DOC PREVIEW
UIUC MATH 241 - Lecture20714

This preview shows page 1-2-3-19-20-38-39-40 out of 40 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Partial Derivatives (§14.3)Tangent Planes and Differentiability (§14.4)Math 241, Spring 2014Jayadev S. AthreyaSpring 2014Jayadev S. Athreya Math 241, Spring 2014Approaching along linesWe need limits from all directions to exist.In 2-variables, lines through (0, 0) are of the formx = at, y = bt(for some a , b ∈ R). Forlim(x,y)→(p1,p2)f (x, y) = L,we need ALL (that is, for all a, b) of the following limits to exist:limt→0f (p1+ at, p2+ bt)and to ALL equal L.Jayadev S. Athreya Math 241, Spring 2014Approaching along linesIn n-variables, lines through (0, 0, . . . , 0) are of the formx1= a1t, x2= a2t, . . . , xn= ant(for some a1, . . . an∈ R). Forlimx→pf (x) = L,we need ALL (that is, for all a1, . . . , an) of the following limits toexist:limt→0f (p + t(a1, . . . , an))and to ALL equal L.Jayadev S. Athreya Math 241, Spring 2014Non-existenceIn particular, to show a limit DOESN’T exist, all you need toshow is that:One of these directional limits doesn’t exist.Two of these directional limits are not equal.Example: the h(x, y) we used above.Jayadev S. Athreya Math 241, Spring 2014Polar coordinatesIn 2 dimensions, we can use polar coordinates to computedirectional limits:Magnitude and Direction For each (x, y) 6= (0, 0) we can write(x, y) = (r cos θ, r sin θ),with r =px2+ y2, θ = tan−1yx∈ [0, 2π)Each ray Can be expressed associated to an angle θ. Theray associated to angle θ is{(r cos θ, r sin θ) : r > 0}.Jayadev S. Athreya Math 241, Spring 2014Limits and Polar CoordinatesLimit along a ray For a function f and a point p = (p1, p2) ∈ R2,and angle θ we can define the directional limitlimr→0f (p1+ r cos θ, p2+ r sin θ).For limit to exist All directional limits must exist and be equal.Warning This is NECESSARY, but NOT SUFFICIENT forlimit to exist. It’s a good way to check that limitDOESN’T exist.Example: h(x, y) =2xyx2+y2=2r2cos θ sin θr2= 2 cos θ sin θ. So thelimit as r → 0 depends on θ!Jayadev S. Athreya Math 241, Spring 2014Important ExampleCan have limit exist and be equal along all lines, but still notexist: ConsiderM(x, y) =x3yx6+ y2as (x, y) → (0, 0).Along axes x = 0 or y = 0 yields M(x, y) = 0, so limit alongaxes is 0.Along other lines x = at, y = bt(a, b 6= 0),limt→0M(at, bt) = limt→0a3bt4a6t6+ b2t2= limt→0a3bt2a6t4+ b2= 0Jayadev S. Athreya Math 241, Spring 2014Important ExampleCan have limit exist and be equal along all lines, but still notexist: ConsiderM(x, y) =x3yx6+ y2as (x, y) → (0, 0).Along cubic y = x3,M(x, x3) =x6x6+ x6= 1/2Jayadev S. Athreya Math 241, Spring 2014Limit RulesThe usual limit rules do work: suppose f , g : Rn→ R, andlimx→pf (x) = L, limx→pg(x) = M.Then:Sums limx→pf (x) + g(x) = L + MProducts limx→pf (x)g(x) = LMQuotients If M 6= 0, limx→pf (x)/g(x) = L/MJayadev S. Athreya Math 241, Spring 2014Continuityf : Rn→ R is continuous at p iflimx→pf (x) = f (p).Most functions we look at will be continuous, but we need to becareful.Jayadev S. Athreya Math 241, Spring 2014ExamplesNone of the below are continuous at 0:f (x, y) =sin(x2+y2)x2+y2g(x, y) =x2−y2x2+y2h(x, y ) =2xyx2+y2If we define f (0, 0) = 1, we make f continuous.Jayadev S. Athreya Math 241, Spring 2014ExamplesNone of the below are continuous at 0:f (x, y) =sin(x2+y2)x2+y2g(x, y) =x2−y2x2+y2h(x, y ) =2xyx2+y2If we define f (0, 0) = 1, we make f continuous.Jayadev S. Athreya Math 241, Spring 2014How functions changeWe will be interested in how functions change, and how toapproximate them by functions we can understand. In1-variable, the derivative played two roles:Rate of change f0(x) tells us how fast the function f waschanging at x.Linear approximation f0(x) is the slope of the tangent line at(x, f (x)), which gives us a good approximation tothe function.Jayadev S. Athreya Math 241, Spring 2014Partial derivativesGiven a function of 2 variables f (x, y), we can hold one of thevariables fixed and let the other one change, to get functions ofone-variable, and can compute derivatives of those functions:∂∂xThe partial derivative with respect to x at (a, b),denoted∂f∂x(a, b) is given by (if the limit exists)∂f∂x(a, b) = limh→0f (a + h, b) − f (a, b)hOther notation Sometimes we write (Dxf )(a, b)Jayadev S. Athreya Math 241, Spring 2014Partial derivativesGiven a function of 2 variables f (x, y), we can hold one of thevariables fixed and let the other one change, to get functions ofone-variable, and can compute derivatives of those functions:∂∂yThe partial derivative with respect to y at (a, b),denoted∂f∂h(a, b) is given by (if the limit exists)∂f∂y(a, b) = limh→0f (a, b + h) − f (a, b)hOther notation Sometimes we write (Dyf )(a, b)Jayadev S. Athreya Math 241, Spring 2014ExampleJayadev S. Athreya Math 241, Spring 2014More Examplesf (x, y) = ex+yf (x, y) = sin(xy)f (x, y) = ex2f (x, y) = x3y2Jayadev S. Athreya Math 241, Spring 2014More Examplesf (x, y) = ex+yf (x, y) = sin(xy)f (x, y) = ex2f (x, y) = x3y2Jayadev S. Athreya Math 241, Spring 2014More Examplesf (x, y) = ex+yf (x, y) = sin(xy)f (x, y) = ex2f (x, y) = x3y2Jayadev S. Athreya Math 241, Spring 2014More Examplesf (x, y) = ex+yf (x, y) = sin(xy)f (x, y) = ex2f (x, y) = x3y2Jayadev S. Athreya Math 241, Spring 2014Slopes of Tangent Lines                                                                                                   


View Full Document

UIUC MATH 241 - Lecture20714

Documents in this Course
Notes

Notes

9 pages

16_05

16_05

29 pages

16.6

16.6

43 pages

16_07

16_07

34 pages

16_08

16_08

12 pages

16_09

16_09

13 pages

exam1

exam1

10 pages

exam2

exam2

7 pages

exam3

exam3

9 pages

15_03

15_03

15 pages

15_04

15_04

13 pages

15_04 (1)

15_04 (1)

13 pages

15_05

15_05

31 pages

15_10

15_10

27 pages

15_07

15_07

25 pages

15_08

15_08

12 pages

15_09

15_09

24 pages

15_10_B

15_10_B

8 pages

16_04

16_04

17 pages

14_01

14_01

28 pages

12_06

12_06

12 pages

12_05

12_05

19 pages

12_04

12_04

26 pages

Lecture1

Lecture1

31 pages

Lecture 9

Lecture 9

41 pages

Lecture 8

Lecture 8

35 pages

Lecture 7

Lecture 7

40 pages

Lecture 6

Lecture 6

49 pages

Lecture 5

Lecture 5

26 pages

Lecture 4

Lecture 4

43 pages

Lecture 3

Lecture 3

29 pages

Lecture 2

Lecture 2

17 pages

m2-1

m2-1

6 pages

-

-

5 pages

Load more
Download Lecture20714
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Lecture20714 and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Lecture20714 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?