Curve sketching 2
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This document provides guidance on curve sketching functions by hand without using a graphing calculator. It outlines the key steps and tools needed for curve sketching, including determining the domain, finding intercepts and critical points, checking for asymptotes, analyzing the first and second derivatives to determine concavity and monotonicity, and connecting the points to sketch the curve. The document also provides examples to demonstrate how to apply these techniques to sketch curves based on information about the function's behavior.
Curve sketching 2
- 2. Motivation Graphing calculators decrease the importance of curve sketching So why a lesson on curve sketching? A calculator graph may be misleading • What happens outside specified window? • Calculator plots, connects points without showing what happens between points • False asymptotes Curve sketching is a good way to reinforce concepts of lessons in this chapter 2
- 3. Tools for Curve Sketching Test for concavity Test for increasing/decreasing functions Critical points Zeros Maximums and Minimums 3
- 4. Strategy Determine domain of function Find y-intercepts, x-intercepts (zeros) Check for vertical, horizontal asymptotes Determine values for f '(x) = 0, critical points Determine f ''(x) • Gives inflection points • Test for intervals of concave up, down Plot intercepts, critical points, inflection points Connect points with smooth curve Check sketch with graphing calculator 4
- 5. Using First, Second Derivatives Note the four possibilities for a function to be … • Increasing or decreasing • Concave up or concave down 5 Positive (increasing function) Negative (decreasing function) Positive (concave up) Negative (concave down) f '(x) f ''(x)
- 6. Try It Out Find as much as you can about the function without graphing it on the calculator 6 3 215 ( ) 18 1 2 f x x x x= − − − 2 1 x y x = − ( ) lnf x x x= −
- 7. Graphing Without the Formula Consider a function of this description • Can you graph it? This function is continuous for all reals • • • • • A y-intercept at (0, 2) 7 '( ) 0 on (- , -6) and (1, 3)f x < ∞ '( ) 0 on (-6, 1) and (3, )f x > ∞ ''( ) 0 on (- , -6) and (3, )f x > ∞ ∞ ''( ) < 0 on interval (-6, 3)f x