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This guide explores derivatives and their role in graphing functions. By applying key rules for finding derivatives—such as the derivative of a constant, power functions, and sums—students can streamline the process. Understanding the relationship between derivatives and slopes aids in identifying critical points like relative maxima and minima and points of inflection on graphs. Through examples and exercises, the concepts are put into practice, enabling students to analyze function behavior effectively and apply findings to real-world scenarios, including cost function analysis.
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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) = xn, where n is any nonzero real number, then f(x) = nxn–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!)
Derivative of a Constant Multiple of a Power If f (x) = axn, where a & n are any nonzero real numbers, then f(x) = (an) xn–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) = x7 b) f (x) = 6x5 c) f (x) = 4x2 – 12x + 7 d) f(x) = 7x6 f(x) = 30x4 f(x) = 8x – 12 f(x) = 42x5 f(x) = 120x3 f(x) = 8 *rewrite*
Knowing derivatives & their connection to slopes can help us graph a curve. relative maximum increasing decreasing increasing relative minimum The graph: - increases: derivative positive - decreases: derivative negative - relative maximum: derivative changes from increasing to decreasing - relative minimum: derivative changes from decreasing to increasing
Knowing derivatives & their connection to slopes can help us graph a curve. relative maximum concave down increasing concave up decreasing increasing relative minimum point of inflection 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, 2nd derivative is 0
Let’s apply & learn this by doing it Ex 2) Given Find 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. Take derivative: Find critical numbers:x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 x = 3, –1 Intervals for inc. & dec. (make a “sign chart”) f (x) + – + Is the derivative (+) or (–)? –1 3 inc. (–∞, –1) and (3, ∞) dec. (–1, 3) relative max at x = –1 and relative min at x = 3 (3, –5)
Ex 2) cont… Take 2nd derivative: f (x) = 2x – 2 Find critical numbers: 2x – 2 = 0 2x = 2 x = 1 Intervals of concavity: (make a “sign chart”) f (x) Graph: – + 1 concave down (–∞, 1) and concave up (1, ∞) point of inflection at x = 1
We can apply max’s & min’s to business models. • Ex 3) If c (x) = –0.04x2 + 32x + 1500 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
Homework #1605 Pg 880 #1–15 odd, 19–29 odd
