Practice Transfer of Ideas in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

The ability to recognize that a technique or concept from one area of mathematics applies, possibly in adapted form, to a different area.

Seeing that the same mathematical structure appears in two apparently different contexts — then using what you know about one to solve the other.

Example 1

easy
The idea of completing the square to solve x^2+6x+5=0 transfers to converting x^2+6x+5 to vertex form. Show both applications.

Example 2

medium
The AM-GM inequality \frac{a+b}{2} \ge \sqrt{ab} was originally about two positive numbers. Transfer the idea to prove: for positive reals x, y, z, x+y+z \ge 3\sqrt[3]{xyz}.

Example 3

easy
The factorisation a^2-b^2=(a-b)(a+b) transfers to factoring x^4-16. Apply it.

Example 4

medium
The proof technique 'assume the hypothesis and derive the conclusion' (direct proof) from logic transfers to proving: 'If f and g are continuous at a, then f+g is continuous at a.' Sketch the transferred argument structure.
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