Defines the derivative as the limit of a difference quotient, showing that the slope of a tangent and an instantaneous rate of change are the same question. Computes derivatives from the definition, interprets their sign and size, and closes the course by naming what calculus does with them.
Subject: Precalculus · 65 slides · symbolic lesson
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Title
Precalculus · Chapter 12 — Introduction to Calculus
§12.4 Derivatives, pp. 1432-1457
Objectives
Five things, each one you can check yourself on paper.
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1432-1457 — the pages these objectives are drawn from
Warm-up
Slope was defined for lines. A curve's steepness changes from point to point.
Discussion prompt
A parabola is steeper in some places than others. How would you say how steep it is at one particular point?
Hint: Slope needs two points. How many do you have?
Answer:
Slope is rise over run, which needs two points. At a single point there is no run and no rise — the ratio would be zero over zero.
But taking a second point nearby gives a slope, and moving it closer gives a better approximation to the steepness there.
So the slope at a point is what those approximations approach — a limit. That is why chapter 12 spent three sections on limits before reaching this question.
Concept
The derivative at a point is the limit of the average rate of change over intervals shrinking to nothing, which is both the tangent's slope and the instantaneous rate.
derivative — the limit of the difference quotient as the interval shrinks to zero, giving the instantaneous rate of change
\[ f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h} \]
Both parts of the quotient go to zero, so the form is always indeterminate — every derivative computation is an exercise in the techniques of §12.2. That is what makes those techniques the section's real prerequisite.
Figure (svg): A curve with several secant lines through a fixed point, each cutting closer, approaching the tangent line
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1432-1437
Section
Section 1
Concept
A secant through two points of a curve has a computable slope. Sliding the second point towards the first makes the secants approach a limiting line, and that is the tangent.
Defining the tangent this way rather than as a line touching at one point matters, because the touching description is imprecise and fails for curves that a genuine tangent crosses. The limit definition is exact.
Figure (svg): A curve with several secant lines through a fixed point, each cutting closer, approaching the tangent line
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1432-1438
Picture it
Three dashed secants and the solid tangent.
Figure (svg): A curve with several secant lines through a fixed point, each cutting closer, approaching the tangent line
Each secant's slope is an ordinary rise over run, computable exactly. The tangent's slope is not directly computable, which is why it has to be defined as what those approach.
Worked example
Watch them approach.
\[ \text{For } f(x)=x^2 \text{ at } x=1, \text{ find secant slopes for } h=1, 0.5, 0.1. \]
Use the width one
Why: Rise over run.
\[ \frac{4 - 1}{1} = 3 \]
Use one half
Why: Closer in.
\[ \frac{2.25 - 1}{0.5} = 2.5 \]
Use one tenth
Why: Closer still.
\[ \frac{1.21 - 1}{0.1} = 2.1 \]
Note the pattern
Why: Approaching two.
Figure (svg): A curve with several secant lines through a fixed point, each cutting closer, approaching the tangent line
\[ 3,\;2.5,\;2.1\;\to\;2 \]
Verify: check the pattern continues
Why: At a width of 0.01 the slope is 2.01, and at 0.001 it is 2.001 — the pattern holds and points at 2. That numerical evidence is what the algebraic computation will confirm exactly.
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1433-1436
Prediction
The second point slides towards the first.
Predict first
What do the secant lines approach?
Correct: The tangent line at the first point.
Why: Each secant turns as its second point moves in, and the limiting position is the tangent. That limit is the definition rather than a description of what is drawn.
Worked example
Drawing is imprecise.
\[ \text{Why define the tangent as a limit rather than as a line touching at one point?} \]
Consider the touching description
Why: Vague about what touching means.
Note a counterexample
Why: A tangent can cross the curve.
Consider the limit description
Why: Exactly specified.
Conclude
Why: The limit is the definition.
Figure (svg): The solution to Worked example why not just draw the tangent shown as a ladder of expressions, one row per legal move
\[ \text{the limit is the definition} \]
Verify: consider an inflection point
Why: At an inflection point the tangent crosses the curve, so the touching description would exclude a line that is genuinely the tangent. The limit definition handles that case without any difficulty.
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1436-1438
Trap
\[ \text{the tangent touches the curve at exactly one point} \]
Define the tangent by its intersection
Why: Touching once is taken as the criterion.
A tangent can cross the curve, and a non-tangent line can meet it once.
The tangent is defined as a limit of secants, which is precise and computable.
At an inflection point the tangent crosses the curve, and a vertical line can meet a parabola once without being tangent to it.
The touching picture is a useful intuition for the ordinary case and not a definition.
Faded example
For the squaring function between one and two.
Fill in the blanks
\frac13=\frac___}}___=___
Why: The secant slope is an ordinary rise over run between two points on the curve. Moving the second point closer to the first makes the slope approach the tangent's.
Sorting
Two points or one.
Sort into buckets
Sort each description.
Step zero
You want the slope of a curve at a point.
Discussion prompt
What do you do, given that slope needs two points?
Hint: Where does the second point go?
Answer:
Take a second point nearby and compute the ordinary secant slope. That is a genuine rise over run.
Then let the second point approach the first and see what the slopes approach. That limit is the answer.
Both steps are necessary: the first makes the computation possible and the second removes the arbitrariness of which second point was chosen. The limit is what makes the answer about one point rather than two.
Section
Section 2
Concept
The geometric question about a tangent's slope and the physical question about an instantaneous rate produce the same difference quotient and the same limit.
That coincidence is why calculus is so widely applicable. A single technique answers questions in geometry, physics, economics and biology, because all of them are asking about a rate of change.
Figure (svg): A contrast between the geometric question about a tangent's slope and the physical question about an instantaneous rate, both answered by the same limit
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1438-1442
Picture it
Different subjects, identical structure.
Figure (svg): A contrast between the geometric question about a tangent's slope and the physical question about an instantaneous rate, both answered by the same limit
The bottom row on each side is the same expression. That is not an analogy but an identity, which is why one calculation serves both fields.
Worked example
The same limit as a slope.
\[ \text{An object's position is } t^2. \text{ Find its speed at time } 1. \]
Write the average speed
Why: Change in position over time.
Note the interval
Why: From one to one plus h.
Shrink the interval
Why: Take the limit.
Compute
Why: The same limit as before.
\[ 2 \]
Figure (svg): A contrast between the geometric question about a tangent's slope and the physical question about an instantaneous rate, both answered by the same limit
\[ 2 \]
Verify: compare with the geometric answer
Why: The slope of the squaring curve at 1 was also 2 — the same number from the same limit. The two questions differ only in what the variables are called, which is exactly the identity this idea claims.
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1439-1441
Prediction
A tangent's slope and an instantaneous rate.
Predict first
How are they related?
Correct: They are the same limit.
Why: Both are a change in output over a change in input with the interval shrunk to zero. The expressions are identical, which is why one calculation answers both questions.
Worked example
The same number, different words.
\[ \text{What does a derivative of } 2 \text{ mean in each context?} \]
Geometrically
Why: The tangent's steepness.
\[ \text{slope } 2 \]
Physically
Why: Distance per unit time.
\[ \text{speed } 2 \]
In economics
Why: Cost per additional unit.
Note the common structure
Why: Output per input.
Figure (svg): The solution to Worked example name the interpretation shown as a ladder of expressions, one row per legal move
\[ \text{output change per unit input change} \]
Verify: check the units
Why: In every case the units are output units per input unit — metres per second, dollars per item, rise per run. That shared structure is what makes one mathematical object serve all of them.
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1441-1442
Error analysis
A student describes the connection.
Annotate
On: \( \text{speed is like a slope, so calculus is a useful analogy for physics} \)
Calling it an analogy understates the connection considerably. The two questions produce the same expression and the same limit, which is why no translation between the fields is needed.
Sorting
The same object in different fields.
Sort into buckets
Sort each interpretation.
Faded example
Position at time one and at time one plus h.
Fill in the blanks
\frac1})}h}
Why: The numerator is the change in position and the denominator the elapsed time, giving an average speed. Shrinking h to zero turns it into the instantaneous speed.
Explain it
Two subjects, one technique.
Discussion prompt
Explain to a classmate why calculus applies so widely.
Hint: What are all the questions about?
Answer:
Every one of them asks about a rate of change — how much one quantity changes per unit change in another.
That ratio is the same expression whatever the quantities are called: rise over run, distance over time, cost over units.
So a single technique for shrinking that ratio to an instant serves all of them without adaptation. A good explanation notes that this is an identity rather than an analogy, which is why no translation between fields is needed.
Section
Section 3
Concept
The difference quotient measures the change in output over an interval of width h, and its limit as that width shrinks to zero is the derivative.
That the form is always indeterminate is the reason §12.2 came first. Every derivative computation from the definition begins by producing zero over zero and then resolving it algebraically — usually by cancelling a factor of the width.
Figure (svg): A card giving the difference quotient with each part labelled, and noting that it is always indeterminate at the limit
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1442-1446
Picture it
Each part labelled, with the indeterminacy flagged.
Figure (svg): A card giving the difference quotient with each part labelled, and noting that it is always indeterminate at the limit
The red line names the structural fact. Every derivative is an indeterminate form by construction, which makes §12.2's techniques the standard first step rather than an occasional one.
Worked example
Expand and cancel.
\[ \text{Simplify the difference quotient for } f(x)=x^2 \text{ at } a. \]
Expand the numerator
Why: The square of a sum.
\[ a ^{2} + 2 a h + h ^{2} - a ^{2} \]
Simplify
Why: The squared terms cancel.
\[ 2 a h + h ^{2} \]
Factor out the width
Why: Common to both terms.
\[ h(2 a + h) \]
Cancel
Why: Legitimate away from zero.
\[ 2 a + h \]
Figure (svg): A card giving the difference quotient with each part labelled, and noting that it is always indeterminate at the limit
\[ 2a+h \]
Verify: check the cancellation is the §12.2 technique
Why: The factor of h vanished in both numerator and denominator, which was the indeterminate form's shared factor. Cancelling it is exactly the factoring technique, applied to a quotient that always has this structure.
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1443-1445
Prediction
Substituting zero for the width immediately.
Predict first
What does the difference quotient give?
Correct: Zero over zero, always.
Why: The numerator becomes the output at a point minus itself, and the denominator becomes zero. That happens for every function, which is why the algebra always comes before the limit.
Worked example
Now substitution works.
\[ \text{Complete the derivative of } x^2 \text{ at } a. \]
Start from the simplified quotient
Why: After cancelling.
\[ 2 a + h \]
Let the width go to zero
Why: Substitution now works.
\[ 2 a + 0 \]
State the derivative
Why: At the general point.
\[ 2 a \]
Check at a specific point
Why: At one.
\[ 2 \]
Figure (svg): The solution to Worked example take the limit shown as a ladder of expressions, one row per legal move
\[ f'(a)=2a \]
Verify: compare with the secant table
Why: The secant slopes at 1 were converging on 2, and the formula gives 2 there — matching. The numerical estimate and the algebraic result agree, which is the relationship between §12.1 and §12.2 appearing once more.
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1445-1446
Trap
\[ \frac{f(a+0)-f(a)}{0} \]
Set the width to zero at the start
Why: The limit is taken before the algebra.
That gives zero over zero with nothing simplified, which says nothing.
Simplify first, then take the limit. The algebra removes the shared factor that causes the indeterminacy.
Only after cancelling does substitution give a meaningful answer.
This is the §12.2 procedure applied to a quotient that is always indeterminate, which is why the order matters every single time.
Faded example
After factoring the numerator.
Fill in the blanks
\frachh}=2a+___ \quad(h\ne 0)
Why: The shared factor of the width cancels, removing the indeterminacy. The restriction records that the cancellation is valid away from zero, which is all a limit needs.
Sorting
Algebra before the limit.
Sort into buckets
Sort each step.
Explain it to yourself
Every derivative starts the same way.
Discussion prompt
Explain why, from the quotient's structure.
Hint: What do both parts do?
Answer:
The numerator is a difference of outputs at two points, and as the points come together that difference goes to zero.
The denominator is the distance between those points, which goes to zero by construction — that is what taking the limit means.
So both vanish and the form is zero over zero every time, for every function. A good explanation notes that this makes §12.2's techniques the standard first step of calculus rather than an occasional tool.
Section
Section 4
Concept
A derivative's sign says whether the quantity is rising or falling, and its magnitude says how fast. A derivative of zero marks a level moment.
The last row is why calculus is the tool for optimisation. A maximum or minimum of a smooth quantity occurs where the derivative vanishes, so finding those points reduces an optimisation problem to solving an equation.
| the derivative | what it means |
|---|---|
| positive | the quantity is increasing |
| negative | it is decreasing |
| large in size | changing rapidly |
| near zero | changing slowly |
| exactly zero | momentarily level: a peak, trough or flattening |
Figure (svg): Three cards giving what a derivative tells you at a point: the sign, the size, and the tangent line
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1446-1452
Picture it
Sign, size, and the zero case.
Figure (svg): Three cards giving what a derivative tells you at a point: the sign, the size, and the tangent line
The third card is the one with the widest consequences. Reducing optimisation to equation-solving is what makes calculus indispensable in economics and engineering.
Worked example
The direction of change.
\[ \text{A population's derivative is } -40 \text{ per year. What does that mean?} \]
Read the sign
Why: Negative.
Read the size
Why: Forty.
Read the units
Why: Per year.
State the meaning
Why: Falling at that rate.
\[ \text{losing } 40 a\text{ year} \]
Figure (svg): Three cards giving what a derivative tells you at a point: the sign, the size, and the tangent line
\[ \text{decreasing at }40\text{ per year} \]
Verify: note the instantaneous qualification
Why: The rate is instantaneous, so it describes the trend at that moment rather than guaranteeing forty fewer over the next year. The rate itself may change, which is why the qualification matters.
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1447-1450
Sorting
Sign and size carry different information.
Sort into buckets
Sort each derivative.
Worked example
A momentarily level point.
\[ \text{A projectile's height has derivative zero at some instant. What is happening?} \]
Read the derivative
Why: Zero.
Interpret geometrically
Why: A horizontal tangent.
Interpret physically
Why: Vertical speed zero.
Identify the point
Why: The top of the flight.
Figure (svg): The solution to Worked example a derivative of zero shown as a ladder of expressions, one row per legal move
\[ \text{maximum height} \]
Verify: check the surrounding behaviour
Why: Just before that instant the height was rising and just after it falls, so the zero marks a peak rather than a flattening. The derivative's sign either side is what distinguishes the possibilities.
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1450-1452
Error analysis
A student interprets a derivative.
Annotate
On: \( \text{the derivative is }-40\text{ per year, so there will be }40\text{ fewer next year} \)
An instantaneous rate is a snapshot, and extrapolating it over a long interval assumes it stays constant. That assumption is often reasonable over short intervals and often wrong over long ones.
Prediction
A smooth quantity has derivative zero at a point.
Predict first
What is happening there?
Correct: It is momentarily level: a peak, trough or flattening.
Why: A zero rate means the quantity is not changing at that instant, which is a horizontal tangent. The behaviour on either side distinguishes a maximum from a minimum or a flattening.
Faded example
A rate of negative forty per year.
Fill in the blanks
\textnegative40\text______\text___
Why: The sign gives the direction and the magnitude gives the rate, with the units carried from the two quantities involved. Both pieces are needed for a full interpretation.
Explain it
A derivative of zero has particular significance.
Discussion prompt
Explain why optimisation problems look for it.
Hint: What happens at a maximum?
Answer:
At a maximum the quantity stops rising and begins falling, so at that instant it is neither — the rate is zero.
The same holds at a minimum, with the directions reversed. So every peak and trough of a smooth quantity has a zero derivative.
That turns an optimisation problem into solving an equation, which is a far more tractable task than searching. A good explanation notes that this is why calculus is the standard tool in economics and engineering, where finding a best value is the recurring question.
Section
Section 5
Concept
The derivative is the first of two central ideas in calculus, and the course has now assembled everything needed to develop it systematically.
The definition computed here is impractical for most functions, which is why calculus spends its early chapters deriving rules that shortcut it. But those rules are proved from this definition, so understanding it is what makes them more than formulas to memorise.
Figure (svg): Three cards giving what a derivative tells you at a point: the sign, the size, and the tangent line
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1452-1457
Picture it
The foundation everything after this builds on.
Figure (svg): Three cards giving what a derivative tells you at a point: the sign, the size, and the tangent line
Every technique in a calculus course is a way of getting these three readings more efficiently, or of using them to answer a further question.
Worked example
The definition is impractical at scale.
\[ \text{Why does calculus develop differentiation rules rather than using the definition?} \]
Consider a simple function
Why: Squaring.
Consider a harder one
Why: A product of two complicated functions.
Note what the rules give
Why: Results without expanding.
Note where they come from
Why: Proved from the definition.
Figure (svg): Three cards giving what a derivative tells you at a point: the sign, the size, and the tangent line
\[ \text{rules shortcut the definition} \]
Verify: note what understanding the definition buys
Why: The rules are derivable rather than arbitrary, so knowing the definition makes them explicable and their conditions predictable. Learning them as formulas alone leaves no way to know when they apply.
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1453-1455
Prediction
The derivative has been defined.
Predict first
What do the following chapters develop?
Correct: Rules that compute derivatives without the definition.
Why: The definition is impractical for complicated functions, so calculus derives shortcuts. Those rules are theorems proved from the definition, which is why understanding it makes them explicable rather than arbitrary.
Worked example
Every chapter contributed.
\[ \text{Which parts of precalculus does this section depend on?} \]
Note the function machinery
Why: Chapters 1 to 4.
Note the algebra
Why: Factoring and rationalising.
Note the limits
Why: Sections 12.1 to 12.3.
Note the geometry
Why: Slope from chapter 2.
Figure (svg): The solution to Worked example what the course prepared shown as a ladder of expressions, one row per legal move
\[ \text{most of the course} \]
Verify: consider what would be missing without them
Why: Without factoring, no difference quotient could be simplified; without limits, the definition could not be stated; without slope, the geometric reading would have no meaning. Each piece is load-bearing rather than decorative.
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1455-1457
Trap
\[ \text{the rules are what matter; the definition is just where they came from} \]
Learn the shortcuts and set the definition aside
Why: The definition is treated as historical background.
The rules' conditions and exceptions then have no explanation.
The rules are theorems proved from the definition, so their conditions come from it too.
Knowing why a rule holds is what tells you when it applies and what happens at the edges of its range.
A rule learned as a formula is a rule you cannot check, which matters as soon as an unfamiliar case appears.
Sorting
The whole course fed into it.
Sort into buckets
Sort each topic.
Faded example
Calculus rests on two.
Fill in the blanks
\textintegrationlimits\text______
Why: The derivative and the integral are the two central ideas, and both are defined as limits. That is why chapter 12 devoted three sections to limits before reaching either.
Explain it to yourself
Every chapter contributed something.
Discussion prompt
Explain what precalculus was preparing.
Hint: What does the derivative need?
Answer:
The function types of chapters 1 to 8 are what derivatives are taken of, and each has its own behaviour to understand first.
The algebra — factoring, rationalising, simplifying — is what resolves the indeterminate form every derivative produces.
And the limit of chapter 12 is the definition itself. A good explanation notes that the name precalculus is accurate: the course is the assembly of everything calculus assumes, and this section is where it is finally put to use.
Comparison
Fill the blanks from memory. One is a limit of the other.
Comparison matrix
| average rate | instantaneous rate | |
|---|---|---|
| needs | two points | one point and a limit |
| geometrically | a secant's slope | a tangent's slope |
| computed by | ordinary division | resolving an indeterminate form |
| over what interval | a positive width | the width shrunk to zero |
The third row is why chapter 12 needed its earlier sections. Every derivative is an indeterminate form, so resolving them is not an occasional technique but the standard first step.
Pattern
Five steps, and the order of the last two is the whole point.
Steps 3 to 5 are §12.2's factoring technique applied to a quotient that is indeterminate by construction. Taking the limit before cancelling gives zero over zero every time.
OpenStax Calculus Volume 1, §3.1 Defining the Derivative §3.1
Check
The definition.
Check your understanding
What is a derivative the limit of?
Answer: A
Why: The quotient is an average rate over an interval, and shrinking that interval to nothing gives the instantaneous rate. Both the tangent's slope and a physical rate are that same limit.
Check
The indeterminate form.
Check your understanding
Why is a difference quotient always indeterminate at the limit?
Answer: A
Why: As the two points come together the difference of outputs vanishes and so does the width between them. That happens for every function, which makes the algebra a standard first step rather than an occasional one.
Check
Interpretation.
Check your understanding
A smooth quantity has derivative zero at a point. What does that mark?
Answer: A
Why: A zero rate means the quantity is not changing at that instant, which is a horizontal tangent. Which of the three it is depends on the derivative's sign either side.
Real world
Marginal cost is a derivative, and it is how production decisions are made.
Discussion prompt
A firm's total cost depends on how much it produces. What does the derivative of that function tell it?
Hint: Cost per what?
Answer:
The cost of producing one more unit — the marginal cost — which is the rate at which total cost changes with output.
Comparing it with the price the unit sells for decides whether producing more is worthwhile: profitable while the price exceeds the marginal cost, and not beyond.
So the profit-maximising output is where those two are equal, which is where the profit function's derivative is zero. The optimisation reading of a zero derivative is doing the work directly, and it is the standard analysis in every economics course.
Commit first
State your confidence along with your answer.
Predict first
Why must the difference quotient be simplified before the limit is taken?
Correct: Substituting first gives zero over zero, which says nothing.
Why: Both parts of the quotient vanish as the width shrinks, so substituting immediately always produces an indeterminate form. Cancelling the shared factor of the width first is what makes the substitution meaningful.
Explain it
The test of understanding is being able to say why, not just what.
Discussion prompt
A classmate asks why the whole of chapter 12 was needed to define a slope at a point.
Hint: What does slope require?
Answer:
Slope is rise over run, which needs two points. At a single point both are zero, and zero over zero says nothing.
So the slope at a point has to be defined as what nearby secant slopes approach — which is a limit, and limits needed defining first.
And every such limit is indeterminate by construction, so the algebraic techniques of §12.2 were needed too. A good explanation notes that the three earlier sections were not preliminaries but prerequisites — the definition cannot even be stated without them.
Exit ticket
One honest answer, so the next lesson can start in the right place.
Predict first
Which idea from this lesson would you most want to see again?
Correct: Any of these is a legitimate answer; the useful one is the honest one.
Why: There is no correct choice here. The second is the observation that makes calculus so widely applicable, and the fourth is what turns a computed number into an answer about the situation it came from.
Connect it up
One page, drawn from memory, is worth more than rereading the section.
Draw it
Draw a curve with three secants closing in on a tangent and label what each slope measures. Beside it, write the difference quotient with its parts labelled and note why it is always indeterminate. Underneath, compute one derivative from the definition showing the cancellation, and write what the sign, the size and a zero each tell you.
If your computation cancels before substituting and your interpretation covers all three readings, the course's final idea is on the page in both its computational and its meaningful form.
Recap
Five things, and they are what the whole course was for.
| if you remember one thing | it should be this |
|---|---|
| about the definition | the limit of an average rate as the interval vanishes |
| about the two questions | a slope and a rate are the same limit, not an analogy |
| about computing | simplify first; the form is indeterminate by construction |
| about meaning | sign for direction, size for rate, zero for a level point |
That completes the course. Precalculus assembles the functions, the algebra and the limits that calculus assumes, and this section is where they are first put to use — differentiation and integration are what a calculus course builds from here.
OpenStax, Precalculus, §12.4 Derivatives §12.4, pp. 1432-1457 — everything on these slides traces back here
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