Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Mathematics of radio engineering wikipedia , lookup
John Wallis wikipedia , lookup
Line (geometry) wikipedia , lookup
Fundamental theorem of calculus wikipedia , lookup
Analytical mechanics wikipedia , lookup
Cartesian coordinate system wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Blaise Pascal 1623 – 1662 Blaise Pascal was a very influential French mathematician and philosopher who contributed to many areas of mathematics. He worked on conic sections and projective geometry and in correspondence with Fermat he laid the foundations for the theory of probability. The First Derivative Test Suppose that c is a critical number of a continuous function f ( x). a) If f ( x) changes from positive to negative at c, then f ( x) has a local maximum at c. b) If f ( x) changes from negative to positive at c, then f ( x) has a local minimum at c. c) If f ( x) does not change sign at c, then f ( x) has no local maximum or minimum at c. Example: Consider y 3 x 4 4 x 3 12 x 2 5. a) Find the critical values for y. b) Find intervals of increasing and decreasing. c) Identify the coordinates of all the extrema, determine if they are local or absolute. 0 0 y 12 x3 12 x 2 24 x 1 Let y 0 0 y x 2 0 20 12 x 3 12 x 2 24 x 0 12 x( x 2 x 2) 0 12 x( x 2)( x 1) 0 f (x ) 2 1 0 1 20 Critical numbers: x 0, x 2, x 1 x 2 3 Example: Consider y 3 x 4 4 x 3 12 x 2 5. a) Find the critical values for y. b) Find intervals of increasing and decreasing. c) Identify the coordinates of all the extrema, determine if they are local or absolute. 0 0 y 12 x3 12 x 2 24 x Let y 0 12 x 3 12 x 2 24 x 0 12 x( x 2 x 2) 0 12 x( x 2)( x 1) 0 Critical numbers: x 0, x 2, x 1 1 0 y x 0 2 By IDT, Increasing: x 1, 0 Decreasing: 2, x , 1 0, 2 y 3x 4 4 x3 12 x 2 5 20 Let x 1 3(1)4 4(1)3 12(1) 2 5 0 f (x ) 2 1 0 1 2 3 Let x 0 3(0)4 4(0)3 12(0)2 5 5 Let x 2 20 x 3(2)4 4(2)3 12(2)2 5 27 By FDT, the coordinates of the relative maximum point is (0,5), and the coordinates of the relative minimum points are ( 1,0) and (2, 27). Because of the end-behavior of the function, (2,27) is an absolute minimum point. Example: Curvature of a Function - Concavity y 20 Concave up f (x ) 2 1 0 1 2 3 x Concave Down 20 Lets take the second derivative of y. y 36 x 2 24 x 24. Let x 0, then y (0) 24. x Let x 1, then y ( 1) 36. Let x 2, then y (2) 72. Example: Analyze y and y for the function y x3 3x 2 4. y 3 x 2 6 x Set y 0 Possible extrema at x 0, 2. 0 3x 6 x 0 x 0, 2 0 y x 2 3x x 2 0 0 2 By IDT, Increasing: x , 0 2, Decreasing: x 0, 2 By FDT, the coordinates of a relative maximum point are (0, 4) and the coordinates of a relative minimum point are (2, 0). Because of end-behavior there are no absolute extrema. Now lets examine concavity, and look for inflection points by setting the second derivative equal to zero. y 6 x 6 Possible inflection point at x 1. 0 6x 6 By CT, y x 6 6x 1 x 0 1 y 0 6 0 6 6 negative y 2 6 2 6 6 positive Concave Upward: x 1, Concave Downward: x ,1 By DIP, there is an inflection point at (1, 2) since the second derivative changes from negative to positive. Given a real number c in the domain of a continuous function y = f (x), First derivative: y(c) is positive Curve is increasing. IDT y(c) is negative Curve is decreasing. y(c) is zero or Possible local maximum or minimum points. undefined FDT Second derivative: y(c) is positive Curve is concave up. y(c) is negative Curve is concave down. y(c) is zero or Possible inflection point (where concavity changes). undefined CT DIP Example: Find the extreme points for y x3 3x 2 4. y 3 x 2 6 x y 3x( x 2) Possible extrema at x = 0, 2. y(0) 0 and y(2) 0. We can use the Second Derivative Test: y 6 x 6 y 0 6 0 6 6 y 2 6 2 6 6 Because the second derivative at x = 0 is negative, the graph is concave down and therefore (0,4) is a local maximum point. Because the second derivative at x = 2 is positive, the graph is concave up and therefore (2,0) is a local minimum point. Examples Examples