IB Math Analysis and Approaches — Core flashcards

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All five topics of IB Mathematics: Analysis and Approaches (first assessment 2021): sequences, logs, functions, trigonometry, probability and calculus, with HL complex numbers, proof, vectors, Bayes and further calculus.

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Learn deck · part of IB Mathematics: Analysis and Approaches · written against: IB Diploma Programme IB Mathematics: Analysis and Approaches subject guide for first assessment 2021; SL and HL in one deck, HL-only cards under subtopic tags starting HL. Updated 2026-10-10.

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An arithmetic sequence starts at 7 and goes up by 4 each term. What is the sum of its first 20 terms?

900

Use $S_n = \frac{n}{2}\left(2u_1 + (n - 1)d\right) = \frac{20}{2}\left(14 + 19 \times 4\right) = 10 \times 90$ ⁠.
Number and Algebra › Sequences Series and Finance
In a geometric sequence $u_4 = 24$ and $u_7 = 192$ ⁠. What are the common ratio and the first term?

$r = 2$ and $u_1 = 3$

Dividing the terms gives $r^3 = \frac{192}{24} = 8$ ⁠, so $r = 2$ ⁠, and then $u_1 = \frac{24}{2^3} = 3$ ⁠.
Number and Algebra › Sequences Series and Finance
For $f(x) = 2x + 1$ and $g(x) = x^2 - 3$ ⁠, what are $(f \circ g)(x)$ and $(g \circ f)(x)$ ⁠?

$(f \circ g)(x) = 2x^2 - 5$ but $(g \circ f)(x) = 4x^2 + 4x - 2$ — $f \circ g$ means apply $g$ first, then $f$ ⁠: $2\left(x^2 - 3\right) + 1$ ⁠.

Watch out: composition is not commutative.
Functions › Functions and Graphs
What is $f^{-1}(x)$ for $f(x) = \frac{3x - 2}{x + 1}$ ⁠, and what is its domain?

$f^{-1}(x) = \frac{x + 2}{3 - x}$ ⁠, with domain $x \neq 3$

Solve $y(x + 1) = 3x - 2$ for $x$ ⁠, then swap the letters. The domain of $f^{-1}$ is the range of $f$ ⁠, which misses the horizontal asymptote $y = 3$ ⁠.
Functions › Functions and Graphs
In triangle ABC, $a = 7$ ⁠, $b = 5$ and $C = 60^\circ$ ⁠. What is $c$ ⁠?

$c = \sqrt{39} \approx 6.24$

Two sides and the included angle call for the cosine rule: $c^2 = 49 + 25 - 2 \times 7 \times 5 \times \cos 60^\circ = 39$ ⁠.
Geometry and Trigonometry › Triangles Circles and Trigonometry
When can the sine rule give two different triangles from the same data?

In the ambiguous case: two sides and a non-included acute angle are given, and the side opposite that angle is shorter than the other given side (but long enough to reach). Both $B$ and $180^\circ - B$ then work.

Example: $A = 30^\circ$ ⁠, $a = 6$ ⁠, $b = 10$ gives $\sin B = \frac{5}{6}$ ⁠, so $B \approx 56.4^\circ$ or $123.6^\circ$ ⁠.
Geometry and Trigonometry › Triangles Circles and Trigonometry
When should you use the regression line of $x$ on $y$ rather than the line of $y$ on $x$ ⁠?

When you want to predict $x$ from a given value of $y$ ⁠. The two least-squares lines minimise different residuals (horizontal versus vertical) and coincide only when $|r| = 1$ ⁠.

Watch out: rearranging the $y$ -on- $x$ equation to find $x$ gives a different, unjustified estimate.
Statistics and Probability › Statistics and Probability
A discrete random variable has $P(X = 0) = 0.2$ ⁠, $P(X = 1) = 0.5$ and $P(X = 2) = k$ ⁠. What are $k$ and $E(X)$ ⁠?

$k = 0.3$ and $E(X) = 1.1$ — probabilities sum to 1, then $E(X) = \sum x P(X = x) = 0 + 0.5 + 0.6$ ⁠.

Why: a game is called fair when the expected gain is exactly zero.
Statistics and Probability › Statistics and Probability
What is the equation of the normal to $y = x^2 - 3x$ at $x = 1$ ⁠?

$y = x - 3$

At $x = 1$ ⁠, $y = -2$ and $\frac{dy}{dx} = 2x - 3 = -1$ ⁠; the normal's gradient is the negative reciprocal, 1, so $y + 2 = 1(x - 1)$ ⁠.
Calculus › Differentiation
How do you differentiate $y = e^{3x}\sin 2x$ ⁠?

$\frac{dy}{dx} = 3e^{3x}\sin 2x + 2e^{3x}\cos 2x$ — product rule, with the chain rule supplying the 3 and the 2.

Example: factorised, $e^{3x}\left(3\sin 2x + 2\cos 2x\right)$ ⁠.
Calculus › Differentiation

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About these cards

Review status: Second-reader reviewed, with corrections applied.

Checked against:

IB Diploma Programme subject guide (topic outline only; no text copied): Mathematics: analysis and approaches guide for first assessment 2021, topics 1–5 (SL and AHL).
OpenStax Calculus Volumes 1–3, OpenStax Precalculus 2e, OpenStax Algebra and Trigonometry 2e, OpenStax Introductory Statistics and OpenStax Introductory Statistics 2e, consulted as references; no text copied.
Also checked against the IB Diploma Programme Mathematics: applications and interpretation guide for first assessment 2021 (for the term “principal axis”) and the Statistics Poland glossary of terms used in official statistics (reference only; no text copied).

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