1 / 30

4.1 Extreme Values for a function

4.1 Extreme Values for a function. Absolute Extreme Values There is an absolute maximum value at x = c iff f(c)  f( x ) for all x in the entire domain. There is an absolute minimum value at x = c iff f(c)  f( x ) for all x in the entire domain .

laken
Download Presentation

4.1 Extreme Values for a function

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. 4.1 Extreme Values for a function • Absolute Extreme Values • There is an absolute maximum value at x = c iff • f(c)  f(x) for all x in the entire domain. • There is an absolute minimum value at x = c iff • f(c) f(x) for all x in the entire domain. • Relative Extreme Values • There is a relative maximum value at x = c iff • f(c)  f(x) for all x in some open interval containing c. • There is a relative minimum value at x = c iff • f(c) f(x) for all x in some open interval containing c.

  2. Maxima and Minima

  3. Locations of Extreme values If f(x) has a maximum or minimum value at x = c it must occur at one of the following locations: • Where f (c) =0 • Where f (c ) is undefined • At an endpoint of a closed interval **Values in the domain of f where f (c) is zero or is undefined are called critical values of the function.***

  4. Maxima and Minima on closed interval for continuous function must exist.

  5. Maximum and minimums

  6. Local Max A curve with a local maximum value. The slope at c, is zero.

  7. 4.3 First Derivative test for Increasing and Decreasing functions • If a function is continuous on [a, b] and differentiable on (a,b) • If f  > 0 at each point of (a,b) then f increases on [a,b]. • If f  < 0 at each point of (a,b) then f decreases on [a,b].

  8. Find the critical points and identify intervals on which f is increasing and decreasing.

  9. First derivative test for Local Extrema At a critical point x = c • f has a local minimum iff changes from negative to positive at c. • f has a local maximum iff changes from positive to negative at c. • There is no extreme value if the sign of f  does not change. Could be a horizontal tangent without direction change.

  10. Concavity Figure 3.24: The graph of f (x) = x3 is concave down on (–, 0) and concave up on (0, ).

  11. Second derivative test for Concavity A graph is concave up on any interval where The second derivative is positive. A graph is concave down on any interval where The second derivative is negative. A point of inflection for a function Occurs where the concavity changes

  12. Second derivative test for extreme values - - - + + + If there is a critical value at x = c and conclusion A local max at x = c A local min at x = c inconclusive

  13. Section 4.3 · Figures 11 Graph of the curve Section 1 / Figure 1 A © 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.

  14. Finding intervals of concave up and concave down and Inflection points The graph of f (x) = x4 – 4x3 + 10.

  15. What derivatives tell us about the shape of a graph

  16. 4. 4 Limits to Infinity (End behavior) Figure 1.42: The blue graph of f (x) = x + e–x looks like the graph of g(x) = x (black) to the right of the y-axis and like the graph of h(x) = e–x (red) to the left of the y-axis. (Example 1) What happens to the value of the function when the value of x increases without bound? What happens to the value of the function when the value of x decreases without bound?

  17. Basic limits to infinity

  18. Figure 1.27: The function in Example 3. Divide each term by highest power of x in the denominator and calculate limits As x gets larger and larger, the function gets closer and closer to 5/3.

  19. Figure 1.27: The function in Example 3. Divide each term by highest power of x in the denominator and calculate limits As x gets larger and larger, the function gets closer and closer to 0.

  20. Figure 1.27: The function in Example 3. Divide each term by highest power of x in the denominator and calculate limits As x gets larger and larger, the function decreases without bound.

  21. Figure 1.27: The function in Example 3. As x gets larger and larger, the function gets closer and closer to 5/3. When the limit to infinity exists, at y = L we can say that the line y = L is a horizontal asymptote.

  22. Horizontal Asymptote Figure 1.27: The function in Example 3.

  23. Limits that are infinite (y increases without bound) An infinite limit will exist as x approaches a finite value when direct substituion produces If an infinite limit occurs at x = c we have a vertical asymptote with the equation x = c.

  24. Slant Asymptote As x gets larger and larger, the graph gets closer and closer to the line Figure 1.29: The function in Example 5(a). As x gets smaller and smaller, the graph gets closer to the same line. You can use long division to rewrite the given function. There is a vertical asymptote at x = - 4 / 7

  25. 4.6Different scales Graphs of the polynomial Section 1 / Figure 1 © 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.

  26. Graphs of derivatives Derivatives of the polynomial Section 1 / Figure 1 © 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.

  27. Section 1 / Figure 1

  28. Section 1 / Figure 1

  29. Section 4.6 · Figures 19, 20 The family of functions Section 1 / Figure 1

  30. The family of functions

More Related