Download Extrema on an Interval

10/5/2016
Extrema on an Interval
Definition of Extrema
Let f be defined on an interval I containing c.
1. f(c) is the minimum of f on I if f(c)≤f(x) for all
x in I.
2. f(c) is the maximum of f on I if f(c)≥f(x) for all
x in I.
Section 3.1 AP Calculus
The minimum and maximum of a function on an
interval are the extreme values, or extrema,
of the function on the interval. The minimum
and maximum of a function on an interval are
also called the absolute minimum and
absolute maximum.
Thm 3.1 The Extreme Value Theorem
If f is continuous on a closed interval [a,b], then f
has both a minimum and a maximum on the
interval.
y   ( x  2) 2  4
Find the extrema of the function on the interval
[-3,3].
y  x3  4 x  2
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Definition of Relative Extrema
1. If there is an open interval containing c on
which f(c) is a maximum, then f(c) is called a
relative maximum of f.
2. If there is an open interval containing c on
which f(c) is a minimum, then f(c) is called a
relative minimum of f.
Definition of a Critical Number
Let f be defined at c. If f’(c)=0 or if f is not
differentiable at c, then c is a critical number
of f.
Finding Extrema on a Closed Interval
1. Find critical numbers in (a,b)
2. Evaluate f at each critical number in (a,b)
3. Evaluate f at each endpoint of [a,b]
4. The least of these values is the minimum, the
greatest is the maximum
Find all relative extrema on (-3,3).
y
4x
x 1
2
Thm 3.2 Relative Extrema Occur Only at Critical
Numbers
If f has a relative minimum or relative maximum
at x=c, then c is a critical number of f.
Find the extrema on [0,4].
y  x 3  12 x
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Find the extrema on [-2,2].
Find the extrema on 6 , 4  .
g ( x)  4  x 2
h(t )   cos(2t )
Rolle’s Theorem and Mean Value
Theorem
Sketch a rectangular coordinate plane on a piece of paper.
Label the points (1, 3) and (5, 3). Using a pencil or pen,
draw the graph of a differentiable function f that starts at
(1, 3) and ends at (5, 3). Is there at least one point on the
graph for which the derivative is zero? Would it be possible
to draw the graph so that there isn’t a point for which the
derivative is zero? Explain your reasoning.
Section 3.2 AP Calc
Thm 3.3 Rolle’s Theorem
Let f be continuous on closed interval [a, b] and
differentiable on the open interval (a, b). If
f(a)=f(b) then there is at least one number c in
(a, b) such that f’(c)=0.
Find 2 x-intercepts of y=x²-3x, show that
f’(x)=0 for some point c on the interval
between the two intercepts.
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Determine if Rolle’s Thm can be applied on the
interval [-1,1] for
f ( x) 
x2 1
x
If so, find all values c in the open interval
(-1,1) s.t. f’(c)=0.
Thm 3.4 The Mean Value Theorem
If f is continuous on [a,b] and differentiable an
(a,b) then there exists a number c in (a,b) s.t.
f ' (c ) 
f (b )  f ( a )
ba
Can Rolle’s Thm be applied? If so find f’(c)=0
for all values c.
f ( x)  ( x  3)( x  1) 2
[-1,3]
Determine whether the MVT can be applied
x 1
on [1/2, 2] for
f ( x) 
x
and find all values of c s.t.
f ' (c ) 
f (b )  f ( a )
ba
The height of an object t seconds after dropped from a
height of 500m is s (t )  4.9t 2  500
a) Find the average velocity during the first 3 seconds
b) Use MVT to verify that at some time during the first 3
seconds of fall the instantaneous velocity equals the avg.
velocity. Find the time when this occurs.
Increasing & Decreasing Functions and
the
First Derivative Test
Section 3.3 AP Calc
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Increasing/Decreasing Functions
A function f is increasing on an interval if for any
two numbers x1 and x2 in the interval, x1 < x2
implies f(x1)<f(x2).
A function f is decreasing on an interval if for
any two numbers x1 and x2 in the interval, x1 <
x2 implies f(x1)>f(x2).
Thm 3.5 Test for Increasing/Decreasing
Functions
Let f be a function continuous on a closed
interval [a,b] and differentiable on (a,b);
a) If f’(x)>0 for all x in (a,b), then f is increasing
on [a,b].
b) If f’(x)<0 for all x in (a,b), then f is decreasing
on [a,b].
c) If f’(x)=0 for all x in (a,b), then f is constant on
[a,b].
Guidelines for finding increasing/decreasing
intervals
1) Find critical numbers on interval, use them to
determine test intervals
2) Determine sign of f’(x) in each test interval
3) Use Thm 3.5 to determine if f is increasing,
decreasing, or constant
What is the derivative if the function is
increasing?
What is the derivative if the function is
decreasing?
Find open intervals where the function is
increasing and decreasing:
f ( x)  x 3  6 x 2  15
First Derivative Test
Let c be a critical number of f that is
continuous on the open interval I containing
c. If f is differentiable on I, except possibly at
c, then f(c) is classified as follows:
1) If f’(x) changes from negative to positive at c,
then f(c) is a relative minimum of f.
2) If f’(x) changes from positive to negative at c,
then f(c) is a relative maximum of f.
3) If f’(x) does not change sign at c, f(c) is neither
a min or a max.
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Use the First Derivative Test to find the relative
g ( x)  x  4
extrema of the function
2
3
Concavity and the
2nd Derivative Test
Consider the function on the interval (0,2 )
Find open intervals where the function is
increasing and decreasing and locate the
extrema. h( x)  sin x cos x
Concavity
Let f be a differentiable function on open
interval I. The graph of f is concave upward on
I if f’ is increasing on the interval, and concave
down on I if f’ is decreasing on the interval.
Section 3.4 AP Calc
Find the intervals of concavity:
3
2
y   x  3x  2
Thm 3.7 Test for Concavity
Let f be a function whose 2nd derivative exists on
an open interval I.
1) If f’’(x)>0 for all x in I, then the graph of f is
concave upward in I.
2) If f’’(x)<0 for all x in I, then the graph of f is
concave downward in I.
Note: 3rd case f’’(x)=0, f linear- concavity not
defined
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Find where the function is concave up and
concave down: g ( x)  x 3 ( x  4)
Thm 3.8 Points of Inflection
If (c,f(c)) is a point of inflection of the graph of f,
then either f’’(c)=0 or f’’ does not exist at x=c.
Note: At inflection points (where concavity
changes) the tangent line crosses the graph of
f.
Find the inflection points and discuss the
concavity:
f ( x) 
x 1
x
Domain x>0
Thm 3.9 2nd Derivative Test
Let f be a function such that f’(c)=0 and the 2nd
Derivative of f exists on I containing c.
1) If f’’(c)>0 then f(c) is a relative min
2) If f’’(c)<0 then f(c) is a relative max
Note: If f’’(c)=0, test fails, must use 1st Derivative
Test
Use the 2nd Derivative Test to find the extrema:
h( x) 
1
8
( x  2) 2 ( x  4) 2
Find the extrema:
f ( x)  x 3  9 x 2  27 x
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4x  3
Given f ( x)  2 x  1
Find the end behavior by completing the table:
Limits at Infinity
Section 3.5 AP Calc
x
-100 -10
-1
x 
each   0 there exists an M > 0 such that
1
10
100
f(x)
lim f ( x) 
lim f ( x ) 
x 
Definition of Limits at Infinity
Let L be a real number:
1) The statement lim f ( x)  L means for
0
x 
2) The statement lim f ( x)  Lmeans for
x  
each   0 there exists an N>0 such that
f ( x)  L   wherever x<N.
f ( x)  L   wherever x > M.
Definition of Horizontal Asymptote
The line y=L is a horizontal asymptote of the
graph of f if lim f ( x)  L or lim f ( x)  L
x  
x 
Properties (If f and g both have a limit that
exists):
lim f ( x)  g ( x)  lim f ( x)  lim g ( x)
x 
x 

x 

lim f ( x)  g ( x)  lim f ( x)  lim g ( x)
x 
x 
x 

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Find the limit:
lim 20 
x 
1

x
Thm 3.10 Limits at Infinity
If r is a positive rational number and c is a real
number, then
c
lim r  0
x  x
Furthermore, if xr is defined when x<0, then
c
lim
0
x   x r
Find the limit:
5x 2
lim 2
x  x  4
Find the limits:
lim
x 
2x  5
4x2  3
2x2  5
x  4 x 2  3
lim
2x3  5
x  4 x 2  3
lim
Guidelines for finding Limits at Infinity of
Rational Functions:
1. If degree numerator less than degree
denominator, limit 0.
2. If degree num. equal to degree den., limit is
ratio of leading coefficients.
3. If degree num. greater then degree den., limit
does not exist.
Find the limit:
lim
x  
 3x  1
x2  x
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Find the limit:
x  cos x
lim
x 
x
Definition Infinite Limits at Infinity
Let f be a func on defined on the interval (a,∞).
1. The statement
means for each
limthere
f ( x) is
 acorresponding
positive number M,
x 
number N>0 such that f(x)>M wherever x>N.
2. The statement
means for
each negative number M, there is a
lim f ( xN>0
)  
corresponding number
such that f(x)<M
x 
wherever x>N.
Find the limit:
5x 2
x  x  3
lim
A Summary of Curve Sketching
Section 3.6 AP Calc
Analyze:
f ( x) 
x2
x
Analyze:
Find the 1st and 2nd Derivative
Find the intercepts, asymptotes, critical
numbers, inflections points, determine
symmetry
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x
x2 1
Analyze:
f ( x) 
Analyze:
f ( x)  3 x 4  6 x 2  53
Analyze:
Analyze:
f ( x) 
x3
x2  4
f ( x)  2( x  2)  cot x,0  x  
A rancher has 200 feet of fencing to enclose two
adjacent rectangular corrals. What dimensions
maximize the area?
Optimization Problems
Section 3.7 AP Calc
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Guidelines for solving applied Min/Max problems
1. ID given quantities and quantities to be determined.
Make Sketch.
2. Write primary equation for quantity to be
max/minimized
3. Reduce primary equation to 1 single variable, using
other equations
4. Determine feasible domain, which values make sense
(+,-)
5. Determine max/min value by calculus techniques,
like the 2nd Derivative Test
Find the point on the graph of the function
closest to the given point
A rectangular page is to contain 36 in² of print.
Margins on each side of the page are 1 1/2’’.
Find the dimensions of the page such that the
least amount of paper is used.
Find the volume of the largest right circular
cylinder that can be inscribed in a sphere with
radius r.
2
f ( x)  x  1
(5,3)
Newton’s Method
-method for approximating zeros of a function
Newton’s Method
Section 3-8 AP Calc
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On interval [a,b], By Intermediate Value
Theorem, If f(a) and f(b) differ in sign (one is +
one is -) there must be a zero on the interval.
The tangent line of an approximate x value close
to c (the zero) happens to cross the x-axis at
about the same point.
Equation of tangent line:
y  f ( x1 )  f ' ( x1 )( x  x1 )
y  f ' ( x1 )( x  x1 )  f ( x1 )  0
x  x1 
Newton’s Method for Approx. Zero’s of a
Function
Let f(c)=0, where f is differentiable on an open
interval containing c. Then to approx. c, use
steps:
1) Make initial estimate “close” to c (graph)
2) Determine New Approx. xn 1  xn  f ( xn )
f ' ( xn )
f ( x1 )
 x2
f ' ( x1 )
New estimate
of zero
is x2
Each successful application of Newton’s Method
is called an iteration.
3) If xn  xn 1 is within the desired interval, let
xn+1 serve as final approx. Otherwise go back
to step 2.
Approximate the zero(s) using Newton’s Method
to find approx. that differ by less than 0.001.
Find the actual zero and compare the results.
When the approximations approach a limit, the
sequence x1, x2, x3, x4,… xn.. is said to
converge.
f ( x)  x 5  x  1
n xn
1
2
f(xn)
f’(xn)
f(xn)/ f’(xn) xn-[f(xn)/ f’(xn)]
Newton’s Method does not always converge
-involves dividing by f’(xn), so if f’(xn)=0 for any x
in the sequence the method fails
-example pg 224
3
4
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Apply Newton’s Method to approximate the xvalue of the intersection of f(x)=x² and
g(x)=cosx
Continue the process until two successive
approx. differ by less than 0.001
n xn
f(xn)
f’(xn)
Use Newton’s Method to obtain a general rule
for approximating the radical
xn a
f(xn)/ f’(xn) xn-[f(xn)/ f’(xn)]
1
2
Use the general rule to approximate
x  3 15
Differentials
Section 3.9 AP Calc
Linear Approximations
- function f, differentiable at c with eq. of
tangent line through point (c, f(c))
y  f (c)  f ' (c)( x  c)
y  f ' (c)( x  c)  f (c)
-by restricting values of x to be close to c, the
values of y can be approximations of the
actual values of f.
Find the tangent line at the point (2, 3/2):
f ( x) 
6
x2
Fill in the table:
x
1.9
1.99
2
2.01
2.1
f(x)
T(x)
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Definition of Differentials
Let y=f(x) represent a function that is
differentiable in an open interval containing x.
The differential of x (dx) is any nonzero real
number, the differential of y (dy) is dy=f’(x)dx
Evaluate and compare ∆y and dy when x=2 and
∆x=dx=0.01 :
f ( x)  2 x  1
differential used as approx. of ∆y
∆y dy and ∆y f’(x)dx


Error propagation:
(error of physical measuring devices)
x = measured value
x+∆x = exact value
∆x = error in measurment
Base of a triangle is 36cm, height is 50cm. The
possible error in measurement is 0.25cm. Use
differentials to approximate the propagated
error in computing the area of the triangle.
f ( x  x)  f ( x )  y
Differential Formulas:
u and v are differentiable functions of x
Constant Multiple: d[cu]=cdu
Sum/Difference: d[u±v]=du±dv
Product: d[uv]=udv+vdu
Quotient: d[u/v]=vdu-udv
v2
Approximate Function Values with Differentials
f(x+∆x) f(x)+dy = f(x)+f’(x)dx

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Use differentials to approximate
25.6
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