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MATA31H3
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Natalie Rose
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Lecture 10

Department

Mathematics

Course Code

MATA31H3

Professor

Natalie Rose

Description

MATA 31
Week 10: NOTES
Textbook: Taalman, Kohl
CALcULUS. Single variable.
APPLICATIONS OF DIFFERENTIATION
10.1 Maximum and Minimum Values
Local (Relative) Extrema.
Let
be a function which is defined on open interval (a,b)
DEFINITION 1
A function for has a local minimum (or relative minimum) at point x-cif there exists
some o> 0
such that fac s fox for all xe (c-o.c +o)
A function fx has a local maximum (or relative maximum) at point r cif there
exists some such that f(c) fi
for all xE (c-a,c o)
How to find points of local extrema
function has local extrema at points where it has either peak orvalley. Such points are
called critical. Points x-c on the following graphs are critical.
valley f(c)
Peak f(c) f (c)-0
f (e)- DNE
f'( DNE
DEFINITION 2
A critical number (point of function f(x) is the point cin the domain of f(x)
such that either f (c)- 0 or f (c) does not exist.
Example 3. Find
critical points of f
(x-1)3
Fermat's Theorem for Local Extrema.
point of
If f(x) has a local extremum at an interior point c and f
exists, then f (c) 0
Proof
ou definitipu docal Max
al mar
CRED
The inverse is not true!!!! f (c) 0 is the necessary condition for f
to attain
extremum, but not a sufficient condition.
Point where f
(c) 0 is only candidate for local extrema.
Example 1.
Example 2.
no extend a
MATA 31 Week 10: NOTES Textbook: Taalman, Kohl CALcULUS. Single variable. APPLICATIONS OF DIFFERENTIATION 10.1 Maximum and Minimum Values Local (Relative) Extrema. Let be a function which is defined on open interval (a,b) DEFINITION 1 A function for has a local minimum (or relative minimum) at point x-cif there exists some o> 0 such that fac s fox for all xe (c-o.c +o) A function fx has a local maximum (or relative maximum) at point r cif there exists some such that f(c) fi for all xE (c-a,c o) How to find points of local extrema function has local extrema at points where it has either peak orvalley. Such points are called critical. Points x-c on the following graphs are critical. valley f(c) Peak f(c) f (c)-0 f (e)- DNE f'( DNE DEFINITION 2 A critical number (point of function f(x) is the point cin the domain of f(x) such that either f (c)- 0 or f (c) does not exist. Example 3. Find critical points of f (x-1)3 Fermat's Theorem for Local Extrema. point of If f(x) has a local extremum at an interior point c and f exists, then f (c) 0 Proof ou definitipu docal Max al mar CRED The inverse is not true!!!! f (c) 0 is the necessary condition for f to attain extremum, but not a sufficient condition. Point where f (c) 0 is only candidate for local extrema. Example 1. Example 2. no extend aGlobal (Absolute) Extrema
DEFINITION 3
A function f(r) has a global maximum (or absolute maximum) at point x -cif
f(c) for all x in he domain of f
A function f(r) has a global minimum (or absolute minimum) at pointx-cif
f(c)2 for all x in the domain of f(x)
The Extreme Value Theorem
If function (r) is continuous on a closed interval [a,bl, the
f
an absolute
maximum value and an absolute minimum value at some numbers in [a,bl
The EVT doesn't work for discontinuous functions.
How ro find absolute maximum and absolute minimum values of function f(x)
on closed interval Ila,b]?
The closed interval method
1. Find the values of fox) at the critical points of fox) in (a,b)
2. Find the values f(a) and f(b) at endpoints of the interval.
3. The largest value from step land 2 is the absolute maximum value
The smallest value from step 1 and 2 is the absolute minimum value
Example 5. Find absolute minimum and absolute maximum of the function
1 Cr. pts.
point
not a Gr Pt
blonel naar 55 fo) at Point 6
Global
us at point 3
10.2 Rolle's Theorem.
Suppose a differentiable function fox) has two points with the same ordinate. From thc
illustrations we can conclude that such function will always have a local extremum with
horizontal tangent between two points with the same ordinate
C
a d b.
a C
Rolle's Theorem
h that f
Global (Absolute) Extrema DEFINITION 3 A function f(r) has a global maximum (or absolute maximum) at point x -cif f(c) for all x in he domain of f A function f(r) has a global minimum (or absolute minimum) at pointx-cif f(c)2 for all x in the domain of f(x) The Extreme Value Theorem If function (r) is continuous on a closed interval [a,bl, the f an absolute maximum value and an absolute minimum value at some numbers in [a,bl The EVT doesn't work for discontinuous functions. How ro find absolute maximum and absolute minimum values of function f(x) on closed interval Ila,b]? The closed interval method 1. Find the values of fox) at the critical points of fox) in (a,b) 2. Find the values f(a) and f(b) at endpoints of the interval. 3. The largest value from step land 2 is the absolute maximum value The smallest value from step 1 and 2 is the absolute minimum value Example 5. Find absolute minimum and absolute maximum of the function 1 Cr. pts. point not a Gr Pt blonel naar 55 fo) at Point 6 Global us at point 3 10.2 Rolle's Theorem. Suppose a differentiable function fox) has two points with the same ordinate. From thc illustrations we can conclude that such function will always have a local extremum with horizontal tangent between two points with the same ordinate C a d b. a C Rolle's Theorem h that fExample 5. Find absolute minimum and absolute maximum of the function
(x) x 27.
+1 at D-1,6)
a cr pot
f 3) 53
tie) 55
attains it max 55 at 6
10.2 Rolle's Theorem.
Suppose a differentiable function f(x) has two points with the same ordinate. From the
de th
h fun
always ha
horizontal tangent between two points with the same ordinate
afte a (a
Rolle's Theorem
differentiable on open interval (a,b) and continuous on closed interval with
unction
f f(b) the

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