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C1.1,2,3 – Function Analysis – Critical Values, Intervals of Increase/Decrease & First Derivative Test IB Math HL/SL - Santowski

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(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. 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) 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 f(x 2 ) when x 1 < x 2 ; the graph of the function is going up to the left (or down to the right) "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

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(B) Review – Graphic Analysis of a Function We have seen functions analyzed given the criteria intervals of increase, intervals of decrease, critical points (AKA turning points or maximum or minimum points) We have also seen graphically how the derivative function communicates the same criteria about a function these points are summarized on the next slide:

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(B) Review – Graphic Analysis of a Function f(x) has a max. at x = -3.1 and f `(x) has an x-intercept at x = -3.1 f(x) has a min. at x = -0.2 and f `(x) has a root at –0.2 f(x) increases on (-, -3.1) & (-0.2, ) and on the same intervals, f `(x) has positive values f(x) decreases on (-3.1, -0.2) and on the same interval, f `(x) has negative values

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(C) Analysis of Functions Using Derivatives – A Summary If f(x) increases, then f `(x) > 0 If f(x) decreases, then f `(x) < 0 At a max/min point, f `(x) = 0 We can also state the converse of 2 of these statements: If f `(x) > 0, then f(x) is increasing If f `(x) < 0, then f(x) is decreasing 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

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(D) First Derivative Test 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?) Since we have done some graphic analysis with functions and their derivatives, in one sense we already now the answer: see next slide

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(E) First Derivative Test - Graphically 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 At a the min (x = -0.2), the fcn changes from decreasing to increasing the derivative changes from negative to positive

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(F) First Derivative Test - Algebraically At a maximum, the fcn changes from being an increasing fcn to a decreasing fcn the derivative changes from positive values to negative values At the minimum, the fcn changes from decreasing to increasing the derivative changes from negative to positive So to state the converses: 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 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 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

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(G) First Derivative Test – Example #1 Find the local max/min values of y = x 3 - 3x + 1 (Show how to use inequalities to analyze for the sign change) f `(x) = 3x 2 – 3 f `(x) = 0 for the critical values 0 = 3x 2 – 3 0 = 3(x 2 – 1) 0 = 3(x – 1)(x + 1) x = 1 or x = -1 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

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(G) First Derivative Test – Example #1 Factor Interval 3(x-1)(x+1) f `(x) f(x) (-∞,-1)+--+inc (-1,1)+-+-dec (1, ∞) ++++inc

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(G) First Derivative Test – Example #1 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) 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) 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 And going another step, we can test the end behaviour of f(x): –lim x-∞ f(x) = -∞ –lim x ∞ f(x) = +∞ 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”)

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(G) First Derivative Test – Example #1 – Graphic Summary

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(H) In Class Examples Ex 2. Find the local max/min values of g(x) = x 4 - 4x 3 - 8x Ex 3. Find the absolute minimum value of f(x) = x + 1/x for x > 0 Ex 4. Find the intervals of increase and decrease and max/min values of f(x) = cos(x) – sin(x) on (-,) Ex 5. Find the critical numbers, intervals of increase & decrease and max/min values of y = csc(x) – cot(x) on (- /2,3/2) Ex 6. Find the intervals of increase/decrease and max/min points of f(x) = x 2 e -x Ex 7. Find the local and absolute maximum & minimum points for f(x) = x(ln(x)) 2

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(I) Internet Links Visual Calculus - Maxima and Minima from UTK Visual Calculus - Maxima and Minima from UTK Visual Calculus - Maxima and Minima from UTK Visual Calculus - Mean Value Theorem and the First Derivative Test from UTK Visual Calculus - Mean Value Theorem and the First Derivative Test from UTK Visual Calculus - Mean Value Theorem and the First Derivative Test from UTK First Derivative Test -- From MathWorld First Derivative Test -- From MathWorld First Derivative Test -- From MathWorld Tutorial: Maxima and Minima from Stefan Waner at Hofstra U Tutorial: Maxima and Minima from Stefan Waner at Hofstra U Tutorial: Maxima and Minima from Stefan Waner at Hofstra U

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(J) Homework Handout from Stewart, 1997, Chap 4.2, p , Q3-6,7-14,25-46,47- 50

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