1
Compute an average rate of change exactly
The quotient (f(a + h) − f(a))/h as a single fraction for a given function, point and step, read as the slope of a secant, with the division carried out rather than dropped.
Course · Differential Equations
Rates of Change and the Derivative
Every differential equation in this Subject is a sentence about a rate, so the rate has to be something you can compute and not only something you can recite. Here it is computed first and named second. The average rate of change over [a, a + h] is a fraction, and the lab prints it as one for a column of halving steps; the derivative is the number that column heads for, stated as a claim the table illustrates and does not prove. For a polynomial the claim is not even needed, because the quotient simplifies to a polynomial in h and the derivative is its constant term. The power, product and chain rules are then checked exactly on polynomials by expanding both sides, and the course ends on the exponential, the one place where a quotient cannot be exact and the lab says so by rounding and labelling every figure.
(f(a + h) − f(a))/h exact, for each h
a polynomial's quotient is a polynomial in h
f′(a) = what the quotients approach
tangent: y = f(a) + f′(a)·(t − a)
(tⁿ)′ = n·tⁿ⁻¹
(f·g)′ = f′·g + f·g′
p′ = 0 where the graph turns; p″ the bend
(bᵗ)′ = bᵗ·(a constant); e makes it 1
What this course is about
- Subject
- Differential Equations
- Lessons
- 11
- Format
- Interactive
Overview
What you will be able to do
By the end you can compute a rate of change exactly, say precisely what the derivative claims, find it for a polynomial by algebra and by rule, and use it to describe how a graph moves.
2
Simplify a difference quotient to a polynomial in h
Expand p(t + h), cancel the h that every surviving term shares, and read off the part that does not depend on h — without setting h to zero first.
3
State what the derivative claims, and what a table shows
A column of exact quotients for halving h, the number it heads for, and one sentence that separates the claim about every h from the seven rows on the page.
4
Write the tangent line and measure what it gets wrong
y = f(a) + f′(a)·(t − a), the estimate it gives at a + h, and the exact error — a fraction, not a feeling about nearness.
5
Differentiate and read the graph from the sign
The power rule, term by term; where p′ is positive, negative or zero; the product and chain rules checked by expanding both sides; and the second derivative as the rate of the rate.
6
Recognise the exponential by its rate
The quotient of bᵗ as bᵗ times a factor that does not depend on t, that factor read off a rounded and labelled table, and e as the base whose factor is 1.
Lessons
11 lessons
Rates come first as fractions: the average rate, then the average rate of a polynomial as a polynomial in the step. Only then is the question asked that gives the course its name, what the quotients approach, and it is answered at a point, with the tangent line, and for one function that is not a polynomial. The second half turns the point into a function and uses it: the rules, the places where the rate vanishes, the rate of the rate, and the exponential.
Average Rate of ChangeThe average rate over a step is a difference divided by the step, an exact fraction that is the slope of a secant.Not marked
The Difference Quotient as a Polynomial in hFor a polynomial the quotient simplifies to another polynomial in h, and the part that does not contain h is the number the next lessons call the rate at the place.Not marked
What the Quotients ApproachHalve the step and the exact quotients head for one number; the lesson states that as a claim, and the table is its evidence.Not marked
The Derivative at a Point and the Tangent LineThe derivative at a point is the number the quotients approach, and the tangent line it defines estimates nearby values with an error you can write down exactly.Not marked
The Derivative of 1/tThe quotient of 1/t is found by combining two fractions, it simplifies to the exact fraction −1/(a(a + h)), and the derivative is −1/a².Not marked
The Derivative as a FunctionDifferentiate a polynomial term by term with the power rule, and read where its graph rises and falls from the sign of the result.Not marked
Where the Rate Is ZeroSolve p′ = 0 exactly, then use the sign of p′ on each side to say whether each zero is a maximum, a minimum or neither.Not marked
The Product RuleThe derivative of a product is f′·g + f·g′, verified here by expanding both sides, and it is not the product of the derivatives.Not marked
The Chain RuleThe derivative of f(g(t)) is f′(g(t))·g′(t): the outer rate at the inner value, times the inner rate.Not marked
The Second DerivativeDifferentiate twice to get p″, the rate of the rate, and read from its sign where the graph bends up and bends down.Not marked
The Exponential and Its RateThe quotient of bᵗ is bᵗ times a factor that does not depend on t, so the rate of an exponential is a constant multiple of itself.Not marked
Recommended background
The Algebra Subject's Lines, Functions and Graphs, for function notation, slope and what a graph is, its Polynomials and Factoring, for expanding a power such as (t + h)³ and cancelling a common factor, and, for the last lesson only, the law b^(a + h) = b^a·b^h from its Exponential and Logarithmic Functions; nothing from calculus, and fractions read comfortably, since every page uses them and the lab does the arithmetic.
Practice suggestions
Compute before you name. Each lesson starts from quotients you can check by hand — a handful of fractions — and only then gives the quantity they head for a name. If a figure on the page surprises you, type the function into the lab and watch the column.
Keep two statements apart: what the table shows and what the lesson claims. A table has a finite number of rows and every claim about a rate is about every step, however small. For polynomials the gap is closed by algebra, and the lessons say where; for the exponential it is not, and the lesson says that too.
Do the cancelling yourself at least once per lesson. Expand (t + h)², subtract, divide by h. The labs print the result, but the habit of seeing the h factor out of every surviving term is what makes the derivative a computation rather than a definition to memorise.
Read every rounded figure as rounded. Wherever the lab prints ≈, the digits are a decimal approximation to a number that has none; everywhere else the figure is an exact fraction and you can check it with pencil.
What it does not cover
Limits as a theory. There is no epsilon-and-delta definition here, no limit laws and no definition of continuity. The derivative is what exact quotients head for, demonstrated as a column and, for polynomials, read off by algebra. A reader who wants the theory is at the start of a first analysis course, which is where it properly begins.
Differentiation of general functions. The quotient rule, implicit differentiation, inverse trigonometric functions and the derivatives of logarithms are left out; the later courses need the power, product and chain rules on polynomials and the stated derivatives of the exponential, sine and cosine, and nothing else.
Optimisation as a craft. Where the rate is zero is found and classified, and the second derivative is read, but word problems about boxes, fences and ladders are not set; they are an application of this course and not a part of it.
The result this course states and does not prove. That the quotients of a function that is not a polynomial settle on one number. The exponential's quotient is shown as a rounded column heading for a constant, and the constant is named; that the column really converges is the claim the table illustrates.