Calculus 2

Overview

• Textbook: Calculus Volume 2 — https://openstax.org/details/books/calculus-volume-2 (CC BY-NC-SA 4.0)
• Supplementary: Paul’s Online Math Notes, 3Blue1Brown
• Prerequisites: Calculus 1
• Estimated time: 4 weeks at 10-12 hrs/week

Key Takeaways

• Master advanced integration techniques that appear in probability and physics
• Understand infinite series and convergence — the foundation of numerical computation
• Approximate functions with Taylor/Maclaurin series (how computers actually compute sin, cos, exp)
• Model dynamic systems with introductory differential equations
• Build intuition for when series converge and how fast — critical for algorithm analysis

How to Study

• Read textbook sections listed under each concept
• Work through essential problems with pencil and paper
• Practice choosing the right integration technique before computing
• For series, focus on building a decision tree for which convergence test to apply
• Review connections to CS topics in the algorithms track

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Concepts & Techniques

Core Insight

Calculus 2 extends integration to harder problems and then pivots to the profound idea that functions can be represented as infinite sums (series). This is how computers approximate transcendental functions and how we analyze the long-run behavior of algorithms.

1. Integration Techniques

Textbook sections: Ch 1, Sections 1.1-1.5

Key techniques:

• Integration by parts: integral of u dv = uv - integral of v du. Choose u by LIATE (Log, Inverse trig, Algebraic, Trig, Exponential).
• Trigonometric integrals: Use identities like sin^2(x) = (1 - cos(2x))/2 to reduce powers.
• Trigonometric substitution: For sqrt(a^2 - x^2), let x = asin(theta). For sqrt(a^2 + x^2), let x = atan(theta). For sqrt(x^2 - a^2), let x = a*sec(theta).
• Partial fractions: Decompose P(x)/Q(x) into simpler fractions when degree(P) < degree(Q).

Worked example:

Evaluate integral of x * e^x dx. By parts: u = x, dv = e^x dx, so du = dx, v = e^x. Result: xe^x - integral of e^x dx = xe^x - e^x + C = e^x(x - 1) + C.

Evaluate integral of 1/(x^2 - 1) dx. Partial fractions: 1/(x^2-1) = 1/[(x-1)(x+1)] = A/(x-1) + B/(x+1). Solving: A = 1/2, B = -1/2. Integral = (1/2)ln|x-1| - (1/2)ln|x+1| + C.

Essential problems: OpenStax 1.1 Exercises: #1-15; 1.3 Exercises: #115-128; 1.4 Exercises: #155-170 Challenge problems: OpenStax 1.5 Exercises: #200-210

2. Applications of Integration

Textbook sections: Ch 2, Sections 2.1-2.4

Key definitions:

• Arc length: L = integral from a to b of sqrt(1 + [f’(x)]^2) dx
• Surface area of revolution: S = 2*pi * integral from a to b of f(x) * sqrt(1 + [f’(x)]^2) dx
• Center of mass: x_bar = M_y / m where M_y = integral of x * rho(x) * f(x) dx

Worked example:

Find the arc length of y = x^(3/2) from x=0 to x=4. f’(x) = (3/2)x^(1/2). L = integral from 0 to 4 of sqrt(1 + 9x/4) dx. Let u = 1 + 9x/4, du = 9/4 dx. L = (4/9) * (2/3) * [u^(3/2)] from 1 to 10 = (8/27)(10*sqrt(10) - 1).

Essential problems: OpenStax 2.1 Exercises: #1-10; 2.2 Exercises: #50-60; 2.4 Exercises: #100-110 Challenge problems: OpenStax 2.4 Exercises: #115-120

3. Parametric Curves and Polar Coordinates

Textbook sections: Ch 3, Sections 3.1-3.4

Key definitions:

• Parametric curve: x = f(t), y = g(t); derivative dy/dx = (dy/dt)/(dx/dt)
• Polar coordinates: x = rcos(theta), y = rsin(theta)
• Polar area: A = (1/2) * integral from alpha to beta of [r(theta)]^2 d(theta)

Key theorems:

• Parametric arc length: L = integral from a to b of sqrt([dx/dt]^2 + [dy/dt]^2) dt. Intuition: at each instant, the speed is the hypotenuse of the velocity components.

Worked example:

Find the area enclosed by one petal of r = cos(2theta). One petal spans theta in [-pi/4, pi/4]. A = (1/2) integral from -pi/4 to pi/4 of cos^2(2theta) d(theta) = (1/2) * (pi/4) = pi/8.

Essential problems: OpenStax 3.1 Exercises: #1-12; 3.3 Exercises: #110-122; 3.4 Exercises: #150-162 Challenge problems: OpenStax 3.4 Exercises: #168-175

4. Sequences and Series

Textbook sections: Ch 5, Sections 5.1-5.6

Key definitions:

• Sequence: {a_n} converges if lim(n->inf) a_n exists and is finite
• Series: Sum from n=1 to inf of a_n; converges if the sequence of partial sums converges
• Power series: Sum of c_n * (x - a)^n with radius of convergence R
• Taylor series: f(x) = Sum from n=0 to inf of f^(n)(a)/n! * (x-a)^n

Key theorems / convergence tests:

• Divergence Test: If lim a_n != 0, the series diverges. Intuition: terms must shrink to zero (necessary but not sufficient).
• Ratio Test: If lim |a_(n+1)/a_n| = L < 1, converges absolutely; L > 1, diverges. Intuition: compare to a geometric series.
• Root Test: If lim |a_n|^(1/n) = L, same conclusion as ratio test.
• Integral Test: If f is positive, continuous, decreasing and a_n = f(n), the series and integral from 1 to inf of f(x) dx converge/diverge together.
• Comparison Test: Compare with a known series (geometric, p-series).
• Alternating Series Test: If |a_n| decreases to 0, the alternating series converges.

Important series:

• Geometric: Sum r^n = 1/(1-r) for |r| < 1
• p-series: Sum 1/n^p converges iff p > 1
• Harmonic: Sum 1/n diverges (p = 1)
• e^x = Sum x^n/n!, sin(x) = Sum (-1)^n * x^(2n+1)/(2n+1)!, cos(x) = Sum (-1)^n * x^(2n)/(2n)!

Worked example:

Determine convergence of Sum from n=1 to inf of n^2/2^n. Apply the ratio test: |a_(n+1)/a_n| = (n+1)^2 / (2*n^2). As n->inf, this approaches 1/2 < 1. The series converges.

Essential problems: OpenStax 5.1 Exercises: #1-14; 5.3 Exercises: #120-138; 5.4 Exercises: #175-190; 5.6 Exercises: #260-275 Challenge problems: OpenStax 5.6 Exercises: #280-288

5. Introduction to Differential Equations

Textbook sections: Ch 4, Sections 4.1-4.3

Key definitions:

• ODE: An equation involving a function and its derivatives
• Separable equation: dy/dx = f(x)*g(y); solve by separating variables and integrating both sides
• First-order linear: dy/dx + P(x)*y = Q(x); solve using integrating factor mu(x) = e^(integral P(x) dx)

Worked example:

Solve dy/dx = 2xy with y(0) = 1. Separate: dy/y = 2x dx. Integrate: ln|y| = x^2 + C. With y(0)=1: C=0. Solution: y = e^(x^2).

Essential problems: OpenStax 4.1 Exercises: #1-10; 4.3 Exercises: #100-115 Challenge problems: OpenStax 4.3 Exercises: #120-125

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Technique Catalog

Technique	When to Use	Key Formula/Idea
Integration by parts	Product of two function types	LIATE priority for choosing u
Trig substitution	sqrt(a^2 +/- x^2) or sqrt(x^2 - a^2)	Substitute x = a*sin/tan/sec
Partial fractions	Rational functions P(x)/Q(x)	Factor denominator, decompose
Ratio test	Series with factorials or exponentials	Compute lim
Comparison test	Series similar to known p-series/geometric	Bound by known convergent/divergent series
Taylor expansion	Approximate f(x) near a point	f(x) = Sum f^(n)(a)/n! * (x-a)^n
Separable ODE	dy/dx = f(x)*g(y)	Separate and integrate both sides

Connections to CS & Algorithms

Math Concept	CS Application	Repo Link
Harmonic series	Average-case hashing analysis, coupon collector	algorithms track
Taylor series	How computers compute sin/cos/exp, floating-point	systems track
Power series / radius	Convergence of iterative algorithms	algorithms track
Differential equations	System modeling, PID controllers, epidemics	systems track
Convergence tests	Analyzing whether iterative methods terminate	ML track

Company Relevance

Company	How This Appears	Difficulty
Quantitative Finance	Series pricing models, stochastic DEs	Hard
Google/DeepMind	Numerical stability of ML training loops	Hard
Signal Processing (Apple, Qualcomm)	Fourier series from Taylor series foundations	Medium
Robotics (Boston Dynamics, Tesla)	Differential equations for control systems	Hard

Chapters

#	Module	Chapter	Kind	Pass
1	M02.1	Taylor series, remainder bounds, range reduction	build	2
2	M02.2	Series convergence, EMA as a geometric series, bias correction	build	2
3	S-M02	Calculus 2 problem set: integration techniques, the Gaussian, series, Taylor, polar, ODEs	solve	2