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16.5 Using Derivatives in Graphing

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Finding the derivative by definition is a bit tiresome, so we can apply some rules to help speed up the process. Please keep in the back of your mind: derivative & slope are synonyms Derivative of a Constant If f (x) = c, where c is a constant, then f (x) = 0 for all x. (*remember f (x) is a horizontal line, so it makes sense the slope is 0, always!) Derivative of a Power If f (x) = x n, where n is any nonzero real number, then f (x) = nx n–1. (*nickname is pop & drop. We pop the exponent to the front & drop the exponent power by 1. This works for positive exponents, negative exponents, & even fractional exponents!)

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Derivative of a Constant Multiple of a Power If f (x) = ax n, where a & n are any nonzero real numbers, then f (x) = (an) x n– 1. Derivative of a Sum If f (x) = g (x) + h (x), then f (x) = g (x) + h (x). Ex 1) Find f (x) and f (x) for each function. (Whiteboards) a) f (x) = x 7 b) f (x) = 6x 5 c) f (x) = 4x 2 – 12x + 7 d) f (x) = 7x 6 f (x) = 30x 4 f (x) = 8x – 12 f (x) = 42x 5 f (x) = 120x 3 f (x) = 8 *rewrite*

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Knowing derivatives & their connection to slopes can help us graph a curve. decreasing relative maximum increasing The graph: - increases: derivative positive - decreases: derivative negative - relative maximum: derivative changes from increasing to decreasing - relative minimum: derivative changes from decreasing to increasing relative minimum increasing

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Knowing derivatives & their connection to slopes can help us graph a curve. concave up decreasing relative maximum increasing concave down The graph: - critical number: where the derivative is 0 (possibilities for relative min’s & max’s) - concave up: second derivative positive - concave down: second derivative negative - point of inflection: point where concavity changes, 2 nd derivative is 0 relative minimum increasing point of inflection

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(1)Take derivative: (2)Find critical numbers: x 2 – 2x – 3 = 0 (x – 3)(x + 1) = 0x = 3, –1 (3)Intervals for inc. & dec. (make a “sign chart”) f (x) Let’s apply & learn this by doing it Ex 2) GivenFind a) critical numbers, b) intervals where function is increasing & decreasing, c) relative min’s & max’s, d) intervals of concave up & concave down, & e) points of inflection. –1 3 ++– Is the derivative (+) or (–)? inc. (–∞, –1) and (3, ∞)dec. (–1, 3) relative max at x = –1 and relative min at x = 3 (3, –5)

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Ex 2) cont… (4)Take 2 nd derivative: f (x) = 2x – 2 (5)Find critical numbers: 2x – 2 = 02x = 2x = 1 (6)Intervals of concavity: (make a “sign chart”) f (x) Graph: 1 +– concave down (–∞, 1) and concave up (1, ∞) point of inflection at x = 1

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We can apply max’s & min’s to business models. Ex 3) If c (x) = –0.04x x is the total cost function for producing mechanical pencils, how many must be produced before costs decrease? What do we want? The relative max because the cost would go from increasing to decreasing here. This means the first derivative c (x) = –0.08x + 32 = 0 –0.08x = –32 x = 400 pencils

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Homework #1605 Pg 880 #1–15 odd, 19–29 odd

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