Calculus II

Integration techniques and applications, differential equations, Taylor series, and parametric and polar curves, with four computer labs. Free, self-paced, and tied to real planet-saving work.

CNES (PIA25623)NASA Blue MarbleNASA/Carla Thomas (AFRC2025-0170-03)NASA/JPL-Caltech

Courses › MATH 111

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About this course

This is a free self-study course outline written by Saving Planets. It is not a university course, it carries no academic credit, and no school is involved.

Its scope follows second-semester college calculus, modeled on the topics and depth of open materials such as OpenStax Calculus Volume 2 and MIT OpenCourseWare 18.01SC (with parametric and polar ideas that lead into 18.02SC Multivariable Calculus), which we link to throughout. Those organizations do not endorse this page.

What we add: every unit connects to real water, energy, ecology and spaceflight problems, every number on the page is re-computed by a checking script before it goes live, and four computer labs put the methods to work on data.

Work through it in order. Each unit has goals, key ideas, equations, step-by-step examples, practice with full answers, common mistakes, flashcards and a quiz. Your progress is saved only in this browser.

How this course helps save the planet

Calculus II is where calculus starts modeling the world: volumes of water in reservoirs, the work to pump it, how a pollutant flushes out of a lake, how a fish population recovers, and the elliptical orbits of satellites that watch it all.

Each unit ends with a short, sourced section showing how its ideas help both people and nature.

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Before you start

Prerequisites

  • Calculus I: limits, derivatives, the chain rule, basic integrals and substitution, the Fundamental Theorem.
  • Trigonometric identities, especially sin²x + cos²x = 1 and the double-angle formulas.
  • Exponentials and logarithms.

How to study

  • Read the unit, then close it and try the worked examples yourself before checking each step.
  • Do every practice problem on paper before opening the answer.
  • For integrals, check your answer by differentiating it.
  • For series, decide which test fits before computing anything.
  • Sketch every region before setting up a volume or area integral.
  • Retake the quizzes a few days later. Spaced practice beats one long session.

Lab equipment (for the wet labs)

  • A computer or tablet with a spreadsheet.
  • A free graphing calculator such as Desmos.
  • Optional: Python with NumPy.

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Suggested 15-week schedule

About 6 to 9 hours a week: reading, practice problems and, in lab weeks, the lab.

WeekUnitTopicsReading
1Techniques of IntegrationIntegration by partsOpenStax 3.1
2Techniques of IntegrationTrigonometric integrals and substitutionOpenStax 3.2–3.3
3Techniques of IntegrationPartial fractions; numerical integrationOpenStax 3.4–3.6
4Applications: Volume, Arc Length, Work & Center of MassAreas, volumes by slicing, disks, washers, shellsOpenStax 2.1–2.3
5Applications: Volume, Arc Length, Work & Center of MassArc length, work, fluid force, centers of mass; Lab BOpenStax 2.4–2.6
6Improper IntegralsImproper integrals and comparison; midterm practiceOpenStax 3.7
7Introduction to Differential EquationsDifferential equations, direction fields, Euler’s methodOpenStax 4.1–4.2
8Introduction to Differential EquationsSeparable and logistic equations; Lab AOpenStax 4.3–4.4
9Sequences & SeriesSequences; infinite series; geometric seriesOpenStax 5.1–5.2
10Sequences & SeriesDivergence, integral and comparison testsOpenStax 5.3–5.4
11Sequences & SeriesAlternating series; ratio and root testsOpenStax 5.5–5.6
12Power Series & Taylor SeriesPower series and their propertiesOpenStax 6.1–6.2
13Power Series & Taylor SeriesTaylor and Maclaurin series; Lab COpenStax 6.3–6.4
14Parametric Equations & Polar CoordinatesParametric equations; polar coordinates and areas; Lab DOpenStax 7.1–7.4
15ReviewReview and final practice setAll units

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Your progress

Tick units and labs as you finish them, and take the quizzes. Progress and quiz scores are saved only in this browser, never sent to us.

Turn on JavaScript to track progress.

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Unit 1 · Weeks 1–3

Techniques of Integration

Integration by parts, trigonometric integrals and substitutions, partial fractions, and numerical integration when no formula exists.

Learning objectives

By the end of this unit you can:

  • Integrate by parts, choosing u and dv well.
  • Evaluate trigonometric integrals and use trigonometric substitution.
  • Decompose rational functions into partial fractions.
  • Approximate integrals with the trapezoid rule and Simpson’s rule, and estimate the error.

Key concepts

  • Integration by parts is the product rule in reverse: ∫u dv = uv − ∫v du. Choose u to get simpler when differentiated (logs, polynomials).
  • Powers of sine and cosine yield to identities such as sin²x = (1 − cos 2x)/2.
  • For √(a² − x²), substitute x = a sin θ; for √(a² + x²), x = a tan θ.
  • A proper rational function splits into simple fractions whose integrals are logs and arctangents.
  • Many integrals, such as ∫e^(−x²) dx, have no elementary antiderivative. Simpson’s rule is often accurate to many digits with few points.

Key equations

∫u dv=uv−∫v du\int u\,dv=uv-\int v\,du
Integration by parts
sin⁡2x=12(1−cos⁡2x)\sin^2x=\tfrac12(1-\cos2x)
Power-reduction identity
1x2−1=1/2x−1−1/2x+1\frac{1}{x^2-1}=\frac{1/2}{x-1}-\frac{1/2}{x+1}
A partial-fraction decomposition
Sn=Δx3[f0+4f1+2f2+⋯+4fn−1+fn]S_n=\frac{\Delta x}{3}\left[f_0+4f_1+2f_2+\dots+4f_{n-1}+f_n\right]
Simpson’s rule (n even)

Worked examples, step by step

Example 1.1 Find ∫₀¹ x eˣ dx.

Show the steps
  1. u = x, dv = eˣ dx, so du = dx, v = eˣ.
  2. ∫x eˣ dx = x eˣ − ∫eˣ dx = (x − 1)eˣ.
  3. From 0 to 1: 0 − (−1) = 1.

Answer: 1.

Example 1.2 Find ∫₂³ dx/(x² − 1).

Show the steps
  1. Partial fractions: ½[1/(x − 1) − 1/(x + 1)].
  2. Antiderivative ½ ln|(x − 1)/(x + 1)|.
  3. ½[ln(2/4) − ln(1/3)] = ½ ln(3/2).

Answer: ½ ln 1.5 ≈ 0.2027.

Example 1.3 Estimate ∫₀¹ e^(−x²) dx with Simpson’s rule, n = 4.

Show the steps
  1. Δx = 0.25; f = 1, 0.93941, 0.77880, 0.56978, 0.36788.
  2. S₄ = (0.25/3)[1 + 4(0.93941) + 2(0.77880) + 4(0.56978) + 0.36788].
  3. S₄ = 0.74686.

Answer: 0.74686, versus the true 0.74682: an error of only 0.00003 from five points.

Practice problems

Work each on paper first, then check.

  1. ∫₀^π x sin x dx

    Show answer

    Parts: [−x cos x + sin x]₀^π = π.

  2. ∫₁ᵉ x ln x dx

    Show answer

    Parts with u = ln x: [x² ln x/2 − x²/4]₁ᵉ = (e² + 1)/4 ≈ 2.097.

  3. ∫₀^(π/2) sin²x dx

    Show answer

    Power reduction: ½[x − sin 2x/2]₀^(π/2) = π/4.

  4. ∫₀¹ √(1 − x²) dx

    Show answer

    x = sin θ gives ∫₀^(π/2) cos²θ dθ = π/4 (a quarter of the unit disk).

  5. ∫₀¹ dx/(1 + x²)

    Show answer

    [arctan x]₀¹ = π/4.

Common misconceptions

  • Myth: Always let u be the first factor you see.

    Actually: Choose u to simplify when differentiated: logs and polynomials first, exponentials and trig as dv.

  • Myth: Partial fractions works on any rational function.

    Actually: Divide first if the top’s degree is not less than the bottom’s.

  • Myth: If I can’t find an antiderivative, the integral has no value.

    Actually: It still has a value; estimate it numerically.

Flashcards

  • Integration by parts

    ∫u dv = uv − ∫v du.

  • Trig substitution

    √(a² − x²): x = a sin θ.

  • Partial fractions

    Split a rational function into simple fractions.

  • Trapezoid rule

    Average of left and right sums; error ∝ Δx².

  • Simpson’s rule

    Parabolic fit; error ∝ Δx⁴.

  • Non-elementary integral

    No formula antiderivative, e.g. ∫e^(−x²) dx.

Self-check quiz

1. For ∫x² eˣ dx, a good first choice of u is…

Show answer

x². x² gets simpler when differentiated.

2. √(9 − x²) suggests the substitution…

Show answer

x = 3 sin θ. 9 − 9 sin²θ = 9 cos²θ.

3. Simpson’s rule fits the curve with…

Show answer

parabolas. Parabolas through each set of three points.

4. ∫ ln x dx =

Show answer

x ln x − x + C. Parts with u = ln x, dv = dx.

How this helps save the planet

The bell-curve integral in Example 1.3 has no formula answer, yet it sits behind every error bar on a climate graph and every uncertainty in a satellite measurement. Numerical integration is how it, and most integrals from real data, actually get computed.

Methods like Simpson’s rule are how irregular measurements, such as river depths across a channel, become totals like flow or volume.

On Saving Planets: Calculus I, Lab D: numerical integration · Rivers hub

Sources: OpenStax Calculus 2, 3.6 Numerical Integration · OpenStax Calculus 2, 3.1 Integration by Parts

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Unit 2 · Weeks 4–5

Applications: Volume, Arc Length, Work & Center of Mass

Slice, add and take the limit: volumes of solids, lengths of curves, work against forces, fluid pressure and centers of mass.

Learning objectives

By the end of this unit you can:

  • Find volumes by slicing, disks, washers and shells.
  • Compute arc length.
  • Compute work done by variable forces, including pumping liquids, and hydrostatic force.
  • Find the center of mass of rods and thin plates.

Key concepts

  • Every application has the same pattern: approximate one thin slice, write its contribution, integrate.
  • Disk method: V = ∫π[f(x)]² dx about the x-axis. Shell method: V = ∫2πx f(x) dx about the y-axis.
  • Arc length adds tiny hypotenuses: L = ∫√(1 + [f′(x)]²) dx.
  • Work = ∫F(x) dx. Pumping: each layer of liquid is lifted its own distance.
  • Pressure grows with depth, so the force on a dam is an integral over depth.
  • Center of mass = moment ÷ mass.

Key equations

V=∫abπ[f(x)]2 dxV=\int_a^b\pi[f(x)]^2\,dx
Disk method
V=∫ab2πx f(x) dxV=\int_a^b2\pi x\,f(x)\,dx
Shell method
L=∫ab1+[f′(x)]2 dxL=\int_a^b\sqrt{1+[f'(x)]^2}\,dx
Arc length
W=∫abF(x) dxW=\int_a^bF(x)\,dx
Work by a variable force
xˉ=∫x ρ(x) dx∫ρ(x) dx\bar x=\frac{\int x\,\rho(x)\,dx}{\int\rho(x)\,dx}
Center of mass of a rod

Worked examples, step by step

Example 2.1 Find the volume when the region under y = √x, 0 ≤ x ≤ 4, is spun about the x-axis.

Show the steps
  1. Disks of radius √x: V = ∫₀⁴ π x dx.
  2. π[x²/2]₀⁴ = 8π.

Answer: 8π ≈ 25.1 cubic units.

Example 2.2 A cylindrical tank of radius 1 m and height 2 m is full of water. How much work does it take to pump all the water out over the top?

Show the steps
  1. A layer at height y has volume π(1)² dy and weight 1,000 × 9.81 × π dy newtons, and is lifted 2 − y meters.
  2. W = ∫₀² 9,810π(2 − y) dy = 9,810π × 2.
  3. = 19,620π J.

Answer: ≈ 61,600 J (61.6 kJ).

Example 2.3 A rod on [0, 2] m has density ρ(x) = 1 + x kg/m. Find its center of mass.

Show the steps
  1. Mass: ∫₀² (1 + x) dx = 2 + 2 = 4 kg.
  2. Moment: ∫₀² x(1 + x) dx = 2 + 8/3 = 14/3.
  3. x̄ = (14/3)/4 = 7/6.

Answer: x̄ = 7/6 ≈ 1.17 m, toward the heavier end.

Practice problems

Work each on paper first, then check.

  1. Use shells to find the volume when the region under y = x², 0 ≤ x ≤ 1, is spun about the y-axis.

    Show answer

    V = ∫₀¹ 2πx · x² dx = 2π/4 = π/2.

  2. Find the arc length of y = (2/3)x^(3/2) from x = 0 to 3.

    Show answer

    y′ = √x, so L = ∫₀³ √(1 + x) dx = (2/3)[(1 + x)^(3/2)]₀³ = (2/3)(8 − 1) = 14/3.

  3. A spring with k = 200 N/m is stretched from rest length by 0.10 m. How much work is done?

    Show answer

    W = ∫₀^0.1 200x dx = 100(0.01) = 1.0 J.

  4. A vertical rectangular gate 10 m wide holds back water 5 m deep. Find the total water force on it.

    Show answer

    F = ∫₀⁵ ρg y (10) dy = 1,000 × 9.81 × 10 × 25/2 = 1.23 × 10⁶ N.

Common misconceptions

  • Myth: Volume of revolution = π × area.

    Actually: You must square the radius inside the integral: ∫π f² dx, not π(∫f dx).

  • Myth: Pumping work is weight × tank height.

    Actually: Each layer moves a different distance, so you must integrate.

  • Myth: The center of mass is always the midpoint.

    Actually: Only for uniform density and a symmetric shape.

Flashcards

  • Disk method

    V = ∫π f(x)² dx.

  • Washer method

    V = ∫π(R² − r²) dx.

  • Shell method

    V = ∫2πx f(x) dx.

  • Arc length

    L = ∫√(1 + f′²) dx.

  • Work

    W = ∫F dx.

  • Center of mass

    Moment divided by mass.

Self-check quiz

1. Shells about the y-axis use the integrand…

Show answer

2πx f(x). Circumference 2πx × height f(x) × thickness dx.

2. Arc length uses…

Show answer

∫√(1 + f′²) dx. Pythagoras on each tiny piece.

3. Hydrostatic pressure on a dam grows…

Show answer

linearly with depth. P = ρgh; the total force then grows with depth squared.

4. The work to stretch a spring by x from rest is…

Show answer

½kx². ∫₀ˣ kt dt.

How this helps save the planet

Water engineers use these integrals directly: the volume of a reservoir from its shape, the energy to pump water to where it is needed, and the force a dam or flood barrier must hold back. Pumping is a significant energy cost in many water systems, so getting the integral right saves both energy and money.

Satellites now measure the height of rivers, lakes and reservoirs from orbit. NASA and CNES’s SWOT mission surveys surface water so that changes in stored water can be estimated, which is slicing-and-adding on a planetary scale.

On Saving Planets: SWOT · Room for the River

Sources: OpenStax Calculus 2, 2.5 Physical Applications · NASA: SWOT mission

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Unit 3 · Week 6

Improper Integrals

Integrals over infinite intervals or across infinite spikes: when they add up to a finite number, and when they do not.

Learning objectives

By the end of this unit you can:

  • Evaluate improper integrals as limits.
  • Decide convergence with the p-test and comparison.
  • Apply improper integrals to escape energy and to average lifetimes.

Key concepts

  • ∫ₐ^∞ f dx means the limit of ∫ₐᵇ f dx as b → ∞. If the limit is finite, the integral converges.
  • ∫₁^∞ dx/xᵖ converges exactly when p > 1. ∫₀¹ dx/xᵖ converges exactly when p < 1.
  • Comparison: if 0 ≤ f ≤ g and ∫g converges, so does ∫f.
  • An infinite region can have finite area.

Key equations

∫a∞f(x) dx=lim⁡b→∞∫abf(x) dx\int_a^{\infty}f(x)\,dx=\lim_{b\to\infty}\int_a^bf(x)\,dx
Improper integral
∫1∞dxxp=1p−1 (p>1)\int_1^{\infty}\frac{dx}{x^p}=\frac1{p-1}\ (p>1)
p-integral
W∞=∫R∞GMmr2 dr=GMmRW_{\infty}=\int_R^{\infty}\frac{GMm}{r^2}\,dr=\frac{GMm}{R}
Work to escape a planet’s gravity

Worked examples, step by step

Example 3.1 Evaluate ∫₁^∞ dx/x².

Show the steps
  1. ∫₁ᵇ x⁻² dx = 1 − 1/b.
  2. As b → ∞, 1/b → 0.

Answer: 1 (converges).

Example 3.2 Does ∫₁^∞ dx/x converge?

Show the steps
  1. ∫₁ᵇ dx/x = ln b.
  2. ln b → ∞.

Answer: No, it diverges, even though 1/x → 0.

Example 3.3 How much energy per kilogram does it take to escape Earth’s gravity from its surface (ignoring air and rotation)?

Show the steps
  1. W/m = ∫_R^∞ GM/r² dr = GM/R.
  2. GM = 3.986 × 10¹⁴ m³/s², R = 6.371 × 10⁶ m.
  3. W/m = 6.26 × 10⁷ J/kg.

Answer: About 62.6 MJ per kilogram, finite even though the distance is infinite. Setting ½v² equal to it gives the 11.2 km/s escape speed.

Practice problems

Work each on paper first, then check.

  1. ∫₀^∞ e^(−x) dx

    Show answer

    [−e^(−x)]₀^∞ = 0 + 1 = 1.

  2. ∫₀¹ dx/√x

    Show answer

    [2√x]₀¹ = 2 (converges; p = ½ < 1).

  3. ∫₂^∞ dx/x³

    Show answer

    [−1/(2x²)]₂^∞ = 1/8.

  4. Does ∫₁^∞ dx/(x² + 1) converge?

    Show answer

    Yes: it is less than 1/x², which converges. In fact it equals π/4.

  5. For exponential decay with rate λ, the mean lifetime is ∫₀^∞ t λe^(−λt) dt. Evaluate it.

    Show answer

    By parts: 1/λ.

Common misconceptions

  • Myth: If f(x) → 0 as x → ∞, the integral converges.

    Actually: 1/x → 0 but ∫₁^∞ dx/x diverges.

  • Myth: An infinitely long region has infinite area.

    Actually: ∫₁^∞ dx/x² = 1.

  • Myth: ∫₋₁¹ dx/x² = −2 by the usual formula.

    Actually: The integrand blows up at 0, so the integral is improper, and it diverges.

Flashcards

  • Improper integral

    Infinite interval or infinite integrand; defined by a limit.

  • Converges

    The limit exists and is finite.

  • p-test (∞)

    ∫₁^∞ x⁻ᵖ converges iff p > 1.

  • p-test (0)

    ∫₀¹ x⁻ᵖ converges iff p < 1.

  • Comparison test

    Smaller than convergent ⇒ converges.

  • Escape energy

    GM/R per kg.

Self-check quiz

1. ∫₁^∞ dx/x^1.5…

Show answer

converges to 2. 1/(p − 1) = 1/0.5 = 2.

2. ∫₁^∞ dx/x…

Show answer

diverges. ln b → ∞.

3. ∫₀^∞ e^(−2x) dx =

Show answer

1/2. [−e^(−2x)/2]₀^∞ = 1/2.

4. An improper integral is defined as…

Show answer

a limit of proper integrals. Take the limit of ∫ₐᵇ.

How this helps save the planet

Improper integrals give the energy to leave a planet, the number behind every launch of a weather or climate satellite. It is finite, which is why spaceflight is possible at all.

Mean lifetimes, also improper integrals, describe how long substances stick around: how long a pollutant persists in a lake, or how long a radioactive tracer used to date ice and ocean water remains.

On Saving Planets: Physics I, Unit 7: orbits · Delta-v calculator

Sources: OpenStax Calculus 2, 3.7 Improper Integrals · NASA Earth fact sheet

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Unit 4 · Weeks 7–8

Introduction to Differential Equations

Equations for rates of change: exponential growth and decay, cooling, mixing, and the logistic model of populations.

Learning objectives

By the end of this unit you can:

  • Verify solutions and solve initial-value problems.
  • Read direction fields and use Euler’s method.
  • Solve separable equations, including mixing and cooling problems.
  • Use the logistic equation and interpret its carrying capacity.

Key concepts

  • A differential equation links a function to its derivatives. Its solution is a function, usually a family fixed by an initial condition.
  • Separable equations: put all y on one side and all x on the other, then integrate.
  • y′ = ky gives y = y₀e^(kt): growth if k > 0, decay if k < 0.
  • Mixing: rate of change of a pollutant = rate in − rate out.
  • Logistic: P′ = rP(1 − P/K). Growth slows as P nears the carrying capacity K; it is fastest at P = K/2.
  • Euler’s method steps along the slope: yₙ₊₁ = yₙ + h f(tₙ, yₙ). Smaller steps, smaller error.

Key equations

dydt=ky ⇒ y=y0ekt\frac{dy}{dt}=ky\ \Rightarrow\ y=y_0e^{kt}
Exponential growth and decay
dTdt=−k(T−Ts)\frac{dT}{dt}=-k(T-T_s)
Newton’s law of cooling
dPdt=rP(1−PK),P(t)=K1+K−P0P0e−rt\frac{dP}{dt}=rP\left(1-\frac PK\right),\quad P(t)=\frac{K}{1+\frac{K-P_0}{P_0}e^{-rt}}
Logistic equation and its solution
yn+1=yn+h f(tn,yn)y_{n+1}=y_n+h\,f(t_n,y_n)
Euler’s method

Worked examples, step by step

Example 4.1 A lake section holds 1,000 m³ of water with 50 kg of a dissolved pollutant. Clean water flows in at 10 m³/day and well-mixed water flows out at the same rate. How long until half the pollutant is gone?

Show the steps
  1. dS/dt = −(10/1,000)S = −S/100.
  2. S(t) = 50e^(−t/100).
  3. Half: e^(−t/100) = ½, so t = 100 ln 2.

Answer: ≈ 69.3 days (a simplified, well-mixed model).

Example 4.2 A fish stock follows P′ = 0.5P(1 − P/1,000) with P(0) = 100. Find P(4).

Show the steps
  1. P(t) = 1,000/(1 + 9e^(−0.5t)).
  2. P(4) = 1,000/(1 + 9e^(−2)) = 1,000/2.218.

Answer: ≈ 451 fish, near the fastest-growth level K/2 = 500.

Example 4.3 Use Euler’s method with h = 0.5 to estimate y(1) for y′ = y, y(0) = 1.

Show the steps
  1. y₁ = 1 + 0.5(1) = 1.5.
  2. y₂ = 1.5 + 0.5(1.5) = 2.25.

Answer: 2.25, versus the exact e ≈ 2.718; halving h shrinks the error.

Practice problems

Work each on paper first, then check.

  1. Coffee at 90 °C cools in a 20 °C room with k = 0.1 per minute. Find its temperature after 10 minutes.

    Show answer

    T = 20 + 70e^(−1) ≈ 45.8 °C.

  2. Solve y′ = xy with y(0) = 2, and find y(1).

    Show answer

    dy/y = x dx, so y = 2e^(x²/2); y(1) = 2e^0.5 ≈ 3.30.

  3. Solve y′ = −2y, y(0) = 5. Find y(1).

    Show answer

    y = 5e^(−2t); y(1) ≈ 0.677.

  4. For P′ = rP(1 − P/K), at what population is growth fastest?

    Show answer

    rP(1 − P/K) is a downward parabola in P with its peak at P = K/2.

Common misconceptions

  • Myth: The solution of a differential equation is a number.

    Actually: It is a function (or family of functions).

  • Myth: Euler’s method gives the exact answer if you are careful.

    Actually: It has a step-size error; it converges only as h → 0.

  • Myth: Logistic populations stop growing at K/2.

    Actually: Growth is fastest at K/2 and slows to zero only as P approaches K.

Flashcards

  • Differential equation

    An equation involving a function and its derivatives.

  • Initial-value problem

    A DE plus a starting value.

  • Separable

    g(y) dy = h(x) dx.

  • Logistic equation

    P′ = rP(1 − P/K).

  • Carrying capacity

    K, the long-run limit of a logistic population.

  • Euler’s method

    yₙ₊₁ = yₙ + h f(tₙ, yₙ).

Self-check quiz

1. The general solution of y′ = 3y is…

Show answer

Ce^(3x). Separate and integrate.

2. In the logistic model, K is the…

Show answer

carrying capacity. The level the population approaches.

3. One Euler step from (0, 2) with y′ = y and h = 0.1 gives…

Show answer

2.2. 2 + 0.1 × 2.

4. dy/dx = x²y is…

Show answer

separable. dy/y = x² dx.

How this helps save the planet

Population models built on the logistic equation are part of how fishery scientists assess stocks and set catch limits that let populations stay healthy, which protects ocean ecosystems and the people who depend on them.

Mixing equations like Example 4.1 are the first step in estimating how quickly a lake or estuary flushes out a pollutant once the source is stopped.

Weather and climate models solve very large systems of differential equations with step-by-step numerical methods, descendants of Euler’s method.

On Saving Planets: Lakes hub · Oceans hub

Sources: OpenStax Calculus 2, 4.4 The Logistic Equation · NOAA Fisheries: population assessments

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Unit 5 · Weeks 9–11

Sequences & Series

When does adding infinitely many numbers give a finite total? Geometric series, the harmonic series and the convergence tests.

Learning objectives

By the end of this unit you can:

  • Find limits of sequences.
  • Sum geometric and telescoping series.
  • Apply the divergence, integral, comparison, alternating-series, ratio and root tests.
  • Distinguish absolute from conditional convergence.

Key concepts

  • A series Σaₙ converges if its partial sums approach a limit.
  • Geometric: Σ arⁿ (n ≥ 0) = a/(1 − r) when |r| < 1.
  • Divergence test: if aₙ does not → 0, the series diverges. The converse is false: the harmonic series Σ1/n diverges.
  • p-series Σ1/nᵖ converges iff p > 1 (by the integral test).
  • Ratio test: if |aₙ₊₁/aₙ| → L < 1 the series converges absolutely; L > 1 diverges; L = 1 says nothing.
  • Alternating series with terms decreasing to 0 converge; Σ(−1)ⁿ⁺¹/n = ln 2.

Key equations

∑n=0∞arn=a1−r (∣r∣<1)\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r}\ (|r|<1)
Geometric series
∑1np converges  ⟺  p>1\sum\frac1{n^p}\ \text{converges}\iff p>1
p-series
L=lim⁡∣an+1an∣L=\lim\left|\frac{a_{n+1}}{a_n}\right|
Ratio test

Worked examples, step by step

Example 5.1 Sum Σ (1/2)ⁿ from n = 0.

Show the steps
  1. Geometric with a = 1, r = ½.
  2. 1/(1 − ½) = 2.

Answer: 2.

Example 5.2 Each year a lake loses 10% of a stored pollutant, and 5 tonnes are added. What level does it settle at?

Show the steps
  1. After many years the amount is 5 + 5(0.9) + 5(0.9)² + …
  2. Geometric: 5/(1 − 0.9).

Answer: 50 tonnes. Cutting the yearly input in half halves the long-run level (a simplified model).

Example 5.3 Does Σ n/2ⁿ converge?

Show the steps
  1. Ratio: [(n + 1)/2ⁿ⁺¹]/[n/2ⁿ] = (n + 1)/(2n) → ½.
  2. ½ < 1.

Answer: Converges (its sum is 2).

Practice problems

Work each on paper first, then check.

  1. Σ 0.9(0.1)ⁿ from n = 0

    Show answer

    0.9/(1 − 0.1) = 1. So 0.999… = 1.

  2. Does Σ 1/n² converge?

    Show answer

    Yes, p = 2 > 1 (the sum is π²/6).

  3. Does Σ n/(n + 1) converge?

    Show answer

    No: the terms → 1, not 0 (divergence test).

  4. Sum Σ 3(2/3)ⁿ from n = 0.

    Show answer

    3/(1 − 2/3) = 9.

  5. Does Σ (−1)ⁿ⁺¹/n converge absolutely?

    Show answer

    No. It converges (to ln 2), but Σ1/n diverges, so the convergence is conditional.

Common misconceptions

  • Myth: If the terms go to 0, the series converges.

    Actually: The harmonic series is the counterexample.

  • Myth: A ratio-test limit of 1 means divergence.

    Actually: L = 1 is inconclusive; use another test.

  • Myth: Rearranging terms never changes a sum.

    Actually: For conditionally convergent series it can.

Flashcards

  • Sequence

    An ordered list a₁, a₂, …

  • Series

    Sum of a sequence; limit of partial sums.

  • Geometric series

    a/(1 − r) for |r| < 1.

  • Harmonic series

    Σ1/n, diverges.

  • Ratio test

    L < 1 converges, L > 1 diverges, L = 1 inconclusive.

  • Conditional convergence

    Converges, but not absolutely.

Self-check quiz

1. Σ (1/3)ⁿ from n = 0 =

Show answer

3/2. 1/(1 − 1/3).

2. The harmonic series Σ1/n…

Show answer

diverges. p = 1, integral test.

3. Σ 1/n^1.01…

Show answer

converges. p > 1.

4. The ratio test for Σ 1/n! gives L =

Show answer

0. 1/(n + 1) → 0, so it converges.

How this helps save the planet

Geometric series describe anything that is partly carried over from one period to the next, as in Example 5.2. That simple model shows a key fact of pollution and of greenhouse gases: the long-run level depends on the input rate, so cutting inputs lowers where things settle.

Real systems are more complex than one geometric series, and scientists use full models, but the series is the right first intuition.

On Saving Planets: Lake Erie algal bloom forecast · Lakes hub

Sources: OpenStax Calculus 2, 5.2 Infinite Series · NOAA GML: trends in atmospheric CO₂

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Unit 6 · Weeks 12–13

Power Series & Taylor Series

Turn functions into infinite polynomials. Taylor series are how calculators compute, and how physicists make hard problems simple.

Learning objectives

By the end of this unit you can:

  • Find the radius and interval of convergence of a power series.
  • Build Taylor and Maclaurin series and polynomials.
  • Bound the error of a Taylor approximation.
  • Use series to approximate integrals and to justify small-angle approximations.

Key concepts

  • A power series Σcₙ(x − a)ⁿ converges inside a radius R and diverges outside; check the endpoints separately.
  • Taylor series: cₙ = f⁽ⁿ⁾(a)/n!. Around a = 0 it is called a Maclaurin series.
  • Key series: eˣ = Σxⁿ/n!, sin x = x − x³/3! + x⁵/5! − …, cos x = 1 − x²/2! + x⁴/4! − …, ln(1 + x) = x − x²/2 + x³/3 − … (|x| < 1).
  • Taylor’s remainder: the error is at most M|x − a|ⁿ⁺¹/(n + 1)!, where M bounds the next derivative.
  • Keeping only the first term turns sin θ into θ, the small-angle approximation behind the simple pendulum formula.

Key equations

f(x)=∑n=0∞f(n)(a)n!(x−a)nf(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n
Taylor series
ex=∑n=0∞xnn!e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}
Exponential series
sin⁡x=x−x33!+x55!−⋯\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots
Sine series
∣Rn(x)∣≤M∣x−a∣n+1(n+1)!|R_n(x)|\le\frac{M|x-a|^{n+1}}{(n+1)!}
Taylor remainder bound

Worked examples, step by step

Example 6.1 Estimate e^0.1 with the first four terms of its series.

Show the steps
  1. 1 + 0.1 + 0.01/2 + 0.001/6.
  2. = 1.105167.

Answer: 1.105167, versus the true 1.105171.

Example 6.2 Estimate sin 0.5 with three terms and bound the error.

Show the steps
  1. 0.5 − 0.125/6 + 0.03125/120 = 0.479427.
  2. Next term bound: 0.5⁷/7! = 1.55 × 10⁻⁶.

Answer: 0.479427, error under 1.6 × 10⁻⁶ (true value 0.479426).

Example 6.3 Find the radius of convergence of Σ xⁿ/3ⁿ.

Show the steps
  1. Ratio: |x|/3 < 1.
  2. Endpoints x = ±3 give terms ±1, which do not → 0.

Answer: R = 3; it converges for −3 < x < 3.

Practice problems

Work each on paper first, then check.

  1. Write the Maclaurin polynomial of degree 4 for cos x and use it for cos 0.2.

    Show answer

    1 − x²/2 + x⁴/24; cos 0.2 ≈ 1 − 0.02 + 0.0000667 = 0.980067 (true 0.980067).

  2. Find the Maclaurin series of 1/(1 − x).

    Show answer

    Σxⁿ for |x| < 1 (geometric).

  3. Find the radius of convergence of Σ xⁿ/n.

    Show answer

    R = 1 (ratio n/(n + 1) → 1).

  4. Use a series to estimate ∫₀¹ e^(−x²) dx with three terms.

    Show answer

    ∫(1 − x² + x⁴/2) dx = 1 − 1/3 + 1/10 = 0.7667. More terms approach 0.7468.

Common misconceptions

  • Myth: A Taylor polynomial is accurate everywhere.

    Actually: It is best near the center a and can be terrible far away or outside the radius of convergence.

  • Myth: Every smooth function equals its Taylor series.

    Actually: Not always, though it holds for eˣ, sin x and cos x everywhere.

  • Myth: The radius of convergence decides the endpoints too.

    Actually: Endpoints must be checked separately.

Flashcards

  • Power series

    Σcₙ(x − a)ⁿ.

  • Radius of convergence

    R such that the series converges for |x − a| < R.

  • Taylor coefficient

    f⁽ⁿ⁾(a)/n!.

  • Maclaurin series

    Taylor series at a = 0.

  • Small-angle approximation

    sin θ ≈ θ, cos θ ≈ 1 − θ²/2.

  • Remainder bound

    M|x − a|ⁿ⁺¹/(n + 1)!.

Self-check quiz

1. The Maclaurin series of eˣ is…

Show answer

Σxⁿ/n!. Every derivative of eˣ at 0 is 1.

2. For small θ, sin θ ≈…

Show answer

θ. The first term of the series.

3. The coefficient of x³ in the series for sin x is…

Show answer

−1/6. −1/3!.

4. Σxⁿ converges for…

Show answer

|x| < 1. Geometric series.

How this helps save the planet

Taylor series are how physicists turn hard equations into usable ones. The pendulum period T = 2π√(L/g) in Physics I relies on sin θ ≈ θ, and the same kind of linearization underlies much of how engineers analyze vibrations in wind turbines, bridges and aircraft.

Satellite orbit and climate models expand complicated effects into series and keep the terms that matter. Knowing the error bound tells you when an approximation is good enough to trust.

On Saving Planets: Physics I, Unit 8: oscillations · Pendulum period calculator

Sources: OpenStax Calculus 2, 6.3 Taylor and Maclaurin Series · OpenStax University Physics 1, 15.4 Pendulums

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Unit 7 · Week 14

Parametric Equations & Polar Coordinates

Describe paths by time and curves by angle, including the ellipse every satellite and planet follows.

Learning objectives

By the end of this unit you can:

  • Graph parametric curves, find dy/dx and arc length.
  • Convert between polar and Cartesian coordinates.
  • Find areas and arc lengths in polar coordinates.
  • Describe a Kepler orbit in polar form.

Key concepts

  • A parametric curve gives x(t) and y(t); dy/dx = (dy/dt)/(dx/dt).
  • Arc length: ∫√(x′² + y′²) dt.
  • Polar: x = r cos θ, y = r sin θ. Area of a polar region: ½∫r² dθ.
  • An orbit around one focus is r = p/(1 + e cos θ), where e is the eccentricity.
  • Kepler’s second law (equal areas in equal times) is the statement that ½r² dθ/dt is constant.

Key equations

dydx=dy/dtdx/dt\frac{dy}{dx}=\frac{dy/dt}{dx/dt}
Slope of a parametric curve
L=∫ab(x′)2+(y′)2 dtL=\int_a^b\sqrt{(x')^2+(y')^2}\,dt
Parametric arc length
A=12∫αβr2 dθA=\tfrac12\int_\alpha^\beta r^2\,d\theta
Polar area
r=a(1−e2)1+ecos⁡θr=\frac{a(1-e^2)}{1+e\cos\theta}
Conic orbit in polar form

Worked examples, step by step

Example 7.1 Find the slope of x = t², y = t³ at t = 1.

Show the steps
  1. dx/dt = 2t, dy/dt = 3t².
  2. dy/dx = 3t²/(2t) = 3t/2 = 3/2 at t = 1.

Answer: 3/2.

Example 7.2 Find the area inside the cardioid r = 1 + cos θ.

Show the steps
  1. A = ½∫₀^(2π) (1 + cos θ)² dθ.
  2. = ½∫(1 + 2cos θ + cos²θ) dθ = ½(2π + 0 + π).

Answer: 3π/2 ≈ 4.71.

Example 7.3 Earth’s orbit has a = 149.6 million km and e = 0.0167. Find its closest and farthest distances from the Sun.

Show the steps
  1. At θ = 0: r = a(1 − e) = 149.6 × 0.9833.
  2. At θ = π: r = a(1 + e) = 149.6 × 1.0167.

Answer: About 147.1 and 152.1 million km, matching NASA’s published perihelion and aphelion.

Practice problems

Work each on paper first, then check.

  1. Find the length of x = cos t, y = sin t for 0 ≤ t ≤ 2π.

    Show answer

    ∫√(sin²t + cos²t) dt = 2π (a circle’s circumference).

  2. Find the area of the circle r = 3.

    Show answer

    ½∫₀^(2π) 9 dθ = 9π.

  3. Convert r = 2 cos θ to Cartesian form.

    Show answer

    r² = 2r cos θ gives x² + y² = 2x: a circle of radius 1 centered at (1, 0), area π.

  4. Eliminate t from x = 2 cos t, y = sin t.

    Show answer

    x²/4 + y² = 1, an ellipse.

Common misconceptions

  • Myth: dy/dx = dy/dt × dx/dt.

    Actually: Divide: (dy/dt)/(dx/dt).

  • Myth: Polar area is ∫r dθ.

    Actually: It is ½∫r² dθ, from thin pie slices.

  • Myth: The Sun is at the center of Earth’s orbit.

    Actually: It is at one focus of the ellipse.

Flashcards

  • Parametric curve

    x = x(t), y = y(t).

  • Parametric slope

    (dy/dt)/(dx/dt).

  • Polar coordinates

    (r, θ); x = r cos θ, y = r sin θ.

  • Polar area

    ½∫r² dθ.

  • Eccentricity

    e = 0 circle, 0 < e < 1 ellipse.

  • Kepler’s second law

    Equal areas in equal times.

Self-check quiz

1. Polar area uses…

Show answer

½∫r² dθ. Each slice is approximately a sector of area ½r² dθ.

2. x = r cos θ, y = …

Show answer

r sin θ. Polar to Cartesian.

3. For an orbit r = p/(1 + e cos θ), e = 0 gives…

Show answer

a circle. r = p is constant.

4. Kepler’s second law says r² dθ/dt is…

Show answer

constant. Equal areas in equal times.

How this helps save the planet

Every Earth-observing satellite follows a path described by the polar orbit equation in this unit. Mission designers choose the orbit’s shape and tilt so instruments revisit the same places under similar lighting, which makes changes on the ground easier to detect.

Parametric equations describe the ground track a satellite traces over Earth, which decides which forests, coasts and ice sheets it can see and when.

On Saving Planets: Orbits guide · Landsat 9

Sources: NASA Earth fact sheet · OpenStax Calculus 2, 7.4 Area and Arc Length in Polar Coordinates

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Computer labs

Four short computer labs apply the units with a spreadsheet: Euler’s method for a population model, Simpson’s rule for a reservoir volume, Taylor-series accuracy, and Kepler’s second law in polar coordinates.

These are the numerical methods scientists and engineers use daily. Sample data is labeled as such; each lab links to its source.

Safety reference: OpenStax Calculus Volume 2 (free textbook)

Lab A: Euler’s method for a recovering fish stock

Week 8

Goal: Solve the logistic equation numerically and compare with the exact solution.

Pn+1=Pn+h rPn(1−PnK)P_{n+1}=P_n+h\,rP_n\left(1-\frac{P_n}{K}\right)
Euler’s method for the logistic equation

Procedure

  1. Set r = 0.5, K = 1,000, P₀ = 100 and h = 1 in a spreadsheet.
  2. Fill down Euler steps to t = 4.
  3. Add a column with the exact solution and the error.
  4. Repeat with h = 0.5 and h = 0.25; how does the error change?

Data table

tEuler P (h = 1)
0100
1145.0
2207.0
3289.1
4391.9

Computed model values (not a real fishery). Exact P(4) = 450.9.

Sample calculation

  1. P₁ = 100 + 0.5 × 100 × 0.9 = 145.0.
  2. P₂ = 145 + 0.5 × 145 × 0.855 = 207.0.
  3. Euler underestimates here (391.9 vs 450.9) because the curve bends upward during early growth; halving h roughly halves the error.

Safety

  • None beyond normal computer use.

Home or virtual alternative

  • A spreadsheet is enough.

OpenStax Calculus 2, 4.2 Direction Fields and Numerical Methods

Planet link: Numerical population models support fishery management. NOAA Fisheries: population assessments

Lab B: Reservoir volume with Simpson’s rule

Week 5

Goal: Estimate a reservoir’s volume from surface-area measurements at different depths.

V=∫0HA(h) dh≈Δh3[A0+4A1+2A2+4A3+A4]V=\int_0^HA(h)\,dh\approx\frac{\Delta h}3[A_0+4A_1+2A_2+4A_3+A_4]
Volume as the integral of cross-sectional area

Procedure

  1. Enter the sample areas at 0, 2, 4, 6 and 8 m above the bottom.
  2. Apply Simpson’s rule with Δh = 2 m.
  3. Compare with the trapezoid rule.

Data table

Height h (m)Surface area A (km²)
00.0
21.2
42.6
64.0
85.6

Sample values for practice, not a real reservoir.

Sample calculation

  1. Simpson: (2/3)[0 + 4(1.2) + 2(2.6) + 4(4.0) + 5.6] = (2/3)(31.6) = 21.07 km²·m.
  2. 1 km²·m = 10⁶ m³, so V ≈ 2.11 × 10⁷ m³ (21 million cubic meters).
  3. Trapezoid: 2[0/2 + 1.2 + 2.6 + 4.0 + 5.6/2] = 21.2 km²·m, close to Simpson.

Safety

  • None beyond normal computer use.

Home or virtual alternative

  • A spreadsheet is enough.

NASA: SWOT mission

Planet link: Surface-water satellites such as SWOT help estimate changes in water stored in lakes and reservoirs.

Lab C: How good is a Taylor polynomial?

Week 13

Goal: Compare Taylor polynomials of sin x with the true value and with the remainder bound.

P5(x)=x−x36+x5120P_5(x)=x-\frac{x^3}{6}+\frac{x^5}{120}
Degree-5 Maclaurin polynomial of sin x

Procedure

  1. Tabulate sin x, P₁ = x, P₃ and P₅ for x from 0 to π in steps of 0.25.
  2. Plot the errors.
  3. Find where the small-angle approximation sin x ≈ x is off by 1%.

Data table

xsin xP₅(x)
0.50.4794260.479427
1.00.8414710.841667
2.00.9092970.933333

Computed values.

Sample calculation

  1. At x = 0.5, P₅ errs by about 10⁻⁶; at x = 2 by 0.024. Accuracy falls fast away from 0.
  2. sin x ≈ x is within 1% up to about x = 0.24 rad (14°), which is why the pendulum formula is used for small swings.

Safety

  • None beyond normal computer use.

Home or virtual alternative

  • Desmos sliders work well.

PhET: Pendulum Lab

Planet link: Knowing an approximation’s error is how engineers decide which simplified model to trust.

Lab D: Kepler’s second law in polar coordinates

Week 14

Goal: Show numerically that an elliptical orbit sweeps area faster near the focus in angle but at a constant rate in time.

A(θ1,θ2)=12∫θ1θ2r(θ)2 dθ,r=a(1−e2)1+ecos⁡θA(\theta_1,\theta_2)=\tfrac12\int_{\theta_1}^{\theta_2}r(\theta)^2\,d\theta,\quad r=\frac{a(1-e^2)}{1+e\cos\theta}
Swept area of an orbit

Procedure

  1. Use a = 1 and e = 0.5.
  2. Compute the swept area from θ = 0 to 0.5 and from θ = π − 0.25 to π + 0.25 (same 0.5 rad wide) with Simpson’s rule.
  3. Explain why equal angles do not give equal areas, and so take unequal times.

Data table

ArcSwept area
θ = 0 to 0.5 (near the Sun)0.0643
θ = π ± 0.25 (far side)0.5513

Computed values for a = 1, e = 0.5.

Sample calculation

  1. Near the Sun r ≈ 0.5, far side r ≈ 1.5: area ∝ r², so the far arc sweeps about 8.6 times more area (0.5513 ÷ 0.0643).
  2. Since area grows at a constant rate in time, the planet spends about 8.6 times longer crossing the far 0.5 rad: it moves slowest at aphelion.

Safety

  • None beyond normal computer use.

Home or virtual alternative

  • A spreadsheet or Desmos is enough.

PhET: Kepler’s Laws

Planet link: Orbit geometry decides when a satellite can see a place on Earth.

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Exam prep practice sets

Time yourself: about 10 minutes per problem. Then compare with the full solutions.

Midterm practice set (Units 1–3)

  1. Find ∫₀¹ x² eˣ dx.

    Show full solution
    1. Parts twice: ∫x²eˣ dx = (x² − 2x + 2)eˣ.
    2. From 0 to 1: e − 2.

    Answer: e − 2 ≈ 0.718.

  2. Find ∫₀¹ x/(x + 1) dx.

    Show full solution
    1. x/(x + 1) = 1 − 1/(x + 1).
    2. [x − ln(x + 1)]₀¹ = 1 − ln 2.

    Answer: 1 − ln 2 ≈ 0.307.

  3. Find the volume when the region under y = x², 0 ≤ x ≤ 2, is spun about the x-axis.

    Show full solution
    1. Disks: ∫₀² π x⁴ dx = π × 32/5.

    Answer: 32π/5 ≈ 20.1.

  4. Evaluate ∫₂^∞ dx/x³.

    Show full solution
    1. [−1/(2x²)]₂^∞ = 1/8.

    Answer: 1/8.

Final practice set (all units)

  1. Solve y′ = −2y, y(0) = 5, and find y(1).

    Show full solution
    1. y = 5e^(−2t).

    Answer: 5e^(−2) ≈ 0.677.

  2. Sum Σ 3(2/3)ⁿ from n = 0.

    Show full solution
    1. Geometric: 3/(1 − 2/3).

    Answer: 9.

  3. Estimate cos 0.2 with the degree-4 Maclaurin polynomial.

    Show full solution
    1. 1 − 0.04/2 + 0.0016/24.

    Answer: 0.980067.

  4. Find the area inside r = 3.

    Show full solution
    1. ½∫₀^(2π) 9 dθ.

    Answer: 9π ≈ 28.3.

  5. Find the interval of convergence of Σ xⁿ/3ⁿ.

    Show full solution
    1. Ratio: |x|/3 < 1; endpoints diverge.

    Answer: −3 < x < 3.

  6. A mixing tank’s pollutant follows S′ = −S/50 with S(0) = 20 kg. When is S = 5 kg?

    Show full solution
    1. S = 20e^(−t/50); e^(−t/50) = 1/4.
    2. t = 50 ln 4.

    Answer: ≈ 69.3 time units.

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Formula sheet

∫u dv=uv−∫v du\int u\,dv=uv-\int v\,du
sin⁡2x=12(1−cos⁡2x), cos⁡2x=12(1+cos⁡2x)\sin^2x=\tfrac12(1-\cos2x),\ \cos^2x=\tfrac12(1+\cos2x)
Sn=Δx3[f0+4f1+2f2+⋯+fn]S_n=\tfrac{\Delta x}3[f_0+4f_1+2f_2+\cdots+f_n]
V=∫πf2 dx, V=∫2πxf dxV=\int\pi f^2\,dx,\ V=\int2\pi x f\,dx
L=∫1+f′2 dxL=\int\sqrt{1+f'^2}\,dx
W=∫F dxW=\int F\,dx
∫1∞x−pdx<∞  ⟺  p>1\int_1^\infty x^{-p}dx<\infty\iff p>1
y′=ky⇒y=y0ekty'=ky\Rightarrow y=y_0e^{kt}
P′=rP(1−P/K)P'=rP(1-P/K)
yn+1=yn+hf(tn,yn)y_{n+1}=y_n+hf(t_n,y_n)
∑arn=a/(1−r)\sum ar^n=a/(1-r)
L=lim⁡∣an+1/an∣L=\lim|a_{n+1}/a_n|
ex=∑xn/n!e^x=\sum x^n/n!
sin⁡x=∑(−1)nx2n+1/(2n+1)!\sin x=\sum(-1)^nx^{2n+1}/(2n+1)!
cos⁡x=∑(−1)nx2n/(2n)!\cos x=\sum(-1)^nx^{2n}/(2n)!
dy/dx=(dy/dt)/(dx/dt)dy/dx=(dy/dt)/(dx/dt)
A=12∫r2 dθA=\tfrac12\int r^2\,d\theta

Constants

ConstantValue
e2.718281828…
π3.141592654…
ln 20.693147…
Earth’s orbital eccentricity0.0167
Earth’s semi-major axis149.6 million km
Density of water1,000 kg/m³

Source: NASA Earth fact sheet

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Graphing and data tools

Use these free tools alongside the course.

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Glossary

Arc length
Length of a curve, ∫√(1 + f′²) dx.
Carrying capacity
Long-run limit K of a logistic population.
Center of mass
Balance point; moment divided by mass.
Conditional convergence
Converges, but the absolute values diverge.
Differential equation
Equation involving derivatives of an unknown function.
Eccentricity
How stretched an orbit is.
Euler’s method
Step-by-step numerical solution of a DE.
Geometric series
Σarⁿ.
Improper integral
Integral with an infinite limit or integrand.
Integration by parts
∫u dv = uv − ∫v du.
Logistic equation
P′ = rP(1 − P/K).
Maclaurin series
Taylor series at 0.
Parametric equations
x and y given as functions of t.
Partial fractions
Splitting a rational function into simpler terms.
Polar coordinates
Position given by distance r and angle θ.
Power series
Σcₙ(x − a)ⁿ.
Radius of convergence
Distance from the center within which a power series converges.
Separable equation
DE that can be written g(y) dy = h(x) dx.
Simpson’s rule
Numerical integration with parabolas.
Taylor series
Σf⁽ⁿ⁾(a)(x − a)ⁿ/n!.

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Self-assessment rubric

No grade or credit is given. Use this to judge honestly where you are in each unit.

LevelWhat it looks like
4 · MasteryChooses the right technique or test unprompted, justifies it, and checks results numerically.
3 · ProficientSolves the practice problems with occasional slips.
2 · DevelopingFollows worked examples but struggles to choose a method.
1 · BeginningApplies formulas without checking conditions; needs to revisit the unit.

Suggested weighting if you want a score

PartWeight
Practice problems30%
Computer labs15%
Midterm practice set25%
Final practice set30%

Aim for level 3 or better in every unit before moving to the next course.

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