2026 Fall Section 3.1 Derivatives and Rates of Change

Overview

Professor Ara Araeipour develops the derivative from the secant-line average rate of change to the tangent-line instantaneous rate, emphasizing the limit definitions and algebra needed to evaluate them. Worked examples cover polynomial and square-root tangent lines, velocity and sunflower growth, and a falling ball, with attention to units and a discrepancy between the stated 415 m drop and the later 450 m calculation.

Key takeaways

The derivative f'(a) is the limit of secant slopes as the interval shrinks to zero, making it both the tangent slope at x = a and the instantaneous rate of change there.
Difference-quotient calculations often begin as 0/0; expand and factor the numerator, cancel the matching interval factor, and only then evaluate the limit.
For f(x) = x² - 3x, the average rate from 2 to 4 is 2, while the instantaneous rate at 4 is 5, showing that an interval average need not equal the endpoint rate.
For S(t) = 2t² - 5t + 40 feet, average velocity from 2 to 4 seconds is 7 ft/s, while instantaneous velocity at 4 seconds is 11 ft/s.
The sunflower model f(x) = 6x - (1/4)x² gives a height of 32 inches at week 8, average growth of 1.5 inches per week from weeks 6 to 12, and instantaneous growth of 1 inch per week at week 10.
The falling-ball example states a 415 m drop but later calculates with 450 m; those inputs are inconsistent, so the reported impact speed of 94.08 m/s corresponds to the 450 m calculation.

Chapters

0:00 Limit and Continuity Review Before Derivatives

Reviews limits as functions approach a value from the left and right, and distinguishes convergent limits from divergence or nonexistence.
Connects matching one-sided limits to a two-sided limit, and a limit equal to f(a) to continuity at x = a.
Recaps horizontal asymptotes y = L for limits at positive or negative infinity and vertical asymptotes x = a for infinite limits near a.

3:00 Secant Slopes as Average Rates of Change

For points P = (a, f(a)) and Q = (x, f(x)), the secant slope is (f(x) - f(a))/(x - a).
Equivalent forms include (f(b) - f(a))/(b - a) and (f(a + h) - f(a))/h, each measuring average change over an interval.
Interprets the secant slope as rise over run, or Δy/Δx.

5:00 Tangent Slopes and the Limit Definition of the Derivative

As Q approaches P, the secant line approaches the tangent line, whose slope is the instantaneous rate of change.
Defines f'(a) as lim x→a of (f(x) - f(a))/(x - a), equivalently lim h→0 of (f(a + h) - f(a))/h.
Finds a tangent-line equation with point-slope form: y - f(a) = f'(a)(x - a).

8:40 Polynomial Difference Quotients at x = 4

For f(x) = x² - 3x, computes the average rate of change from x = 2 to x = 4 as 2.
Uses the h-definition at x = 4, expands (4 + h)², factors and cancels h, then evaluates the limit to get f'(4) = 5.
Checks the equivalent x-definition by factoring x² - 3x - f(4), illustrating why the shared factor x - 4 cancels before substitution.

14:55 Tangent Lines for x² and a Second Polynomial Derivative

For y = x² at (1, 1), evaluates the difference quotient by factoring x² - 1 and obtains tangent slope 2.
Uses point-slope form to derive the tangent line y = 2x - 1, then confirms it passes through (1, 1).
For f(x) = x² - 8x + 9 at x = 5, expands the h-difference quotient and finds the instantaneous rate f'(5) = 2.

23:30 Average and Instantaneous Velocity from Position

For position S(t) = 2t² - 5t + 40 feet, calculates average velocity from 2 to 4 seconds as 7 ft/s.
Applies the derivative definition at t = 4, expands S(4 + h), cancels h, and obtains instantaneous velocity 11 ft/s.
Emphasizes that velocity units come from distance divided by time, so the result is expressed in feet per second.

28:35 Square-Root Difference Quotient and Tangent at x = 16

For f(x) = √x at x = 16, uses the conjugate to simplify the difference quotient rather than substituting into its initial 0/0 form.
Finds the tangent slope f'(16) = 1/8 and uses the point (16, 4) to derive y = (1/8)x + 2.
Verifies the line by substituting x = 16, while distinguishing that point check from the separate slope calculation.

33:14 Sunflower Height, Average Growth, and Instantaneous Growth

Models sunflower height in inches by f(x) = 6x - (1/4)x², where x is weeks after planting; at week 8 the height is 32 inches.
Computes average growth from week 6 to week 12 as 3/2 inches per week.
Uses the derivative definition at week 10, simplifying the difference quotient to find an instantaneous growth rate of 1 inch per week.
Treats units as part of each application answer, distinguishing inches of height from inches per week of growth.

42:48 Falling-Ball Velocity and the Drop-Height Calculation

For S(t) = 4.9t², derives velocity v(t) = 9.8t from the limit definition and gets 49 m/s after 5 seconds.
To estimate impact speed, sets the distance equal to the drop height, solves for positive t, then substitutes that time into v(t).
The setup first states a 415 m drop, but the later calculation uses 450 m, yielding t ≈ 9.6 seconds and a reported impact speed of 94.08 m/s.
Retains exact expressions during calculations and rounds only as required, with speed reported in meters per second.

Transcript excerpt

0:07 Good morning, class. Here's chapter 3, derivatives, section 3.1, derivatives and rates of change. So, in this chapter, we're going to cover derivatives and rates of change, the derivative as a function, and then to use definitions, and then we can use the easy way out using formulas. Basic differentiation formulas, the product and quotient rules, the chain rule which has two parts, implicit differentiation and related rates,

0:56 derivatives of the log functions and differentials. So, again, reminding you of what we have done, we are interested in limit of f of x as x approaches a equals L. When the limit of a sub n as n approaches infinity is L, we say a sub n converges to it. If it's infinity, negative infinity, or DNE, does not exist, we say it diverges. To find limits, simply plug in and observe. In all these three cases, to look for limit of f of x as x approaches a, we approach a from the left and right, those are the branches, and it moves us towards this point on the graph toward this

1:44 uh y coordinate, and therefore the limit of f of x equals as x approaches a equals L in all graphs. The difference f of a equals L makes it continuous at a, f of a equals something else, discontinuous, f of a equals b. In this case, f of a is on is not defined, it's missing. Definition for a horizontal asymptote, when the limit of f of x as x approaches negative or positive infinity is L, the line y equals L gives us the horizontal asymptote. When the limit of f of x as x approaches a or a from the left or right is negative or positive infinity gives us the vertical asymptote, VA, x equals a.

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Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Professor Ara Araeipour.