Monotonicity and the first derivative test (Calc 1; Lecture 2-7; Fall 26)

Overview

The lecture develops a practical calculus workflow: identify critical points, use derivative signs to determine where a function increases or decreases, and compare function values to find absolute extrema on a closed interval. Examples include an inverse-derivative calculation, a logarithm-and-square-root domain review, a cubic and a rational function sign analysis, and the first derivative test for classifying local maxima and minima.

Key takeaways

A critical point must belong to the function’s domain: 1/x is undefined at 0, so 0 is not a critical point, while 0 is a critical point for |x| and x^(1/3) because those functions are defined there but their derivatives are not.
For a continuous function on a closed, bounded interval, absolute extrema are guaranteed; evaluate the function at every in-interval critical point and both endpoints, then compare the values.
Derivative signs give monotonicity interval by interval: f′>0 means increasing and f′<0 means decreasing, so one test value suffices between consecutive critical points.
The first derivative test classifies extrema from sign changes: positive-to-negative yields a local maximum, negative-to-positive yields a local minimum, and no sign change yields neither.
For 12x^(1/3)−3x^(4/3) on [−1,2], the candidate values are −15, 0, 9, and approximately 7.559; the absolute minimum is −15 at x=−1 and the absolute maximum is 9 at x=1.

Chapters

0:00 Domain Review: Square Roots, Logarithms, and Denominators

Reviews the domain of a function involving √(1−x) and a logarithm in the denominator.
Checks the square-root condition, the logarithm’s input, and values that make the denominator zero.
Reinforces three common domain hazards: invalid square roots, invalid logarithms, and division by zero.

4:40 Inverse Derivative at Zero Using f⁻¹(0)

Uses (f⁻¹)'(a)=1/f′(f⁻¹(a)) to evaluate the inverse derivative at 0.
For f(x)=3x−7+e^(x−2), identifies f⁻¹(0)=2 by substitution.
Computes f′(2)=3+1=4, giving (f⁻¹)'(0)=1/4.

6:52 Critical Points: Zero Derivatives and In-Domain Undefined Derivatives

Defines critical points as in-domain locations where f′ is zero or undefined; includes interval boundaries when making an extrema candidate list.
Contrasts 1/x, which is undefined at 0 along with its derivative, with |x| and x^(1/3), whose derivatives are undefined at points in their domains.
For f(x)=log(x⁴+1)−6 arctan(x²), differentiates and factors the numerator to find critical points at 0 and ±√3.

14:50 Absolute Extrema: The Closed-Interval Guarantee

States that a continuous function on a closed, bounded interval [a,b] must attain both an absolute maximum and an absolute minimum.
Outlines the candidate-list method: find critical points in the interval, evaluate the function at each candidate, then compare the outputs.
Connects mathematical understanding with practice, emphasizing that working problems—not only hearing explanations—builds skill.

19:38 Absolute Extrema Example on [−1, 2]

For f(x)=12x^(1/3)−3x^(4/3), checks the endpoints −1 and 2 along with critical points inside the interval.
Finds f′(x)=4x^(−2/3)(1−x), so x=1 makes the derivative zero and x=0 makes it undefined while remaining in the function’s domain.
Compares f(−1)=−15, f(0)=0, f(1)=9, and f(2)≈7.559; the absolute minimum is −15 at −1 and the absolute maximum is 9 at 1.

28:39 Increasing and Decreasing Intervals from Derivative Signs

Defines increasing behavior as outputs rising from left to right and decreasing behavior as outputs falling from left to right.
Uses the tangent-line interpretation: f′>0 indicates increasing behavior, while f′<0 indicates decreasing behavior.
Finds where f′ is zero or undefined, splits the domain at those critical points, and tests one convenient value in each interval.
Notes that conventions differ on including interval endpoints; the required notation may depend on the instructor or problem.

34:15 Cubic Monotonicity: Sign Chart at −1 and 3

For y=x³−3x²−9x+17, computes y′=3x²−6x−9=3(x−3)(x+1).
Uses critical points −1 and 3 to divide the real line into three intervals and tests easy values such as −2, 0, and 4.
Finds the function increasing on (−∞,−1) and (3,∞), and decreasing on (−1,3).

40:32 Rational Monotonicity: h(y)=(y+1)/(y²+8)

Applies the quotient rule to obtain h′(y)=−(y²+2y−8)/(y²+8)².
Factors the numerator as −(y+4)(y−2); the denominator is always positive, so the critical points are −4 and 2.
Tests the intervals around −4 and 2 to find increasing behavior on (−4,2) and decreasing behavior on (−∞,−4) and (2,∞).

46:17 First Derivative Test: Classifying Local Extrema

Classifies a critical point by checking the derivative signs immediately to its left and right.
A positive-to-negative sign change means the function rises then falls, giving a local maximum; a negative-to-positive change gives a local minimum.
Positive-to-positive and negative-to-negative sign patterns indicate no local extremum.
Applies the test to the cubic: −1 is a local maximum and 3 a local minimum; for the rational function, −4 is a local minimum and 2 a local maximum.

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