16.5 Using Derivatives in Graphing

1. 16.5 Using Derivatives in Graphing

2. 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!)

3. 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*

4. 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

5. 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

6. 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)

7. 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 

8. 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

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