Lesson 56 First Derivative T est h SL

Lesson 56 First Derivative T est h SL 1 t a IB M wski o Sant 21 0 2 / 2/20 1

(A) Terms 0 Given a function, f(x), that is defined on a given interval and let c be a number in the domain 0 f(c) is the ABSOLUTE or GLOBAL maximum of f(x) on the interval if f(c) > f(x) for every x in the interval 0 Now, sketch an example of what has just been described. 2/20/2021 IB Math SL 1 - Santowski 2

(A) Terms 0 Given a function, f(x), that is defined on a given interval and let c be a number in the domain 0 f(c) is the ABSOLUTE or GLOBAL minimum of f(x) on the interval if f(c) < f(x) for every x in the interval 0 Now, sketch an example of what has just been described. 2/20/2021 IB Math SL 1 - Santowski 3

(A) Important Terms Recall the following terms as they were presented in a previous lesson: turning point: points where the direction of the function changes maximum: the highest point on a function minimum: the lowest point on a function local vs absolute: a max can be a highest point in the entire domain (absolute) or only over a specified region within the domain (local). Likewise for a minimum. 0 increase: the part of the domain (the interval) where the function values are getting larger as the independent variable gets higher; if f(x 1) < f(x 2) when x 1 < x 2; the graph of the function is going up to the right (or down to the left) 0 decrease: the part of the domain (the interval) where the function values are getting smaller as the independent variable gets higher; if f(x 1) > f(x 2) when x 1 < x 2; the graph of the function is going up to the left (or down to the right) 0 "end behaviour": describing the function values (or appearance of the graph) as x values getting infinitely large positively or infinitely large negatively or approaching an asymptote 0 0 0 2/20/2021 IB Math SL 1 - Santowski 4

(B) Review – Graphic Analysis of a Function 0 We have seen functions analyzed given the criteria intervals of increase, intervals of decrease, critical points (AKA turning points or maximum or minimum points) 0 We have also seen graphically how the derivative function communicates the same criteria about a function these points are summarized on the next slide: 2/20/2021 IB Math SL 1 - Santowski 5

(B) Review – Graphic Analysis of a Function 0 f(x) has a max. at x = -3. 1 and f `(x) has an x-intercept at x = -3. 1 0 f(x) has a min. at x = -0. 2 and f `(x) has a root at – 0. 2 0 f(x) increases on (- , -3. 1) & (-0. 2, ) and on the same intervals, f `(x) has positive values 0 f(x) decreases on (-3. 1, -0. 2) and on the same interval, f `(x) has negative values 2/20/2021 IB Math SL 1 - Santowski 6

(C) Analysis of Functions Using Derivatives – A Summary 0 If f(x) increases, then f `(x) > 0 0 If f(x) decreases, then f `(x) < 0 0 At a max/min point, f `(x) = 0 0 We can also state the converse of 2 of these statements: 0 If f `(x) > 0, then f(x) is increasing 0 If f `(x) < 0, then f(x) is decreasing 0 The converse of the third statement is NOT true if f `(x) = 0, then the function may NOT necessarily have a max/min so for now, we will call any point that gives f `(x) = 0 (i. e. produces a horizontal tangent line) a CRITICAL POINTS or EXTREME POINTS 2/20/2021 IB Math SL 1 - Santowski 7

(D) First Derivative Test 0 So if f `(x) = 0, how do we decide if the point at (x, f(x)) is a maximum, minimum, or neither (especially if we have no graph? ) 0 Since we have done some graphic analysis with functions and their derivatives, in one sense we already now the answer: see next slide 2/20/2021 IB Math SL 1 - Santowski 8

(E) First Derivative Test - Graphically 0 At the max (x = -3. 1), the fcn changes from being an increasing fcn to a decreasing fcn the derivative changes from positive values to negative values 0 At a the min (x = -0. 2), the fcn changes from decreasing to increasing the derivative changes from negative to positive 2/20/2021 IB Math SL 1 - Santowski 9

(F) First Derivative Test - Algebraically 0 At a maximum, the fcn changes from being an increasing fcn to a decreasing fcn the derivative changes from positive values to negative values 0 At the minimum, the fcn changes from decreasing to increasing the derivative changes from negative to positive 0 So to state the converses: 0 If f `(x) = 0 and f the sign of if `(x) changes from positive to negative, then the critical point on f(x) is a maximum point 0 If f `(x) = 0 and f the sign of if `(x) changes from negative to positive, then the critical point on f(x) is a minimum point 0 So therefore, if the sign on f `(x) does not change at the critical point, then the critical point is neither a maximum or minimum we will call these points STATIONARY POINTS 2/20/2021 IB Math SL 1 - Santowski 10

(G) First Derivative Test – Example #1 0 Find the local max/min values of y = x 3 - 3 x + 1 (Show to use inequalities to analyze for the sign change) 0 0 0 f `(x) = 3 x 2 – 3 f `(x) = 0 for the critical values 0 = 3 x 2 – 3 0 = 3(x 2 – 1) 0 = 3(x – 1)(x + 1) x = 1 or x = -1 0 Now, what happens on the function, at x = + 1? let’s set up a chart to se what happens with the signs on the derivative so that we can determine the sign on the derivative so that we can classify the critical points 2/20/2021 IB Math SL 1 - Santowski 11

(G) First Derivative Test – Example #1 Factor 3 (x-1) (x+1) f `(x) f(x) (-∞, -1) + - - + inc (-1, 1) + - dec (1, ∞) + + inc Interval 2/20/2021 IB Math SL 1 - Santowski 12

(G) First Derivative Test – Example #1 0 Since the derivative changes signs from +ve to –ve, the critical point at x = -1 is a maximum (the original function changing from being an increasing fcn to now being a decreasing fcn) 0 Since the derivative changes signs from -ve to +ve, the critical point at x = 1 is a minimum (the original function changing from being a decreasing fcn to now being an increasing fcn) 0 Then, going one step further, we can say that f(-1) = 3 gives us a maximum value of 3 and then f(1) = -1 gives us a minimum value of -1 0 And going another step, we can test the end behaviour of f(x): 0 lim x -∞ f(x) = -∞ 0 lim x ∞ f(x) = +∞ 0 Therefore, the point (-1, 3) represents a local maximum (as the fcn rises to infinity “at the end”) and the point (1, -1) represents a local minimum (as the fcn drops to negative infinity “at the negative end”) 2/20/2021 IB Math SL 1 - Santowski 13

(G) First Derivative Test – Example #1 – Graphic Summary 2/20/2021 IB Math SL 1 - Santowski 14

(H) In Class Examples 0 Ex 2. Find the local max/min values of g(x) = x 4 - 4 x 3 - 8 x 2 - 1 0 Ex 3. Find the absolute minimum value of f(x) = x + 1/x for x > 0 2/20/2021 IB Math SL 1 - Santowski 15

(I) More Examples 0 Determine the absolute extrema for the following function and interval: 0 g(x) = 2 x 3 + 3 x 2 – 12 x + 4 on [-4, 2] 0 f(x) = x 4 – 4 x 3 + 4 x 2 0 f(x) = x 4 – 4 x 3 0 f(x) = 3 x 5 - 25 x 3 + 60 x 0 f(x) = 3 x 4 - 16 x 3 + 18 x 2 + 2 2/20/2021 IB Math SL 1 - Santowski 16

(J) Internet Links 0 Visual Calculus - Maxima and Minima from UTK 0 Visual Calculus - Mean Value Theorem and the First Derivative Test from UTK 0 First Derivative Test -- From Math. World 0 Tutorial: Maxima and Minima from Stefan Waner at Hofstra U 0 http: //www. math. hmc. edu/calculus/tutorials/extre ma/ 2/20/2021 IB Math SL 1 - Santowski 17

(K) Homework 0 Handout from Stewart, 1997, text 2/20/2021 IB Math SL 1 - Santowski 18