mathematics

Increment theorem

Increment theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Increment theorem rather than just read about it. In short: In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function y = f(x) is differentiable at x and that Δx is infinitesimal. Then Δ y = f ′ ( x ) Δ x + ε Δ x {\displaystyle \Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x} for some infinitesimal ε, where Δ y = f ( x + Δ x ) − f ( x ) . {\displaystyle \Delta y=f(x+\Delta x)-f(x).} If Δ x ≠ 0 {\textstyle \Delta x\neq 0}…

Key takeaways

Increment theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Increment theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Increment theorem from memory before moving on to harder problems.

Reference excerpt

In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function y = f(x) is differentiable at x and that Δx is infinitesimal. Then

Δ y = f ′ ( x ) Δ x + ε Δ x {\displaystyle \Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x}

for some infinitesimal ε, where

Δ y = f ( x + Δ x ) − f ( x ) . {\displaystyle \Delta y=f(x+\Delta x)-f(x).}

If Δ x ≠ 0 {\textstyle \Delta x\neq 0} then we may write

Δ y Δ x = f ′ ( x ) + ε , {\displaystyle {\frac {\Delta y}{\Delta x}}=f'(x)+\varepsilon ,}

which implies that Δ y Δ x ≈ f ′ ( x ) {\textstyle {\frac {\Delta y}{\Delta x}}\approx f'(x)} , or in other words that Δ y Δ x {\textstyle {\frac {\Delta y}{\Delta x}}} is infinitely close to f ′ ( x ) {\textstyle f'(x)} , or f ′ ( x ) {\textstyle f'(x)} is the standard part of Δ y Δ x {\textstyle {\frac {\Delta y}{\Delta x}}} . A similar theorem exists in standard Calculus. Again assume that y = f(x) is differentiable, but now let Δx be a nonzero standard real number. Then the same equation

Δ y = f ′ ( x ) Δ x + ε Δ x {\displaystyle \Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x}

holds with the same definition of Δy, but instead of ε being infinitesimal, we have

lim Δ x → 0 ε = 0 {\displaystyle \lim _{\Delta x\to 0}\varepsilon =0}

(treating x and f as given so that ε is a function of Δx alone).

See also Nonstandard calculus Elementary Calculus: An Infinitesimal Approach Abraham Robinson Taylor's theorem

References Howard Jerome Keisler: Elementary Calculus: An Infinitesimal Approach. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html Robinson, Abraham (1996). Non-standard analysis (Revised ed.). Princeton University Press. ISBN 0-691-04490-2.

Worked examples

Example 1 — a first encounter with Increment theorem

Start with the simplest possible case. Write down what Increment theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Increment theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Increment theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Increment theorem

In research: Increment theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Increment theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Increment theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Nonstandard analysis, Theorems in calculus, so understanding it makes those chapters shorter.
In everyday life: Look for Increment theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Increment theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Increment theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Increment theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Increment theorem in simple terms?

In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function y = f(x) is differentiable at x and that Δx is infinitesimal. Then Δ y = f ′ ( x ) Δ x + ε Δ x {\displaystyle \Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x} for some infinitesimal ε, wh…

Why does Increment theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Increment theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Increment theorem.