function and the graph of its derivative, we
will use graphing technology to generate graphs of
functions and simultaneously generate a graph of
its derivative
• Then we will connect concepts like max/min,
increase/decrease, concavities on the original
function to what we see on the graph of its
derivative
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(D) Example #1
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(D) Example #1
•
•
•
•
•
•
Points to note:
(1) the fcn has a minimum at x=2
and the derivative has an xintercept at x=2
(2) the fcn decreases on (-∞,2) and
the derivative has negative values
on (-∞,2)
(3) the fcn increases on (2,+∞) and
the derivative has positive values
on (2,+∞)
(4) the fcn changes from decrease
to increase at the min while the
derivative values change from
negative to positive
(5) the function is concave up and
the derivative fcn is an increasing
fcn
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(E) Example #2
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(E) Example #2
•
f(x) has a max. at x = -3.1 and f `(x) has an xintercept at x = -3.1
•
f(x) has a min. at x = -0.2 and f `(x) has a root at –
0.2
•
f(x) increases on (-, -3.1) & (-0.2, ) and on the
same intervals, f `(x) has positive values
•
f(x) decreases on (-3.1, -0.2) and on the same
interval, f `(x) has negative values
•
At the max (x = -3.1), the fcn changes from being
an increasing fcn to a decreasing fcn  the
derivative changes from positive values to
negative values
•
At a the min (x = -0.2), the fcn changes from
decreasing to increasing  the derivative
changes from negative to positive
•
f(x) is concave down on (-, -1.67) while f `(x)
decreases on (-, -1.67)
•
f(x) is concave up on (-1.67,  ) while f `(x)
increases on (-1.67, )
•
The concavity of f(x) changes from CD to CU at
x = -1.67, while the derivative has a min. at x = 1.67
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(F) Internet Links
• Watch the following animations which serve to
illustrate and reinforce some of these ideas we saw
in the previous slides about the relationship
between the graph of a function and its derivative
• (1) Relationship between function and derivative
function illustrated by IES
• (2) Moving Slope Triangle Movie
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(G) Matching Function Graphs
and Their Derivative Graphs
• To further visualize the relationship
between the graph of a function and the
graph of its derivative function, we can run
through some exercises wherein we are
given the graph of a function  can we
draw a graph of the derivative and vice
versa
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(G) Matching Function Graphs and
Their Derivative Graphs
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(G) Matching Function Graphs and
Their Derivative Graphs - Answer
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(G) Matching Function Graphs and Their
Derivative Graphs – Working Backwards
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(G) Matching Function Graphs and Their
Derivative Graphs – Working Backwards
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(H) Continuity and Differentiability
• Recall the fundamental idea that a derivative at a
point is really the idea of a limiting sequence of
secant slopes (or tangent line) drawn to a curve at
a given point
• Now , if a function is discontinuous at a given
point, try drawing secant lines from the left and
secant lines from the right and then try drawing a
specific tangent slope at the point of discontinuity
in the following diagrams
• Conclusion  you can only differentiate a
function where is it is continuous
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(H) Continuity and Differentiability
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(H) Continuity and Differentiability
• Follow this link to One-sided derivatives
from IES Software
• And then follow this link to Investigating
Differentiability of Piecewise Functions
from D. Hill (Temple U.) and L. Roberts
(Georgia College and State University
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(H) Continuity and Differentiability
• One other point to add that comes from our study of the last two
examples => even if a function is continuous, this does not always
guarantee differentiability!!!!
• If a continuous function as a cusp or a corner in it, then the function is
not differentiable at that point => see graphs on the next slide and
decide how you would draw tangent lines (and secant lines for that
matter) to the functions at the point of interest (consider drawing
tangents/secants from the left side and from the right side)
• As well, included on the graphs are the graphs of the derivatives (so
you can make sense of the tangent/secant lines you visualized)
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(H) Continuity and Differentiability
•
•
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Continuous functions are non-differentiable
under the following conditions:
 The fcn has a “corner” (ex 1)
 The fcn has a “cusp” (ex 2)
 The fcn has a vertical tangent (ex 3)
This non-differentiability can be seen in that
the graph of the derivative has a discontinuity
in it!
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(I) Internet Links
• Work through these interactive applets from
maths online Gallery - Differentiation 1 wherein
we are given graphs of functions and also graphs
of derivatives and we are asked to match a
function graph with its derivative graph
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(J) Interpretation of Derivatives
• What follows in the next slides are various questions
which involve interpretations of derivatives  what do
they really mean in the context of “word problems”
• Here you are expected to verbally convey your
understanding of derivatives along the lines of rates of
change, rate functions, etc...
• Most of the following “applications” are independent of
algebra and graphs
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(J) Interpretation of Derivatives
• Ex. 1. The cost C in dollars of building a house of
A square feet in area is given by the function C =
f(A). What is the practical interpretation of the
function dC/dA or f `(A)? One option on
interpreting is to consider units (dollars per square
foot).
• Ex. 2 You are told that water is flowing through a
pipe at a constant rate of 10 litres per second.
Interpret this rate as the derivative of some
function.
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(J) Interpretation of Derivatives
• Ex. 3. If q = f(p) gives the number of pounds of sugar
produced when the price per pound is p dollars, then what are
the units and meaning of f `(3) = 50?
• Ex. 4. The number of bacteria after t hours in a controlled
experiment is n = f(t). What is the meaning of f `(4)? Suppose
that there is an unlimited amount of space and nutrients for
the bacteria. Which is larger f `(4) or f `(8)? If the supply of
space and nutrients were limited, would that affect your
conclusion?
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(J) Interpretation of Derivatives
• Ex. 5. The fuel consumption (measured in litres per hour)
of a car traveling at a speed of v km/hr is c = f(v). What is
the meaning of f `(v)? What are its units? Write a sentence
that explains the meaning of the equation f `(30) = -0.05.
• Ex. 6. The quantity (in meters) of a certain fabric that is
sold by a manufacturer at a price of p dollars per meter is
Q = f(p). What is the meaning of f `(16). Its units are? Is f
`(16) positive or negative? Explain.
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(J) Interpretation of Derivatives
• Ex. 7. A company budgets for research and development
for a new product. Let m represent the amount of money
invested in R&D and T be the time until the product is
ready to market.
• (A) Give reasonable units for T and m. What is T ' in these
units.
• (B) What is the economic interpretation of T '?
• (C) Would you expect T ' to be positive or negative?
Explain?
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(J) Interpretation of Derivatives
• Ex. 8 Interpret each sentence as a statement about a function and its
derivative. In each case, clearly indicate what the function is, what
each variable means and appropriate units. Make a sketch of a graph
that best reflects the context.
•
•
•
•
(A) The price of a product decreases as more of it is produced.
(B) The increase in demand for a new product decreases over time.
(C) The work force is growing more slowly than it was five years ago.
(D) Health care costs continue to rise but at a higher rate than 4 years
ago.
• (E) During the past 2 years, Canada has continued to cut its
consumption of imported oil
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(K) Homework
• Textbook, p201-204
• Q1,2,4,6 (explanations required)
• Q7-14 (sketches required)
• Q1-19 (word problems)
• photocopy Hughes-Hallett p 115
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