These are my complete notes for Derivative Tests in Differential Calculus.
I color-coded my notes according to their meaning - for a complete reference for each type of note, see here (also available in the sidebar). All of the knowledge present in these notes has been filtered through my personal explanations for them, the result of my attempts to understand and study them from my classes and online courses. In the unlikely event there are any egregious errors, contact me at jdlacabe@berkeley.edu.
Summary of Derivative Tests (Differential Calculus)
?. Derivative Tests.
# Concavity:
Which way the graph bends or shapes, either 'up' or 'down', as defined below.
a) Concave up on an open Interval I if y' is increasing on I. This is the part of the curve on the position graph where the tangent line is on the *bottom* of the curve.
b) Concave down on an open Interval I if y' is decreasing on I. This is the part of the curve on the position graph where the tangent line is on the *top* of the curve.
# Concavity Test:
a) Concave up on any interval where y'' > 0.
b) Concave down on any interval where y'' < 0.
# Point of Inflection: A point on the graph where a tangent line exists and where the concavity changes (i.e., where the y'' changes sign).
#
Rule .
Properties of the Second Derivative (y''):
1. y'' = 0 does not always ive an inflection point. Sometimes both sides are the same sign, like in a parabola, or continually increasin or decreasing, like with x³. For example, with the function f(x) = x⁴, f'(x) = 4x³ and f''(x) = 12x². Although f''(x) = 0 at x=0, it cannot be considered an inflection point as the concavity doesn't change: f''(x) remains positive before and after.
2. y'' ≠ 0 can still be an inflection point, surprisingly. In the example f(x) = ∛x = x¹/³, f'(x) = (1/3)x⁻²/³, f''(x) = (-2 / 4∛x⁵). As evident, f''(0) is DNE, but it is also shockingly still an inflection point. An inflection point can occur where f''(x) = 0, or where f''(x) is DNE, since the only requirement is wherever f''(x) changes signs, regardless of continuity. To check, use the second derivative test.
1. y'' = 0 does not always ive an inflection point. Sometimes both sides are the same sign, like in a parabola, or continually increasin or decreasing, like with x³. For example, with the function f(x) = x⁴, f'(x) = 4x³ and f''(x) = 12x². Although f''(x) = 0 at x=0, it cannot be considered an inflection point as the concavity doesn't change: f''(x) remains positive before and after.
2. y'' ≠ 0 can still be an inflection point, surprisingly. In the example f(x) = ∛x = x¹/³, f'(x) = (1/3)x⁻²/³, f''(x) = (-2 / 4∛x⁵). As evident, f''(0) is DNE, but it is also shockingly still an inflection point. An inflection point can occur where f''(x) = 0, or where f''(x) is DNE, since the only requirement is wherever f''(x) changes signs, regardless of continuity. To check, use the second derivative test.
