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Lesson 40 — Year 1 Consolidation: Integration Workshop

Final workshop for Year 1. Problems combining functions, trigonometry, analytic geometry, vectors, matrices, combinatorics, and probability.

Used in: Capstone 1.º ano EM · Equiv. Math I+II japonês revisão · Equiv. Abitur-Vorbereitung alemão

\text{Functions} + \text{Trig} + \text{Analytic Geo} + \text{Matrices} + \text{Combinatorics} + \text{Probability}

Choose your door

Rigorous notation, full derivation, hypotheses

Year 1 synthesis

You have completed 40 lessons covering the foundations: functions, rate of change, trigonometry, sequences, intuitive limit, analytic geometry, vectors, linear systems, matrices, determinants, combinatorics, and probability.

What you know now

Rigorous mathematical language: sets, intervals, notation.
Modeling with functions: linear, quadratic, exponential, logarithmic, trigonometric.
Pre-calculus: average rate of change, intuitive limit of a sequence — ready for the formal calculus of Term 5-6.
Analytic geometry: algebra-geometry connection via Descartes.
Introductory linear algebra: vectors, matrices, systems, determinants.
Combinatorics + probability: rigorous counting, Bayes.

What's coming in Year 2

Term 5: formal limits ( $\eps$ - $\delta$ ).
Term 6: derivatives and rules (chain, product, quotient).
Term 7: derivative applications (optimization, related rates).
Term 8: descriptive statistics and deeper probability.

How to approach this workshop

Reserve 8 closed-book hours to solve the 40 problems. Check against the answer key (~25% have worked solutions). If you get < 50%, re-read the corresponding lessons; between 70-90%, you're ready for Year 2; above 90%, read complementary material from the reference books.

Map by term

Term	Main topics	Lessons
1	Sets, intervals, basic functions	1-10
2	Trigonometry, sequences, intuitive limit	11-20
3	Analytic geometry, conics, vectors	21-30
4	Matrices, combinatorics, probability	31-40

Exercise list

40 exercises · 10 with worked solution (25%)

Application 25Modeling 12Challenge 2Proof 1

Ex. 40.1Application
Maximum domain of $f(x) = \log_2(x^2 - 9)$ . (Ans: $x < -3$ or $x > 3$ .)
Solve online
Ex. 40.2Application
Compute the average rate of change of $f(x) = x^2 - 3x$ on $[1, 4]$ . (Ans: $2$ .)
Solve online
Ex. 40.3ApplicationAnswer key
Composition: $f(x) = x^2$ , $g(x) = 2x + 1$ . $(f \circ g)(2)$ and $(g \circ f)(2)$ . (Ans: $25$ and $9$ .)
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Ex. 40.4Application
Inverse of $f(x) = (x - 3)/2$ . (Ans: $f^{-1}(x) = 2x + 3$ .)
Solve online
Ex. 40.5Application
Solve $4^x = 32$ . (Ans: $x = 5/2$ .)
Solve online
Ex. 40.6ModelingAnswer key
Bacteria double every 30 min. 100 initially. After 3h?
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Ex. 40.7Modeling
Investment of $1,000 at 10% p.a. compounded. After 7 years?
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Ex. 40.8Application
$\lim_{n \to \infty} (3n + 5)/(n + 1) = ?$ (Ans: $3$ .)
Solve online
Ex. 40.9Application
Quadratic $f(x) = x^2 - 6x + 5$ — vertex and roots.
Solve online
Ex. 40.10Modeling
Cost function $C(q) = 200 + 8q + 0.1 q^2$ . Average cost at $q = 50$ ?
Solve online
Ex. 40.11ApplicationAnswer key
Compute $\sin(\pi/4) + \cos(\pi/3)$ . (Ans: $\frac{\sqrt 2}{2} + \frac{1}{2}$ .)
Solve online
Ex. 40.12Application
Solve $\sin x = \sqrt 3/2$ on $[0, 2\pi)$ . (Ans: $\pi/3$ and $2\pi/3$ .)
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Ex. 40.13Application
Triangle $a = 5, b = 7, C = 60°$ — compute $c$ (law of cosines). (Ans: $c = \sqrt{39}$ .)
Solve online
Ex. 40.14Modeling
Tide: $h(t) = 2 + 1.5 \sin(\pi t/6)$ . Maximum and minimum.
Solve online
Ex. 40.15Application
AP with $a_1 = 3, r = 5$ . $a_{20}$ and $S_{20}$ .
Solve online
Ex. 40.16Application
GP with $a_1 = 2, q = 3$ . $S_8$ .
Solve online
Ex. 40.17ApplicationAnswer key
Infinite sum $1 + 1/2 + 1/4 + \ldots$ . (Ans: $2$ .)
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Ex. 40.18Application
Sum $1 + 2 + 3 + \ldots + 200$ (Gauss). (Ans: $20\,100$ .)
Solve online
Ex. 40.19Modeling
1 m pendulum. Approximate period $T = 2\pi\sqrt{L/g}$ , $g = 9.8$ . (Ans: $\approx 2.01$ s.)
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Ex. 40.20Modeling
Exponential decay: 5-day half-life. How much of 100g remains after 25 days? (Ans: $100/2^5 = 3.125$ g.)
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Ex. 40.21Application
Distance between $(2, 3)$ and $(8, 11)$ . (Ans: $10$ .)
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Ex. 40.22Application
Equation of the line through $(0, 4)$ and $(2, 0)$ . (Ans: $y = -2x + 4$ .)
Solve online
Ex. 40.23Application
Equation of the circle with center $(2, -1)$ radius $5$ . (Ans: $(x-2)^2 + (y+1)^2 = 25$ .)
Solve online
Ex. 40.24Application
$(3, 4) \cdot (1, 2)$ and angle between them. (Ans: product $= 11$ .)
Solve online
Ex. 40.25Modeling
Block on $30°$ ramp, mass 10 kg, no friction. Acceleration? (Ans: $g \sin 30° = 4.9$ m/s².)
Solve online
Ex. 40.26Application
Determinant of $\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$ . (Ans: $5$ .)
Solve online
Ex. 40.27ApplicationAnswer key
Solve $\begin{cases} 2x + y = 7 \\ x - 3y = -2 \end{cases}$ via Cramer. (Ans: $x = 19/7, y = 11/7$ .)
Solve online
Ex. 40.28Application
$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ . $A^{-1}$ ? (Ans: $\begin{pmatrix} -2 & 1 \\ 3/2 & -1/2 \end{pmatrix}$ .)
Solve online
Ex. 40.29ModelingAnswer key
Plane at 600 km/h heading north with 80 km/h east wind. Resultant velocity.
Solve online
Ex. 40.30ApplicationAnswer key
Conic $x^2/16 + y^2/9 = 1$ — vertices, foci.
Solve online
Ex. 40.31ApplicationAnswer key
$5!$ . (Ans: $120$ .)
Solve online
Ex. 40.32ApplicationAnswer key
$\binom{8}{3}$ . (Ans: $56$ .)
Solve online
Ex. 40.33Application
Anagrams of "PROBLEMA" — 8 distinct letters. (Ans: $40\,320$ .)
Solve online
Ex. 40.34Modeling
Mega-Sena: $P(\text{hit 6})$ . (Ans: $1/\binom{60}{6}$ .)
Solve online
Ex. 40.35Modeling
Toss 3 coins. $P(\text{3 heads})$ . (Ans: $1/8$ .)
Solve online
Ex. 40.36Modeling
$X \sim \text{Bin}(6, 0.5)$ . $P(X = 3)$ . (Ans: $20/64 = 5/16$ .)
Solve online
Ex. 40.37Modeling
Rare disease $P(D) = 0.001$ , sensitivity 99%, specificity 95%. $P(D|+)$ via Bayes.
Solve online
Ex. 40.38Challenge
Solve: $\sin x + \cos x = 1$ on $[0, 2\pi)$ .
Solve online
Ex. 40.39ChallengeAnswer key
Largest equilateral triangle inscribed in a unit circle. (Side: $\sqrt 3$ .)
Solve online
Ex. 40.40Proof
Prove that $\binom{n}{p} = \binom{n-1}{p-1} + \binom{n-1}{p}$ (Pascal) combinatorially.

Sources

Workshop combines sources from the entire Year 1. Main:

Active Calculus — Matt Boelkins · 2024, ed. 2.0 · EN · CC-BY-NC-SA · §1, §8.
Algebra and Trigonometry — OpenStax · 2022, 2nd ed · EN · CC-BY · §1-§11.
College Algebra — OpenStax · 2022, 2nd ed · EN · CC-BY · §1-§11.
Linear Algebra Done Right — Axler · 2024, 4th ed · EN · CC-BY-NC · ch. 1, 3, 6, 10.
OpenIntro Statistics — Diez et al. · 2019, 4th ed · EN · CC-BY-SA · ch. 3.
Precalculus / College Algebra / Trigonometry — Stitz, Zeager · 2013, v3 · EN · CC-BY-NC-SA · §1-§11.

Full catalog at /livros.

Next: Year 2 — Differential Calculus

Term 5 starts at /aulas/ano-2/trim-5/licao-41-limite-formal — formal $\eps$ - $\delta$ limit, derivative, applications.