University:
Kentucky State University
Course:
MAT 115 | College Algebra
Academic year:

2023
Views:

492

Pages:

7
Author:

garciavelascogr15

**§4.3 How Derivatives Affect the Shape of a Graph** * What does f' say about the graph of f? (i) If f'(x) > 0 on I = (a,b), then f is increasing on I. (ii) If f'(x) < 0 on I = (a,b), then f is decreasing on I. * The First Derivative Test: Suppose that c is a critical number of a continuous function f, then: 1. If f' changes from positive to negative at c, then f has a local maximum at c. 2. If f' changes from negative to positive at c, then f has a local minimum at c. [Image of graphs showing the first derivative test for local maxima and minima] * No maximum or minimum [Image of a graph showing no local maximum or minimum] **Example: Determine the intervals on which the function is increasing or decreasing, and then use the First Derivative Test to identify the local maximum and minimum values.** **1. f(x) = 2x³ + 3x² - 72x + 17** f'(x) = 6x² + 6x - 72 = 0 ⇒ (x+4)(x-3) = 0 ⇒ x = -4, x = 3. [Image of a number line with critical points -4 and 3, and signs of f'(x) indicated] f increasing on (-∞, -4) U (3, ∞). f decreasing on (-4, 3). f(-4) = 225 local max and f(3) = -118 local min. **2. f(x) = x⁴ - 2x² + 1. (Exercise)** **3. f(x) = x + 2sin(x), 0 ≤ x ≤ 2π** f'(x) = 1 + 2cos(x) = 0 ⇒ cos(x) = -1/2 ⇒ x = 2π/3, 4π/3. [Image of a number line with critical points 2π/3 and 4π/3, and signs of f'(x) indicated] f increasing on (0, 2π/3) U (4π/3, 2π). f decreasing on (2π/3, 4π/3). f(2π/3) = 2π/3 + √3 local max. f(4π/3) = 4π/3 - √3 local min. **#14** f(x) = cos²x - 2sinx; 0 ≤ x ≤ 2π. **Soln:** f'(x) = 2cosx(-sinx) - 2cosx = 0 ⇒ -2cosx(sinx + 1) = 0 ⇒ cosx = 0 or sinx = -1. ⇒ x = π/2, 3π/2. [Image of a number line with critical points π/2 and 3π/2, and signs of f'(x) indicated] f decreasing (0, π/2) U (3π/2, 2π). f increasing (π/2, 3π/2). f(π/2) = -2 local min and f(3π/2) = 2 local max. **What does f'' say about the graph of f?** (i) If f''(x) > 0 on I, then f' is increasing on I and the graph of f is concave upward on I. (ii) If f''(x) < 0 on I, then f' is decreasing on I and the graph of f is concave downward on I. **Defn:** A point P on a curve y = f(x) is called an inflection point if f is continuous and the curve changes from concave upward to downward or from concave downward to upward at P. **Example:** (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)-(c) to sketch the graph. **1. f(x) = x^4 - 4x^3** **Soln:** f'(x) = 4x^3 - 12x^2 = 0 ⇒ 4x^2(x-3) = 0 ⇒ x = 0 and x = 3 C.N.s. [Image of a number line with critical points 0 and 3, and signs of f'(x) indicated] (a) f decreasing (-∞,0) U (0,3). f increasing (3,∞). (b) f(3) = -27 local min value.

How Deriatives Affect the Shape of a Graph

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