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ChaptersAll 17 chapters · read online or download each one
- Start hereHow to read these notes · contents · symbols and notationRead
- 1Exponents & Logarithms1.1, 1.5, 1.7Open
- 2Sequences & Series1.2–1.4, 1.8–1.10Open
- 3Proofs1.6, 1.15Open
- 4Lines & Quadratics1.16, 2.1–2.4, 2.6, 2.7Open
- 5Functions & Graphs2.5, 2.8–2.11, 2.13–2.16Open
- 6Polynomials1.11, 2.12Open
- 7Shapes & Measures3.1–3.4Open
- 8Trigonometry3.5–3.11Open
- 9Vectors3.12–3.18Open
- 10Statistics4.1–4.4, 4.10Open
- 11Probability4.5, 4.6, 4.11, 4.13Open
- 12Probability Distributions4.7–4.9, 4.12, 4.14Open
- 13Limits & Derivatives5.1–5.4, 5.6–5.8, 5.12, 5.13, 5.15Open
- 14Derivative Applications5.8, 5.9, 5.14Open
- 15Integral Calculus5.5, 5.9–5.11, 5.15–5.17Open
- 16Differential Equations5.18, 5.19Open
- 17Complex Numbers1.12–1.14Open
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Hasan K.
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Every problem is a door.
One question a week, from the notes. Try it before you open the door.
Question of the week
Chapter 11, Challenge Question 35
A prize is behind one of three doors. You pick a door. The host, who knows where the prize is, opens one of the other two doors to show it is empty, then lets you switch to the last closed door. Should you switch?
Hint. Your first pick is right only $\tfrac13$ of the time. What does switching give you in the other $\tfrac23$?
Answer. Yes, switch. Switching wins with probability $\tfrac23$; staying wins only when your first pick was right, with probability $\tfrac13$.
Chapter 1, question 15
Solve $\log_x 8 = \log_4 x$.
Hint. Write both logarithms with natural logs: $\dfrac{\ln 8}{\ln x} = \dfrac{\ln x}{\ln 4}$.
Answer. $(\ln x)^2 = \ln 8 \cdot \ln 4 = 6(\ln 2)^2$, so $\ln x = \pm\sqrt{6}\,\ln 2$ and $x = 2^{\pm\sqrt{6}}$: $x \approx 5.46$ or $x \approx 0.183$. A base only needs $x > 0$, $x \neq 1$, so both are valid.
Chapter 1, question 34
A sheet of paper is $0.1$ mm thick. Fold it in half, then in half again, and keep going. How many folds until it is thicker than the distance to the Moon, $384\,000$ km?
Hint. Each fold doubles the thickness: after $n$ folds it is $0.1 \times 2^n$ mm.
Answer. $42$ folds. After $41$ folds it is about $2.2 \times 10^{5}$ km; after $42$ it is about $4.4 \times 10^{5}$ km, past the Moon.
Chapter 11, Reflections
How many people must be in a room before it is more likely than not that two of them share a birthday?
Hint. It is easier to find the probability that all the birthdays are different, then subtract it from $1$.
Answer. Only $23$. With $23$ people, the probability that all birthdays differ is about $0.493$, so a shared birthday has probability about $0.507$.
Chapter 14, section 14.3
Cut a square of side $x$ cm from each corner of a $20 \times 20$ cm card, then fold up the sides to make an open box. Which $x$ gives the largest volume?
Hint. The volume is $V(x) = x(20 - 2x)^2$ for $0 < x < 10$. Solve $V'(x) = 0$.
Answer. $x = \tfrac{10}{3}$ cm, giving the largest volume $V = \tfrac{16000}{27} \approx 593$ cm$^3$.
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Every problem is a door. Open one.