Transcript BCC.01.8 – What Derivatives Tell us About Functions

5.1 - Analyzing a
Polynomial Fcn - Intervals
of Increase and Decrease
MCB4U - Santowski
(A) Important Terms
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Recall the following terms as they were presented in a previous lesson:
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.
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)
"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
(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 and
increase/decrease on the original function to
what we see on the graph of its derivative
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(D) Example #1
(D) Example #1
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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
(E) Example #2
(E) Example #2
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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) Internet Links
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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
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(1) Relationship between function and derivative function
illustrated by IES
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(2) Moving Slope Triangle Movie
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(3) Interactive Applet showing relationship between
derivative, function, and antiderivative from Cut-theKnot.org
(G) Matching Function Graphs
and Their Derivative Graphs
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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
(G) Matching Function Graphs and
Their Derivative Graphs
(G) Matching Function Graphs and
Their Derivative Graphs - Answer
(G) Matching Function Graphs and Their
Derivative Graphs – Working Backwards
(G) Matching Function Graphs and Their
Derivative Graphs – Working Backwards - Answer
(H) Internet Links
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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
(I) Increasing & Decreasing
Analysis – Algebraic Methods
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Ex 1. Find the interval on which f(x) = 1 – 5x + 4x2 is increasing and
decreasing.
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Our solution will be derived using algebraic methods with calculus
 so we will start by taking the derivative
f`(x) = 8x – 5
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From our previous graphic analysis, we know that a positive first
derivative means the original function is increasing  so we need to
find where f `(x) > 0 or 8x – 5 > 0  which happens when x > 5/8
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Likewise a negative first derivative means that the original function is
decreasing, so f `(x) < 0 or 8x – 5 < 0  which happens when x <
5/8
(I) Increasing & Decreasing
Analysis – Algebraic Methods
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Ex 2. On what interval does the function f(x) = x3 + 6x2 + 9x + 2 increase
and decrease?
Start with the derivative  f `(x) = 3x2 + 12x + 9 and then find out where
the derivative is positive (f `(x) > 0) and negative (f `(x)<0)
 To help us with inequalities, we will factor the quadratic
 f `(x) = 3(x2 + 4x + 3) = 3(x + 1)(x + 3)
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So then (x+1)(x+3)>0 when BOTH factors (x+1) and (x+3) are positive, so x
must be greater than -1 as well as x>-3, which happens only when x>-1
 Or (x+1)(x+3)>0 when BOTH factors (x+1) and (x+3) are negative, so x
must be less than -1 as well as x<-3, which happens only when x<-3
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We can try to visualize these inequalities using a number line
(I) Increasing & Decreasing
Analysis – Algebraic Methods
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As an alternative to a number line, we can
set up an “inequality/sign rectangle” as
follows:
 f `(x) = 3(x+1)(x+3)
(x+1)
(x+3)
f ‘(x)
f(x)
(-∞,-3)
-ve
-ve
+ve
Inc
(-3,-1)
-ve
+ve
-ve
Dec
(-1,+∞)
+ve
+ve
+ve
inc
Factors
interval
(I) Increasing & Decreasing
Analysis – Algebraic Methods
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Ex 3. Find the intervals of increase and decrease of P(x)
= x4 – 4x3 – 8x2 – 1
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So P`(x) = 4x3 – 12x2 - 16x
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So now factor so we can find where the positive and
negative intervals on the derivative function are.
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P`(x)=4x(x2 - 3x - 4) = 4x(x – 4)(x + 1)
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So now we have 3 factors to consider in terms of the
sign of the derivative  easier to set up a chart
(I) Increasing & Decreasing
Analysis – Algebraic Methods
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P`(x)=4x(x2 - 3x - 4) = 4x(x – 4)(x + 1)
Factor
Interval
4x
(x-4)
(x+1)
P`(x)
P(x)
(-∞,-1)
-ve
-ve
-ve
-ve
Dec
(-1,0)
-ve
-ve
+ve
+ve
Inc
(0,4)
+ve
-ve
+ve
-ve
Dec
(4,+∞)
+ve
+ve
+ve
+ve
inc
(I) Increasing & Decreasing
Analysis – Algebraic Methods
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ex 4. ROXY Music predicts that every dollar increase in the price of
its CDs will cause sales to decrease by 10 000 CDs per year. The
store now sells 300 000 CDs at an average price of $15 each every
year.
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(a) Develop a mathematical model for the sales revenue for the
store.
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(b) Using this model, determine when the revenue will increase and
when it will decrease
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(c) The manager is thinking about raising the average CD price by
$2. At this new price, what is the rate of change of revenue?
(I) Homework
from Stewart, 1997, Calculus –
Concepts and Contexts, Chap 2.10, p180,
Q1-4,6,8,11,12,15-18,23
 Handout