Extreme values of a function & applications of derivative


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Extreme values are essential when working with calculus .This presentation will provide the basic guide line to tackle them.

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Extreme values of a function & applications of derivative

  1. 1. By Nofal Umair
  2. 2. Extreme values are termed “extrema” Absolute Extrema: the point in question represents either the maximum or minimum value of the function over the domain. Relative Extrema: the point in question represents either the maximum or minimum value of the function on a specified segment of the domain (or in the “hood”). Let’s take a look at an example and consider several points on a function.
  3. 3. Take a look at this diagram. Notice the difference between local and absolute extrema.
  4. 4. Although the concepts are easy to understand, be attuned to the subtlety of the questions asked … Example 1: on the closed interval find any relative extrema for y = sin x and y = cos x Example 2: Same as above but on the open interval , 2 2       , 2 2       
  5. 5. The Extreme Value Theorem: If a function f is continuous on a closed interval [a, b], then f has both a maximum and a minimum on the interval. Question: So, by looking at a graph of a function, how can you find its extrema? What’s true about the curve at its max and min? What’s the one analytical tool we’ve been studying all year?
  6. 6. To fully answer the question, we need to define some terms and give you one theorem. Critical Points – a point on the interior of the domain of a function f at which ○ f’ = 0 or ○ f’ does not exist (is undefined). Theorem: if a function f has a local maximum and/or minimum at some interior point “c” of its domain, and if f’(c) exists, then f’(c) = 0 SO, HOW DO YOU FIND EXTREMA??? 1. Find all critical points (values) 2. Check the endpoints of the specified domain
  7. 7. A little Practice Find the extrema of on the interval [-1, 2]. 1. Find the critical numbers in f 2. Evaluate f at each critical number 3. Evaluate f at each endpoint 4. Compare. Least number is minimum, greatest number is maximum. Answers are: 1. Critical numbers at x = 0 and x = 1 2. f(0) = 0 and f(1) = -1 3. Left endpoint has height of 7, right endpoint has height of 16 4. Min = -1, max = 16 4 3 ( ) 3 4f x x x 
  8. 8. Why Need Of Derivative??????  Derivatives have arisen from the need to manage the risk arising from movements in markets beyond our control ,which may severely impact the revenues and costs of the firm.  Derivatives can be used in a number of ways in everyday life, especially with optimization. Example: The growth rate of any company , the profit or loss it made, etc
  9. 9. CURVES  You can easily graph any function by knowing three things.  1) ZEROS AND UNDEFINED SPOTS  2) MAXIMUM AND MINIMUM POINTS  3) CONCAVITYAND INFLECTION POINTS.
  10. 10. What the First Derivative Tells Us:  Suppose that a function f has a derivative at every point x of an interval I. Then: increases on I if ( ) 0 for all in I.f f x x  decreases on I if ( ) 0 for all in I.f f x x 
  11. 11. What This Means:  In geometric terms, the first derivative tells us that differentiable functions increase on intervals where their graphs have positive slopes and decrease on intervals where their graphs have negative slopes.  WHAT HAPPENS IF THE FIRST DERIVATIVE IS ZERO?
  12. 12. When The First Derivative is Zero  A derivative has the intermediate value property on any interval on which it is defined.  It will take on the value zero when it changes signs over that interval.  Thus, when the derivative changes signs on an interval, the graph of f(x) must have a horizontal tangent.
  13. 13. Relative Maxima and Minima  If the derivative changes sign, there may be a local max or min, as shown here.  More on this later.
  14. 14. Concavity  Concave down—”spills water”  Concave up—”holds water”  The graph of is concave down on any interval where and concave up on any interval where ( )y f x 0y  0y 
  15. 15. Points of Inflection  A point on the curve where the concavity changes is called a point of inflection.  If the second derivative is zero for some x, we may be able to find a point of inflection.  It IS possible for the second derivative to be zero at a point that is NOT a point of inflection.  A point of inflection may occur where the second derivative fails to exist.
  16. 16. Inflection Points  You can tell where the function changes concavity by finding the inflection points.  Evaluate the function at those values where the second derivative is zero;  Take a look at the graph of the original function: 4 2( ) 2f x x x 
  17. 17. The Graph
  18. 18. An Interesting Example Suppose that the yield, r, in the % of students in a one hour exam is given by: r = 300t (1−t). Where 0 < t < 1 is the time in hours. 1. At what moments is the yield zero? 2. At what moments does the yield increase or decrease? 3. When is the biggest yield obtained and which is?
  19. 19. ERRORS AND APPROXIMATIONS  We can use differentials to calculate small changes in the dependent variable of a function corresponding to small changes in the independent variable. e.g