Practice Analogical Reasoning in Math

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

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Quick Recap

Drawing conclusions about a new situation by recognizing its structural similarity to a better-understood situation.

This is like that, so maybe what works there will work here.

Showing a random 20 of 50 problems.

Example 1

medium

A derivative is like a speedometer (instantaneous rate). For position

s(t)=t^2

, the 'speedometer' reading at

t=4

s'(4)

. Compute it.

Example 2

hard

By analogy with arithmetic mean and geometric mean of two numbers, identify the harmonic mean.

Example 3

medium

Even/odd is to integers as ___ is to functions.

Example 4

easy

Derivative is to slope as integral is to ___?

Example 5

easy

The analogy between sets and logic:

A \cup B

(union) corresponds to

p \lor q

(OR), and

A \cap B

(intersection) corresponds to

p \land q

(AND). Use this analogy to conjecture a set version of De Morgan's law

\neg(p \lor q) \equiv \neg p \land \neg q

Example 6

hard

Vector space is to basis as field is to ___? Give the most direct analog.

Example 7

challenge

Analogy is not proof: two systems behaving alike does not transfer theorems. But a valid structural analogy: complex multiplication adds angles. Multiplying numbers at angles

30°

and

40°

gives a result at what angle?

Example 8

challenge

Map-is-not-the-territory: a model approximates reality. Linear approximation of

f(x)=x^2

near

x=3

uses the tangent line

y=9+6(x-3)

. Estimate

f(3.1)

with it.

Example 9

easy

Map is to territory as model is to ___? Both are simplified stand-ins. A model represents reality. If a map scale is

1:1000

, a

5

cm map distance is how many cm in reality?

Example 10

medium

Sum is to integral as product is to ___? Express the analogous operator name and write the symbol.

Example 11

medium

The Pythagorean theorem

a^2+b^2=c^2

for right triangles in the plane has a 3D analog for the diagonal of a box with edges

a,b,c

. State it.

Example 12

easy

Logarithm is to multiplication as ___ is to addition: an inverse-flattening tool. Since logs turn products to sums,

\log_{10}(10\times100)

equals which sum's value?

Example 13

medium

In arithmetic, the number

0

is the identity for addition (

a+0=a

). By analogy, identify the identity element for set union and for set intersection.

Example 14

hard

Use the analogy: rate is to distance as marginal cost is to ___? Use a one-sentence justification.

Example 15

medium

Arithmetic has addition and multiplication. By analogy, what operations does set theory have, and what arithmetic laws transfer? Identify two laws that hold and one that does not.

Example 16

medium

Probability is like a slice of a pie (fraction of the whole). If an event covers

90°

of a

360°

pie, its probability is?

Example 17

easy

Circle is to

\pi

as ___? The circle's circumference-to-diameter ratio is

\pi

. To

1

decimal, give

\pi

Example 18

medium

Numerator is to a fraction as dividend is to division. In

\frac{12}{4}

, the numerator

12

plays the dividend role; give the quotient.

Example 19

easy

Equation is to equals sign as inequality is to ___?

Example 20

easy

In arithmetic, subtraction undoes addition. In linear algebra, what undoes adding a vector

\vec v

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Related Concepts

Transfer of Ideas