Differential Calculus for Social Science 201-103 • CEGEP, Montreal

Revision sheet: differentials and linear approximation (201-103)

Q: What is the difference between the differential and the actual change?

The differential is the change along the tangent, that is, a forecast made by assuming the slope stays constant. The actual change is the change along the curve, that is, what really happens. The two are almost equal over a small step and part ways faster the larger the step and the more curved the curve.

Q: How do I know whether a linear approximation overestimates or underestimates?

Look at the sign of the second derivative. If it is positive, the curve is convex, the tangent lies below, and the approximation underestimates. If it is negative, the curve is concave, the tangent lies above, and the approximation overestimates. This check needs no extra calculation and is often worth a mark.

Q: Why do uncertainties add up even for a quotient?

Because an uncertainty has no known sign. A change does, which is why it subtracts in a quotient. With uncertainties, nothing guarantees that the two errors go in opposite directions, so you keep the worst case and add the absolute values.

Q: How can a pay raise be a pay cut?

When it is smaller than inflation. The real wage is the nominal wage divided by the price index, so its rate of change is the difference of the two rates. A wage that rises by two percent while prices rise by three percent amounts to a loss of purchasing power of about one percent.

Q: When is a change significant?

When it exceeds the margin of error of the indicator. If an unemployment rate falls by two tenths of a point while the margin is half a point, the drop is not significant and the only honest conclusion is that nothing can be concluded. Every published figure should come with its margin.

Q: On which variable should the measurement effort be invested?

On the one whose exponent times relative precision is the largest, not the one whose value is the largest. A variable raised to the third power triples its relative error in the result, so measuring it twice as well pays three times more than improving a variable that only enters to the first power.

This is the most concrete chapter of the course: it answers the question everyone asks in front of a published figure, namely how far it can be taken seriously. An unemployment rate that moves by $0.2$ points with a margin of $0.5$ points does not move, and saying so earns marks.

This sheet gives the differential and the linearization, the error they make and its sign, the rules for propagating uncertainties, and the nine mistakes that 201-103 papers repeat every term.

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The thread of the chapter

The differential travels along the TANGENT, reality travels along the CURVE: the whole chapter consists of knowing by how much the two part ways, and from what step that gap stops being negligible.

This chapter is part of 201-103, Differential Calculus for Social Science

The essentials

Two ways forward: the tangent and the curve

• $dy=f'(a)\,dx$ is the change along the TANGENT. It is a forecast.
• $\Delta y=f(a+dx)-f(a)$ is the actual change, along the CURVE. It is the truth.
• LINEARIZATION: $L(x)=f(a)+f'(a)(x-a)$ . It is the equation of the tangent line, and nothing else.
• The error made is approximately $\dfrac{1}{2}\,|f''(a)|\,(\Delta x)^{2}$ : it grows like the SQUARE of the step, so doubling the step quadruples the error.
• A linearization sees no maximum, no ceiling and no turning point: curvature is exactly what it ignores.

The two round points on the right sit at the same abscissa: the higher one is

f(a+dx)

on the curve, the lower one is the forecast

f(a)+dy

on the tangent.

Over a small step, $dy$ and $\Delta y$ are almost equal and the approximation is excellent. The word “small” only makes sense relative to the curvature, never in absolute terms.

The sign of $f''$ says in advance which way you are wrong

• $f''>0$ , convex curve: the tangent lies BELOW the curve, so $dy$ UNDERESTIMATES the actual change.
• $f''<0$ , concave curve: the tangent lies ABOVE, so $dy$ OVERESTIMATES.
• This is not a theoretical remark: it lets you announce the direction of the error without computing the exact value.
• In context, a convex cost function always makes you underestimate the extra cost, and diminishing returns always make you overestimate the gain.

The direction of the error fits in one sentence and is often worth a whole mark: “since $f''<0$ , this estimate is an upper bound on the actual increase”.

How errors propagate

• Product: $\dfrac{dR}{R}=\dfrac{dp}{p}+\dfrac{dq}{q}$ . Relative errors ADD.
• Power: a power MULTIPLIES the relative error by its exponent. A $1$ percent error on $r$ makes $3$ percent on $r^{3}$ .
• Quotient: CHANGES subtract, but UNCERTAINTIES add in absolute value. The worst case never cancels out.
• Margin on a rate $\dfrac{U}{A}$ : $|du|\le\dfrac{|dU|}{A}+\dfrac{U\,|dA|}{A^{2}}$ .
• NEVER comment on a change smaller than the margin of error, and every published rate goes with its margin.

When

p

and

q

increase slightly, the area

R=pq

gains the two strips

q\,dp

and

p\,dq

; the corner

dp\,dq

is negligible. Divided by

R

\frac{dR}{R}=\frac{dp}{p}+\frac{dq}{q}

, relative errors add.

The measurement effort goes to the variable whose exponent times relative precision is the largest, never to the one whose value is the largest.

The rules in table form

Each row reads from left to right: the assumptions, then the result. A red cell is not an answer, it is the finding that the form settles nothing and the instruction to rewrite it. Every case is followed by a worked example.

Which formula, according to the form of the indicator

The last column says what happens to the errors. The red case is the cancellation everyone hopes for and that never happens.

Indicator	What you write	What happens to the errors
$R=pq$	$\dfrac{dp}{p}+\dfrac{dq}{q}$	they ADD
Example: $2$ percent on the price and $3$ percent on the quantity give $5$ percent on revenue.
$W=\dfrac{S}{I}$ , changes	$\dfrac{S'}{S}-\dfrac{I'}{I}$	they SUBTRACT
Example: A wage that rises by $2$ percent with inflation at $3$ percent gives $-1$ percent real wage.
$W=\dfrac{S}{I}$ , uncertainties	$\left\|\dfrac{dS}{S}\right\|+\left\|\dfrac{dI}{I}\right\|$	they ADD
Example: Margins of $0.3$ and $0.4$ percent give a margin of $0.7$ percent on the real wage.
$A=r^{3}$	$3\,\dfrac{dr}{r}$	the exponent MULTIPLIES
Example: A $1$ percent error on $r$ makes $3$ percent on $A$ .
linearization error	$\dfrac{1}{2}\|f''(a)\|(\Delta x)^{2}$	it grows like the SQUARE
Example: With $\|f''\|=0.4$ : a step of $0.5$ gives $0.05$ , a step of $1$ gives $0.20$ .
uncertainties of a quotient, subtracted	the cancellation	does not exist rule that does not exist
Example: With $0.3$ and $0.4$ percent, the true margin is $0.7$ percent, not $0.1$ percent. What to do: Add the absolute values: an uncertainty has no known sign, so nothing guarantees that it cancels.

Rows 2 and 3 are about the SAME indicator and give opposite rules. This is the most rewarding distinction of the chapter: a change has a sign, an uncertainty does not.

The mistakes that cost marks

These are the errors I correct most often in session. Each one costs marks on a paper, even when the reasoning behind it is right.

1. Confusing the differential with the actual change

1.5 marks, and a forecast that is wrong in the wrong direction

What not to write

“ $dy=f'(a)\,dx$ , so the function increases by exactly $2.7$ .”

What to write

“ $dy=2.7$ is the change along the TANGENT. The actual change is $\Delta y=f(6)-f(3)=4.05$ : the approximation underestimates by $1.35$ .”

From

x=3

x=6

: the tangent rises by

dy=2.7

(orange), the curve by

\Delta y=4.05

(green). The differential is the forecast along the tangent, not the actual change.

Why: The differential is a linear forecast, not a result. The two almost coincide over a small step and diverge fast over a large one.

2. Not saying which way the approximation is wrong

1 mark, systematically asked in reasoning questions

What not to write

“The linear approximation gives $51{,}480$ dollars, roughly the right value.”

What to write

“Since $f''<0$ on this interval, the tangent lies above the curve: $51{,}480$ dollars is an OVERESTIMATE of the actual value.”

On the left the curve is convex and the line stays below; on the right it is concave and the line passes above. The sign of

f''

is therefore enough to announce the direction of the error.

Why: The sign of the second derivative gives the direction of the error for free, with no extra calculation. Not using it means leaving a mark on the table.

3. Extending a linearization too far

2 marks of interpretation, and an absurd percentage as an answer

What not to write

“The model grows by $2$ points per year, so in ten years the adoption rate will reach $125$ percent.”

What to write

“A linearization is local. The model has a ceiling at $100$ percent, which the tangent ignores completely: beyond two or three years, the forecast no longer makes sense.”

The two curves start from the same point and part ways very fast: the curve flattens under the dotted line, the straight line crosses it without noticing.

Why: A line has no ceiling and no turning point. Curvature is exactly the information a linearization throws away, so everything curvature decides escapes it.

4. Subtracting the uncertainties of a quotient

2 marks, and a conclusion presented as certain when it is not

What not to write

“The wage is known to within $0.3$ percent and the index to within $0.4$ percent, so the real wage is known to within $0.1$ percent.”

What to write

“Uncertainties ADD in absolute value: the margin on the real wage is $0.3+0.4=0.7$ percent.”

The two errors can go in the same direction: the real wage is known to within

0.7

percent (green), not

0.1

(red). An uncertainty never cancels, it adds up.

Why: A change has a known sign, an uncertainty does not. Nothing guarantees that the two errors go in opposite directions, so you keep the worst case.

5. Forgetting that the exponent multiplies the relative error

1.5 marks, and a margin announced twice too small

What not to write

“The radius of the urbanized area is known to within $2$ percent, so its area is too.”

What to write

“The area varies as the SQUARE of the radius, so its relative error is $2\times 2=4$ percent.”

A radius slightly too large adds a whole ring:

dA\approx 2\pi r\,dr

, so

\frac{dA}{A}=2\,\frac{dr}{r}

. Two percent of error on

r

makes four on the area.

Why: Differentiating $\ln A=2\ln r$ gives $\dfrac{dA}{A}=2\,\dfrac{dr}{r}$ . The exponent comes out as a factor, and it does so for a fractional or negative exponent too.

6. Commenting on a change smaller than the margin

2 marks of interpretation, and it is the commentary question you will be asked

What not to write

“The unemployment rate went from $5.4$ to $5.2$ percent: the labour market is improving.”

What to write

“The drop of $0.2$ points is smaller than the margin of $0.5$ points: this change is not significant, and nothing can be concluded from it.”

The drop of

0.2

points (red) falls inside the margin of

\pm 0.5

points (orange): it cannot be told apart from zero, and nothing can be concluded.

Why: This is the method point specific to Social Science. A figure without its margin is not a result, and a change under the margin is not a change.

7. Putting the measurement effort on the biggest value

2 marks, and an inefficient data collection plan

What not to write

“The population is the biggest number in the calculation, so it is the one to measure most precisely.”

What to write

“Compare the products exponent times relative precision: the variable whose product is the largest deserves the effort, whatever its size.”

Why: In the total relative error, only the contribution of each variable counts, that is, its exponent times its own relative precision. The raw value appears nowhere.

8. Confusing a nominal raise with a real raise

2 marks, and a conclusion with the wrong sign

What not to write

“The wage rose by $2$ percent, so purchasing power did too.”

What to write

“With inflation at $3$ percent, the REAL wage changes by $2-3=-1$ percent: purchasing power falls despite the nominal raise.”

Why: The real wage is a quotient, $W=\dfrac{100\,S}{I}$ , so its relative rate is the DIFFERENCE of the two. A raise below inflation is a cut.

9. Treating linearization as a new formula to memorize

no marks directly, but one more formula to remember and to mix up

What not to write

“I have to learn the linearization formula on top of the tangent formula.”

What to write

“ $L(x)=f(a)+f'(a)(x-a)$ IS the equation of the tangent line. One object, two names depending on how you use it.”

Why: The confusion costs time in the exam and pushes you to look for a formula where it is enough to write the tangent line, which you have already seen in the derivative chapter.

Which method to choose

Which formula to write, according to what the question asks

Only four questions in this chapter, and each has its formula. The verb of the question tells them apart without ambiguity.

If “estimate”, “approximate”, “without a calculator” → linearization $L(x)=f(a)+f'(a)(x-a)$ , with a nearby, convenient $a$

Example: estimate $\sqrt{101}$ from $a=100$
If “by how much does it change”, “what effect would a rise of ... have” → differential $dy=f'(a)\,dx$

Example: $dy=0.9\times 3=2.7$
If “what error is made”, “is the approximation good” → $\dfrac{1}{2}|f''(a)|(\Delta x)^{2}$ , then compare with the order of magnitude of the result

Example: $0.5\times 0.4\times 1^{2}=0.20$
If “with what margin”, “can we conclude that” → propagation of uncertainty, with ABSOLUTE VALUES added

Example: $0.3+0.4=0.7$ percent
If “what precision must be required on” → inverse problem: share the target among the variables, each weighted by its exponent

Example: for $1$ percent on $r^{3}$ , you need $0.33$ percent on $r$

this is the hardest question of the chapter, and it often comes last

The first two branches give the same calculation and differ in presentation: one answers with a value, the other with a change. Reading the verb keeps you from answering a different question.

Is this approximation acceptable?

Three checks, to be done in this order, before handing in an approximate value.

If the step is small compared with the scale of the phenomenon → the approximation is good, and the error is of order $(\Delta x)^{2}$

Example: forecasting one year from a model fitted over ten
If the step is large, or the curvature strong → the error becomes visible: put a number on it before concluding

Example: doubling the step quadruples the error
If the function has a ceiling, a maximum or a turning point in the interval → the linearization is UNUSABLE, whatever the size of the step

Example: a logistic curve extended past its inflection point
If the predicted change is smaller than the margin of error → conclude nothing, and SAY so explicitly

Example: a drop of $0.2$ points for a margin of $0.5$
If the question asks for bounds rather than a value → give the value AND the direction of the error, taken from the sign of $f''$

Example: “overestimate, since $f''<0$ ”

An approximation handed in without its domain of validity loses half its marks in Social Science, even when the number is right.

How the answer is expected to be written

A marker ticks steps. Here they are in order, with the concluding sentence expected word for word.

Writing up a linear approximation

When to use it: As soon as the question asks you to estimate a nearby value, to predict an effect, or to calculate without a machine.

1 Choose the point $a$ : close to the requested value, and convenient to compute.
2 Compute $f(a)$ and $f'(a)$ , saying so.
3 Write the linearization $L(x)=f(a)+f'(a)(x-a)$ , naming it as the tangent line.
4 Substitute and give the approximate value with its units.
5 Give the DIRECTION of the error from the sign of $f''(a)$ , and an idea of its size.

Concluding sentence

“Take $a=3$ . Then $f(3)=2.35$ and $f'(3)=0.9$ , so $L(x)=2.35+0.9(x-3)$ and $L(6)=5.05$ . Since $f''>0$ , the tangent lies below the curve: $5.05$ UNDERESTIMATES the actual value, which is $6.40$ .”

The trap: Giving the approximate value without saying which way it is wrong. The sign of $f''$ is already known and the sentence takes ten words.

Marking: 1 mark for the choice of $a$ , 1 mark for $f(a)$ and $f'(a)$ , 1 mark for the linearization, 1 mark for the direction of the error.

Writing up a margin on an indicator

When to use it: As soon as the question gives measurement precisions, or asks whether an observed change is significant.

1 Write the indicator as a product or a quotient, and say which.
2 Write the relation between RELATIVE errors, with absolute values added, whatever the operation.
3 Substitute the given precisions and obtain the relative margin.
4 Convert to an absolute margin if the question asks, by multiplying by the value of the indicator.
5 Compare the observed change with the margin, and conclude explicitly on significance.

Concluding sentence

“The real wage is the quotient $W=\dfrac{100\,S}{I}$ . Its uncertainties add: $\left|\dfrac{dW}{W}\right|\le 0.3+0.4=0.7$ percent. The computed drop of $1$ percent exceeds this margin: it is significant.”

The trap: Subtracting the uncertainties because the indicator is a quotient. CHANGES subtract, UNCERTAINTIES add, and that is exactly what the question tests.

Marking: 1 mark for the form of the indicator, 1 mark for the addition rule, 1 mark for the margin, 1 mark for the conclusion on significance.

Check before you hand in

Five minutes of checking recover more marks than one more problem started in a hurry.

Compare the forecast with the exact value

When the calculator can give the exact value, compute it and compare. The gap must go in the direction announced by the sign of $f''$ .

$L(6)=5.05$ against $f(6)=6.40$ : the approximation underestimates, which is consistent with $f''>0$ . An overestimate would signal a sign error.
The order of magnitude of the error

Compute $\dfrac{1}{2}|f''|(\Delta x)^{2}$ and check that the observed gap is of the same order. A gap ten times larger signals an error in the derivative.

With $|f''|=0.3$ and $\Delta x=3$ : $0.5\times 0.3\times 9=1.35$ , exactly the observed gap.
The margin must grow, never shrink

After combining several measurements, check that the margin you obtain is larger than each of the starting margins. A margin that shrinks signals a subtraction.

$0.3$ and $0.4$ give $0.7$ , which is indeed larger than both. Getting $0.1$ signals a cancellation that does not exist.
The sign of the actual change

On a quotient, reread the comparison of the two rates and check that the sign you got matches. A raise below inflation always gives a negative result.

$2$ percent against $3$ percent inflation: the real wage must fall. A positive result would be inconsistent.
Significance, before any sentence

Before writing the slightest interpretation, compare the change with the margin. If it is smaller, the only correct sentence is that nothing can be concluded.

A drop of $0.2$ points for a margin of $0.5$ points cannot be commented on, even if the trend looks clear.

The typical problem, taken apart

Nominal wage, price index and purchasing power

An annual wage is $52{,}000$ dollars and rises by $2.0$ percent. Over the same period, the price index goes from $100$ to $103$ .

The real wage is $W=\dfrac{100\,S}{I}$ . Compute its relative change by approximation and then exactly, and say whether the conclusion holds up against margins of $0.3$ percent on the wage and $0.4$ percent on the index.

Step 1

The real wage is a QUOTIENT: $W=\dfrac{100\,S}{I}$ , where $S$ is the nominal wage and $I$ the price index.

Why

Naming the form of the indicator decides everything that follows: for a quotient, changes subtract and uncertainties add.

Step 2

Changes: $\dfrac{dW}{W}=\dfrac{dS}{S}-\dfrac{dI}{I}=2.0-3.0=-1.0$ percent.

Why

The rule for relative rates gives the answer in one line, without ever computing the value of $W$ . This is the expected route in Social Science.

Step 3

Exact value: $S$ becomes $52{,}000\times 1.02=53{,}040$ dollars, and $W$ goes from $52{,}000$ to $\dfrac{100\times 53{,}040}{103}\approx 51{,}495$ dollars.

Why

The exact calculation serves as a check. It is not always asked for, but it is always possible when both values are given.

Step 4

Exact change: $\dfrac{51{,}495-52{,}000}{52{,}000}\approx -0.97$ percent, against $-1.00$ percent by approximation.

Why

The gap of $0.03$ points is the linearization error. It is small because the changes are small: this is exactly the validity condition of the method.

Step 5

Direction of the error: the approximation announces a slightly larger drop than reality, so it OVERESTIMATES the loss of purchasing power.

Why

Saying the direction of the gap, and not only its size, is what separates a complete answer from a number dropped on the page.

Step 6

Margins: these are UNCERTAINTIES, so they add in absolute value. $\left|\dfrac{dW}{W}\right|\le 0.3+0.4=0.7$ percent.

Why

This is the step where papers go wrong: the quotient subtracted the changes two lines above, and it now adds the uncertainties. Both rules coexist.

Step 7

Comparison: the drop of $1.0$ percent exceeds the margin of $0.7$ percent.

Why

Without this comparison, the conclusion has no value. With it, it becomes a defensible statement.

Step 8

Conclusion in words: purchasing power has indeed fallen, by about $1$ percent, and this drop is significant given the margins.

Why

The final sentence must contain the direction, the order of magnitude and the significance. Each is worth marks on its own.

The conclusion, written out

“ $\dfrac{dW}{W}=2.0-3.0=-1.0$ percent, and the exact calculation gives $-0.97$ percent. Since the combined margin is $0.3+0.4=0.7$ percent, smaller than the observed drop, we can conclude: despite a nominal raise of $2$ percent, purchasing power has fallen by about $1$ percent.”

The classic mistake on this problem: Subtracting the margins because the indicator is a quotient. You would get $0.1$ percent, a precision seven times too optimistic, and the conclusion would be presented as far more solid than it is.

Learn by heart

• $dy=f'(a)\,dx$ follows the TANGENT; $\Delta y=f(a+dx)-f(a)$ follows the CURVE.
• $L(x)=f(a)+f'(a)(x-a)$ IS the equation of the tangent line, it is not one more formula.
• The linearization error is approximately $\dfrac{1}{2}|f''(a)|(\Delta x)^{2}$ : it grows like the SQUARE of the step.
• $f''>0$ means you UNDERestimate, $f''<0$ means you OVERestimate. The sign is enough, no calculation needed.
• Relative errors of a product: they ADD. A power MULTIPLIES them by its exponent.
• For a quotient, CHANGES subtract but UNCERTAINTIES add.
• Never comment on a change smaller than the margin of error.
• The measurement effort goes to the largest product of exponent times relative precision, never to the biggest value.

Frequently asked questions

What is the difference between the differential and the actual change?

The differential is the change along the tangent, that is, a forecast made by assuming the slope stays constant. The actual change is the change along the curve, that is, what really happens. The two are almost equal over a small step and part ways faster the larger the step and the more curved the curve.

How do I know whether a linear approximation overestimates or underestimates?

Look at the sign of the second derivative. If it is positive, the curve is convex, the tangent lies below, and the approximation underestimates. If it is negative, the curve is concave, the tangent lies above, and the approximation overestimates. This check needs no extra calculation and is often worth a mark.

Why do uncertainties add up even for a quotient?

Because an uncertainty has no known sign. A change does, which is why it subtracts in a quotient. With uncertainties, nothing guarantees that the two errors go in opposite directions, so you keep the worst case and add the absolute values.

How can a pay raise be a pay cut?

When it is smaller than inflation. The real wage is the nominal wage divided by the price index, so its rate of change is the difference of the two rates. A wage that rises by two percent while prices rise by three percent amounts to a loss of purchasing power of about one percent.

When is a change significant?

When it exceeds the margin of error of the indicator. If an unemployment rate falls by two tenths of a point while the margin is half a point, the drop is not significant and the only honest conclusion is that nothing can be concluded. Every published figure should come with its margin.

On which variable should the measurement effort be invested?

On the one whose exponent times relative precision is the largest, not the one whose value is the largest. A variable raised to the third power triples its relative error in the result, so measuring it twice as well pays three times more than improving a variable that only enters to the first power.

Practise it

Corrected exercises: Differentials and linear approximation, 201-103

A method is proved on a paper, not on a sheet. The set for the same chapter takes each of these traps into a problem, with the solution written out step by step.

11 corrected exercises
110 points
150 minutes

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Revision sheet: differentials and linear approximation (201-103)

The essentials

Two ways forward: the tangent and the curve

The sign of f′′f''f′′ says in advance which way you are wrong

How errors propagate

The rules in table form

Which formula, according to the form of the indicator

The mistakes that cost marks

1. Confusing the differential with the actual change

2. Not saying which way the approximation is wrong

3. Extending a linearization too far

4. Subtracting the uncertainties of a quotient

5. Forgetting that the exponent multiplies the relative error

6. Commenting on a change smaller than the margin

7. Putting the measurement effort on the biggest value

8. Confusing a nominal raise with a real raise

9. Treating linearization as a new formula to memorize

Which method to choose

Which formula to write, according to what the question asks

Is this approximation acceptable?

How the answer is expected to be written

Writing up a linear approximation

Writing up a margin on an indicator

Check before you hand in

The typical problem, taken apart

Nominal wage, price index and purchasing power

Learn by heart

Frequently asked questions

What is the difference between the differential and the actual change?

How do I know whether a linear approximation overestimates or underestimates?

Why do uncertainties add up even for a quotient?

How can a pay raise be a pay cut?

When is a change significant?

On which variable should the measurement effort be invested?

Corrected exercises: Differentials and linear approximation, 201-103

See also

Looking for a 201-103 tutor in Montreal?

The sign of $f''$ says in advance which way you are wrong