Overview Graphing More Accuratel First and second derivatives Critical points Etrema
6.1 Overall shape of the graph of a function Up with hope, down with dope, Increasing functions have positive slope Using our previous method, we can sketch f () = 3 3 2 :?
6.1 Overall shape of the graph of a function Sketch: f () = 2 + 1 Blow-up discontinuit: = 1 f () = (+2) (+1) 2 f () = 2 (+1) 3
6.1 Overall shape of the graph of a function Concavit Slopes are increasing f () > 0 concave up Slopes are decreasing f () < 0 concave down
6.1 Overall shape of the graph of a function Mnemonic + +
6.1 Overall shape of the graph of a function Concavit Where is the function concave up? Concave down?
6.1 Overall shape of the graph of a function Sketch: f () = 1 4 4 1 3 3 3 2 Consider behaviour near origin and far awa from origin First derivative: increasing and decreasing; find f () at special points Second derivative: concavit
6.1 Overall shape of the graph of a function Using info from first derivative: 2 3 16 3 63 4
6.1 Overall shape of the graph of a function Using info from second derivative: 1 19 3 1+ 19 3 2 3 16 3 63 4
6.1 Overall shape of the graph of a function Sketch graphs with the following properties, or eplain that none eist. concave up concave down increasing decreasing
6.1 Overall shape of the graph of a function Sign Changes in a Factored Function f () = ( 1) ( 2) 2 ( 3)
6.1 Overall shape of the graph of a function Sign Changes in a Factored Form Eample 1: f () = ( 3)( 1) 2 ( + 2) 3 ( + 5) 4 Where is f () positive? Where is it negative? 5 4 3 2 1 0 1 2 3 4 5
6.2 Special points on the graph of a function Special Points: Roots / zeroes critical point (i.e. f () = 0 or f () DNE) maima, minima CP that is not an etremum
6.2 Special points on the graph of a function Each of the following functions have a critical point at = 0. Match the derivatives with their graphs. (a) f () negative when < 0 f () negative when > 0 (c) f () negative when < 0 f () positive when > 0 (b) f () positive when < 0 f () positive when > 0 (d) f () positive when < 0 f () negative when > 0 I II III IV
6.2 Special points on the graph of a function Match the second derivatives with their graphs. (a) f () negative (c) f () negative when < 0 f () positive when > 0 (b) f () positive (d) f () positive when < 0 f () negative when > 0 I II III IV
6.2 Special points on the graph of a function Suppose = a is a critical point of the continuous function f (). That is, f (a) = 0 or f () does not eist at a. (a) f (a) changes from neg to pos I. inflection point (b) f (a) changes from pos to neg (c) f (a) = 0 (d) (e) f (a) changes from neg to pos f (a) changes from pos to neg II. III. IV. local ma local min not a local etrema (f) f (a) positive and f (a) = 0 (g) f (a) negative and f (a) = 0 V. could be local ma, local min, or neither
6.2 Special points on the graph of a function Tests The information from the previous questions forms a method for finding local etrema. First, find all critical points (where f () = 0 or f () doesn t eist). Local etrema can ONLY occur at critical points. Now, we have to classif those critical points: the could be local maes, local mins, or neither. Two options: Using the second derivative: If f () > 0, the CP is a local If f () < 0, the CP is a local If f () = 0, Using the first derivative: If f () does not change signs at the CP, then If f () changes from increasing to decreasing, the CP is a local If f () changes from decreasing to increasing, the CP is a local
6.2 Special points on the graph of a function f () = 1 4 4 2 3 + 9 2 2 + 1 1 Find all critical points of f () 2 Are the local maima, local minima, or neither? Challenge: sketch the function.
6.2 Special points on the graph of a function f () = 1 4 4 2 3 + 9 2 2 + 1 Sketch. 31 4 15 4 1 1 3 local min at (0, 1) also global min inflection points at (3, 31 4 ) and (1, 11 4 )
6.3 Sketching the graph of a function Global etrema over a closed interval Where do global etrema occur? a b
6.3 Sketching the graph of a function Cell Division A cell of age t hours has a probabilit 1 of dividing given b the function where a is the constant 9 6 3 2π. P(t) = at t 3 + 16, Frpm time t = 0 to time t = 10, when is the cell likeliest to divide? When is it least likel to divide? 1 Actuall, we re giving a probabilit distribution function... it s a little more complicated than what we said. Still, a higher value of P(t) means a cell age that is more reproduce-, so we just want to find the ma and min of P(t).
6.3 Sketching the graph of a function Sketch f () = ( 1)2 3 Include the following, if the eist: Roots (zeroes) Discontinuities Asmptotes Local and global etrema (maes and mins) Inflection points and concavit
6.3 Sketching the graph of a function Sketch f () = ( 1)2 3 f () = ( 1)( 3) 4 f () = 2(2 6+6) 5
6.3 Sketching the graph of a function Sketch f () = ( 1)2 3
6.3 Sketching the graph of a function Sketch: f () = 10 3 + 16 f () = 20(8 3 ) ( 3 +16) 2 f () = 602 ( 3 32) ( 3 +16) 3