Research

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PDFs

Measure Theory — Lecture 79:14 PMLemma 7.2. Let f₁, f₂, … be… PDF

Calculus — Chapter 3: Derivatives12:57 AMChapter 3: Derivatives PDF

October 7, 2026 at 10:55 AM

NoteSessionsBrief

#Measure Theory — Lecture 7

Lemma 7.2. Let $f_1, f_2, \ldots$ be non-negative measurable functions. Then

\int \sum_{n=1}^{\infty} f_n \, d\mu = \sum_{n=1}^{\infty} \int f_n \, d\mu

Proof. Let $g_N = f_1 + \cdots + f_N$ . The $g_N$ increase to $\sum f_n$ , so $\int \sum f_n = \lim_N \int g_N$ $= \lim_N \sum_{n \le N} \int f_n = \sum_n \int f_n$ . ∎

End

Listening

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How it works

Hard reading, one passage at a time.

Buddy reads the page you’re on, so you can ask about exactly the part that loses you.

Ask about “this”.

On your Mac, scroll to the part that loses you and ask. Buddy answers about the page you’re on.

Highlight it, and it explains.

On iPad, mark a passage with Apple Pencil and Buddy offers to explain it, in a few plain sentences.

Bring your papers and notes.

PDFs, GoodNotes, Notability, Apple Notes, Markdown and images. Pen Buddy shows you how to export from each app, then brings it in.

Keep what you learn.

Ask Buddy to add it to your notes and it writes it into the note you have open, in the right place.

courses.example.edu/fin301/week-4/black-scholes

Screen shared Stop

Week 4: The Black–Scholes model

The Black–Scholes model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From it, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options.

The key idea is to hedge the option by buying and selling the underlying asset in just the right way, which removes risk. This hedge implies there is only one right price for the option.

Black–Scholes formula

The value of a call option for a non-dividend-paying underlying stock is:

C = N(d_1)\,S_t - N(d_2)\,K e^{-r(T-t)}

d_2 = d_1 - \sigma\sqrt{T-t}

where N is the cumulative distribution function of the standard normal distribution, S is the spot price, K the strike price, r the risk-free rate and σ the volatility of returns.

End

Listening

Chapter 3

Derivatives

The derivative of a function at a point is the slope of its tangent line there. We find it as a limit: the slope between two nearby points, as the distance between them shrinks to zero.

When this limit exists, we say f is differentiable at a. Every differentiable function is continuous, but the reverse is not true: |x| is continuous at 0, yet has no single tangent there.

In the next section we use the limit to derive the power rule, and then build the product and chain rules on top of it.

End

Listening

All Notes

Naming Organic Compounds11:02 AMFind the longest carbon chain… Note

The Chain Rule10:55 AMWhen one function sits inside… Note

Exam Plan — Week 610:29 AMChain rule practice set Note

Two Pointers12:56 AMOn a sorted array, walk one p… Note

Black–Scholes11:02 AMN(d2): the risk-neutral chance… Note

Binary Search11:02 AMdef lowerbound(nums, target): Note

Coin Change11:02 AMFewest coins that make up am… Note

Integration by Parts10:29 AMPick u with LIATE — the first o… Note

October 7, 2026 at 10:55 AM

NoteSessionsBrief

#The Chain Rule

When one function sits inside another, differentiate the outside, keep the inside, then multiply by the derivative of the inside.

\frac{d}{dx}f(g(x)) = f'(g(x))\cdot g'(x)

Worked examples

\frac{d}{dx}\sin(x^2) = \cos(x^2)\cdot 2x

\frac{d}{dx}(3x+1)^5 = 15(3x+1)^4

All Notes

Naming Organic Compounds11:02 AMFind the longest carbon chain… Note

Derivatives11:40 AMRates of change, and the rules… Note

The Chain Rule10:55 AMWhen one function sits inside… Note

Exam Plan — Week 610:29 AMChain rule practice set Note

Two Pointers12:56 AMOn a sorted array, walk one p… Note

Black–Scholes11:02 AMN(d2): the risk-neutral chance… Note

Binary Search11:02 AMdef lowerbound(nums, target): Note

Coin Change11:02 AMFewest coins that make up am… Note

October 7, 2026 at 11:40 AM

NoteSessionsBrief

#Derivatives

Rates of change, and the rules that make them quick.

Rules

Power rule: $\frac{d}{dx}x^n = nx^{n-1}$
Product rule: $(uv)' = u'v + uv'$

Examples

\frac{d}{dx}\,3x^4 = 12x^3

Practice

Problems 1–8
Review the product rule

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FAQ

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Can Pen Buddy help me read research papers?

Yes. Open the paper as a PDF in your library, or in your browser on the Mac with your screen shared, and read with Buddy beside you. Scroll to the part that loses you and ask about “this”: Buddy answers about the page you’re on, in a few plain sentences.

Can it explain a step in a proof?

Yes. Ask why a step works, out loud. For hard questions Buddy takes a moment to think it through, then answers in a sentence or two, with the maths on a card. If you’d rather find it yourself, ask for a hint instead.

Does it work with PDFs on iPad?

Yes. Highlight a passage with Apple Pencil and Buddy offers to explain it. Ask it to show you, and it draws a diagram, timeline or tree on your page.

Can it look things up?

Yes. When your page and your notes don’t have the answer, Buddy can search the web, then tell you what it found in a sentence or two.

How is it different from chatting with a PDF?

Nothing to upload or paste. Buddy reads along with the page you’re on, you ask out loud, it answers briefly, and what matters ends up in your notes, written into the note you have open.

Will it remember what I’ve read?

Yes. Each session is written into your notes with a summary and review questions, and Buddy keeps a short memory of past sessions, so you can pick up a paper where you left off.

Are my papers private?

Your PDFs and notes stay on your device and never go to Pen Buddy’s servers. When you ask a question, only what Buddy needs to answer is sent to the AI services that do its thinking.

Which plan is for research?

The Research plan: 2,500 credits a month for $49.99, room for long papers, proofs and deep thinking every day. You can start on the free plan, with 25 credits a month.

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