1 / 25

Calculus

Math Review with Matlab:. Calculus. Limits. S. Awad, Ph.D. M. Corless, M.S.E.E. D. Cinpinski E.C.E. Department University of Michigan-Dearborn. Limits. Why Use Limits? Determinate and Indeterminate Functions General Limits Limit Command Indeterminate Example Determinate Example

vaponte
Download Presentation

Calculus

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Math Review with Matlab: Calculus Limits S. Awad, Ph.D. M. Corless, M.S.E.E. D. Cinpinski E.C.E. Department University of Michigan-Dearborn

  2. Limits • Why Use Limits? • Determinate and Indeterminate Functions • General Limits • Limit Command • Indeterminate Example • Determinate Example • Discontinuous Example • Restricted Constant Example

  3. Why Use Limits? • Many functions are not defined at certain points • However values of the function arbitrarily close to these points may be well defined • Limits are often used to determine the value of a function around (close to) an undefined point • Limits can also be used to approximate values around points of discontinuity in a function

  4. Determinate and Indeterminate • A limit is said to be Determinateif the function is defined at the limit point • The limit is still considered determinate if the result f(a)=±¥ • A limit is said to be Indeterminate if the function is NOT defined at the limit point • Two common examples of indeterminatelimits include:

  5. General Limits • In general, the limit at an indeterminate limit point can be evaluated as: • Where D is an infinitesimally small number and: • As a±D becomes arbitrarily close to a, then f(a±D) becomes arbitrarily close to the limit l

  6. Right and Left Limits • Consider D is an infinitesimally small positive number The Limit from the Right is defined as: The Limit from the Left is defined as: • If the terms right or left are not specified when evaluating a limit, it is assumed that: • If they are not equal then the right or left limit must be specified before evaluating

  7. Limit Command limit(f,x,a) takes the limit of the symbolic expression f as x approaches a limit(f,a) uses findsym(f) as the independent variable and takes the limit as it approaches a limit(f) uses findsym(f) as the independent variable and takes the limit as it approaches 0 limit(f,x,a,'left') takes the limit from the left limit(f,x,a,'right') takes the limit from the right

  8. Indeterminate Example • Analyze the function: • f(x) is obtained by dividing the two continuously definedfunctions plotted to the right • However f(x) is undefined at x = 0 since:

  9. Defined Defined Undefined Defined Limit Around 0 • Even though f(x) is undefined at x=0 • f(x) is defined as x approaches 0from the left • f(x) is defined as x approaches 0from the right

  10. Evaluate Limit Using Delta • The limit can be determined by setting delta to a small value and interpreting the results » delta=sym('1/1000'); » lim_left =eval(sym('sin(-delta)/(-delta)')); » lim_right=eval(sym('sin(delta)/delta')); » vpa(lim_left,10) ans = .9999998333 » vpa(lim_right,10) ans = .9999998333

  11. Indeterminate Verify Using Limit Command » f_zero=sin(0)/0 f_zero = NaN • Use the limit command to verify the previous results » syms x » f=sin(x)/x; » limit_left=limit(f,x,0,'left') limit_left = 1 » limit_right=limit(f,x,0,'right') limit_right = 1 » limit=limit(f) limit = 1 • The limit from the left is equal to the limit from the right

  12. Plot of sin(x) / x • The ezplot command can be used to plot the function and graphically verify the result » ezplot(f,[-15,15]) » grid on

  13. Determinate Limit Example • Given the determinate symbolic function f: • Use Matlab to find the limit of f as x approaches 0 • Use Matlab to find the limit of f as x approaches a

  14. Limit as x approaches 0 • Since x is the Matlab default independent variable and the default limit point is 0 the limit can simply be found using: » syms a x b » f=sin(a*x+b); » lim1 = limit(f) lim2 = sin(b)

  15. Limit as x approaches a » lim2=limit(f1,a) lim2 = sin(a^2+b) • Since x is the Matlab default independent variable and the default limit point is 0 the limit can simply be found using: • Of course the independent variable can be explicitly specified to return the same results » lim2=limit(f1,x,a) lim2 = sin(a^2+b)

  16. Discontinuous Example • Given the function f(x): • Use Matlab to determine the following limits of f as x approaches -1, 0, and +1 • Graphically verify the results by using Matlab to plot f(x) over the region of interest

  17. Limits at Defined Points • Use Matlab to evaluate the limits at x=-1 and x=0 » syms f x » f=1/(x-1); » f_neg1=limit(f,x,-1) f_neg1 = -1/2 » f_zero=limit(f,x,0) f_zero = -1 • Matlab returns a numerical result for each, implying that the limits from the right andleft are the same

  18. Limit at Discontinuity • Finding the limit as x approaches 1 returns NaN (Not a Number) » f_pos1=limit(f,x,1) f_pos1 = NaN Discontinuity at x=1 • This implies that the function is discontinuous at x=1 and that the limit from the left does not equal the limit from the right

  19. Left and Right Limits • Must explicitly find the limit from the left and from the right » f_pos1_lft=limit(f,x,1,'left') f_pos1_lft = -inf » f_pos1_rght=limit(f,x,1,'right') f_pos1_rght = inf

  20. Verify Results • Use Matlab to plot f(x) and verify the previous results » ezplot(f) » axis([-3 3 -5 5]) » grid on

  21. Restriction on Constants • Sometimes it is useful to put restrictions on symbolic constants when evaluating the limits on symbolic expressions x=sym('x','real') restricts x to be real x=sym('x','positive') restricts x to be real and positive x=sym('x','unreal') puts no restrictions on x and can be used to undo any previous restrictions

  22. Restricted Constant Example • Given the continuous function of: • Determine the limit of f(x) asx approaches infinity for the following cases: • Where k2 is any real number • Where k2is a positive real number • We will not discuss the case where k2 is considered complex

  23. Limit for Real k2 • If k2 is assumed to be any real number, the limit is dependent upon the sign of k2 » syms k1 k2 x » f=k1*atan(k2*x); » k2=sym('k2','real'); » lim_k2real = limit(f,x,inf) lim_k2real = 1/2*signum(k2)*pi*k1

  24. Limit for Positive k2 • The limit as x goes to infinity when k2 is known to be a positive real number returns a simpler result » k2=sym('k2','positive'); » lim_k2pos = limit(f,x,inf) lim_k2pos = 1/2*pi*k1 • The result is consistent with the more general result • This may be useful when simplifying symbolic expressions

  25. Summary • General use of limits • Determinate and indeterminate functions • Limits from the right and left • Evaluating limits of symbolic expressions • Restrictions on symbolic constants

More Related