Class Notes (839,242)
Canada (511,223)
Mathematics (870)
MATA31H3 (103)
Lecture 10

MATA31H3 Lecture 10: W10 Complete Lectures

12 Pages
63 Views

Department
Mathematics
Course Code
MATA31H3
Professor
Natalie Rose

This preview shows pages 1,2 and half of page 3. Sign up to view the full 12 pages of the document.
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
More Less
Unlock Document

Only pages 1,2 and half of page 3 are available for preview. Some parts have been intentionally blurred.

Unlock Document
You're Reading a Preview

Unlock to view full version

Unlock Document

Log In


OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit