Transcript C1.1,2,3 – Function Analysis – Critical Values, Intervals

B.5.1 – Critical Values, Intervals of
Increase/Decrease & First Derivative Test
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Lesson Objectives
• 1. Define the terms increasing and decreasing
• 2. Use calculus methods to determine the intervals
in which a function increases or decreases
• 3. Use calculus methods to determine critical
numbers of a function
• 4. Apply the concepts of increase, decrease and
critical numbers to a real world problem
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Fast Five
• 1. Determine f’(2) if f(x) = x - sinx. Explain what
is happening in f(x) at x = 2
• 2. T or F. If f’(b) = 0, then f(x) has a max or min at
x = b. Justify.
• 3. Solve x2 - x - 6 > 0
• 4. Explain the difference between an absolute
maximum and a local maximum
• 5. Explain how you would use Calculus to prove
that f(x) has a maximum point at x = a
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(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
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(B) Review – Graphic Analysis
of a Function
• We have seen functions analyzed given the
criteria intervals of increase, intervals of
decrease, critical points (AKA turning
points or maximum or minimum points)
• We have also seen graphically how the
derivative function communicates the same
criteria about a function  these points are
summarized on the next slide:
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(B) Review – Graphic Analysis
of a Function
• f(x) has a max. at x = -3.1 and f `(x)
has an x-intercept 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
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(C) Analysis of Functions Using
Derivatives – A Summary
• If f(x) increases, then f `(x) > 0
• If f(x) decreases, then f `(x) < 0
• At a max/min point, f `(x) = 0
• We can also state the converse of 2 of these statements:
• If f `(x) > 0, then f(x) is increasing
• If f `(x) < 0, then f(x) is decreasing
• The converse of the third statement is NOT true  if f `(x) = 0, then
the function may NOT necessarily have a max/min  so for now, we
will call any point that gives f `(x) = 0 (i.e. produces a horizontal
tangent line) a CRITICAL POINTS or EXTREME
POINTS
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(D) First Derivative Test
• So if f `(x) = 0, how do we decide if the
point at (x, f(x)) is a maximum, minimum,
or neither (especially if we have no graph?)
• Since we have done some graphic analysis
with functions and their derivatives, in one
sense we already now the answer:  see
next slide
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(E) First Derivative Test Graphically
• 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
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(F) First Derivative Test Algebraically
•
At a maximum, the fcn changes from being an increasing fcn to a decreasing
fcn  the derivative changes from positive values to negative values
•
At the minimum, the fcn changes from decreasing to increasing  the
derivative changes from negative to positive
•
So to state the converses:
•
If f `(x) = 0 and f the sign of if `(x) changes from positive to negative, then the
critical point on f(x) is a maximum point
If f `(x) = 0 and f the sign of if `(x) changes from negative to positive, then the
critical point on f(x) is a minimum point
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•
So therefore, if the sign on f `(x) does not change at the critical point, then the
critical point is neither a maximum or minimum  we will call these points
STATIONARY POINTS
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(G) First Derivative Test – Ex #1
• Find the local max/min values of y = x3 - 3x + 1 (Show how to use
inequalities to analyze for the sign change)
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f `(x) = 3x2 – 3
f `(x) = 0 for the critical values
0 = 3x2 – 3
0 = 3(x2 – 1)
0 = 3(x – 1)(x + 1)
x = 1 or x = -1
• Now, what happens on the function, at x = + 1?  let’s set up a chart
to se what happens with the signs on the derivative so that we can
determine the sign on the derivative so that we can classify the critical
points
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(G) First Derivative Test – Ex #1
Factor 
3
(x-1)
(x+1)
f `(x)
f(x)
(-∞,-1)
+
-
-
+
inc
(-1,1)
+
-
+
-
dec
(1, ∞)
+
+
+
+
inc
Interval 
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(G) First Derivative Test – Ex #1
•
Since the derivative changes signs from +ve to –ve, the critical point at x = -1
is a maximum (the original function changing from being an increasing fcn to
now being a decreasing fcn)
•
Since the derivative changes signs from -ve to +ve, the critical point at x = 1 is
a minimum (the original function changing from being a decreasing fcn to now
being an increasing fcn)
•
Then, going one step further, we can say that f(-1) = 3 gives us a maximum
value of 3 and then f(1) = -1 gives us a minimum value of -1
•
And going another step, we can test the end behaviour of f(x):
 lim x-∞ f(x) = -∞
 lim x ∞ f(x) = +∞
•
Therefore, the point (-1,3) represents a local maximum (as the fcn rises to
infinity “at the end”) and the point (1,-1) represents a local minimum (as the
fcn drops to negative infinity “at the negative end”)
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(G) First Derivative Test – Example
#1 – Graphic Summary
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(H) In Class Examples
• Ex 2. Find the local max/min values of g(x) = x4 - 4x3 - 8x2 - 1
• Ex 3. Find the absolute minimum value of f(x) = (x + 1)/x for x > 0
• Ex 4. Find the intervals of increase and decrease and max/min values
of f(x) = cos(x) – sin(x) on (-,)
• Ex 5. Find the intervals of increase/decrease and max/min points of
f(x) = x2e-x
• Ex 6. Find the local and absolute maximum & minimum points for f(x)
= x(ln(x))2
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
(H) In Class Examples
• Find the local and absolute maximum & minimum points of
f (x)  x  4x  8x 1 on [2,5]
4
3
2
• (1) Use limits to algebraically determine the end behaviour of the
graph. Justify what x values represent the “ends” of the function
• (2) Use the first derivative test to find all max/mins
• (3) Use the second derivative to verify your max/mins from the FDT
• (4) Explain which method you prefer to test for max/min (FDT or
SDT)
• (5) Use the GDC to see the graph and determine the correctness of
your solution
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(H) In Class Examples
• Find the local and absolute maximum & minimum points of
x2 1
f (x) 
x
• (1) Use limits to algebraically determine the end behaviour of the
graph. Justify what x values represent the “ends” of the function
• (2) Use the first derivative test to find all max/mins
 the second derivative to verify your max/mins from the FDT
• (3) Use
• (4) Explain which method you prefer to test for max/min (FDT or
SDT)
• (5) Use the GDC to see the graph and determine the correctness of
your solution
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(H) In Class Examples
• Find the local and absolute maximum & minimum points for
f (x)  x  ln x
2
• (1) Use limits to algebraically determine the end behaviour of
the graph. Justify what x values represent the “ends” of the
function
• (2)
Use the first derivative test to find all max/mins
• (3) Use the second derivative to verify your max/mins from the
FDT
• (4) Explain which method you prefer to test for max/min (FDT
or SDT)
• (5) Use the GDC to see the graph and determine the correctness
of your solution
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(H) In Class Examples
• Find the local and absolute maximum & minimum points for
f (x)  sin x  cos 2x on [0,2 ]

• (1) Use limits to algebraically determine the end behaviour of
the graph. Justify what x values represent the “ends” of the
function
• (2) Use the first derivative test to find all max/mins
• (3) Use the second derivative to verify your max/mins from the
FDT
• (4) Explain which method you prefer to test for max/min (FDT
or SDT)
• (5) Use the GDC to see the graph and determine the correctness
of your solution
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(H) In Class Examples
• Prove that the following function is a
bounded function
x
f (x)  e
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(H) In Class Examples
• Graph the following functions on the GDC,
find the max/min values and then verify
algebraically
cos
x x
t(

)

• (i) f (x)  e
(ii)
2  sin
3

 local max and mins of
• Find the absolute and
 x 2
1  x  0
f (x)  
2
0  x 1
2  x
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(H) In Class Examples
• Sketch a graph of a function that is continuous on
x = 3 and has an absolute max at 0, an absolute
min at 3 a local max at 1 and a local min at 2
• Sketch a graph of a function that has a local
maximum at x = 2 and is:
• (i) differentiable at x = 2
• (ii) continuous, but not differentiable at x = 2
• (iii) discontinuous at x = 2
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(H) In Class Examples
• An object with a weight of W is dragged along a
horizontal plane by a force acting along a rope
attached to the object. If the rope makes an angle
of  with the plane, then the magnitude of the
force is given by
W
F   
 sin   cos 
where  is a positive constant and where 0< </2.
Show that Fis minimized when tan  = 
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(H) In Class Examples
• The rabbit population on a small island is
approximated by
t22

P(t)  e
3
• (i) Determine when the population was
increasing

• (ii) Determine when the rate of decrease of
rabbits was at its maximum.
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(H) In Class Examples
• For my business, Math Inc, the cost function, C(x) = 5500
+ 2.3x + 0.012x2, the demand function is p(x) = 100 - 0.2x
• (i) Determine the production level that maximizes my
profit
• (ii) Determine the production levels in which my revenues
are increasing
• (iii) Determine the production level(s) at which my
average costs is minimized
• (iv) Determine the production level in which my marginal
profits are decreasing
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(I) Internet Links
• Visual Calculus - Maxima and Minima from UTK
• Visual Calculus - Mean Value Theorem and the
First Derivative Test from UTK
• First Derivative Test -- From MathWorld
• Tutorial: Maxima and Minima from Stefan Waner
at Hofstra U
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(J) Homework
• Textbook, S5.1, p278 - 282
• Q7,8 (graphs)
• Q9-26 (algebra, as needed plus variety)
• Q34,37,39,41,42,44,46,48 (word problems)
• NOTE: I will be marking word problems for
solutions and answers
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