📈 Calculus I

What you’ll learn

1. Why Calculus ExistsExplain why calculus was invented and who built it.Newton (1665–66) and Leibniz (1670s) independently created the mathematics of continuous change — Newton for motion and gravity, Leibniz with the notation we still use. Their work capped a line running from Archimedes and was made rigorous by Cauchy and Weierstrass in the 1800s.
2. Functions: The Raw MaterialDefine functions, domain, range, and composition as the objects calculus studies.A function pairs each input with exactly one output, and its graph makes change visible. Real systems are webs of dependencies — output responding to input — and composition chains those machines together, setting up the chain rule to come.
3. The Central Problem: Change at an InstantPose the instantaneous-rate problem and the shrinking-window strategy that solves it.Instantaneous speed cannot be computed by direct division — zero distance over zero time — yet speedometers report it. The resolution is to average over shrinking intervals and take the trend's limit, the core move of differential calculus.
4. Limits: Sneaking Up on a ValueDefine the limit and evaluate limits at removable holes.A limit is the value a function approaches, indifferent to what happens at the point itself: (x²−1)/(x−1) has a hole at x = 1 yet limit 2. The squeeze instinct is ancient — Archimedes trapped π between polygon bounds two millennia before the formal definition.
5. Continuity and Limits at InfinityDefine continuity and use limits at infinity to describe long-run behavior.Continuity means limit and value agree — no holes, jumps, or gaps — and most physical quantities have it, while counts and step-priced quantities do not. Limits as x → ∞ capture equilibrium: decay curves flattening toward asymptotes.
6. The Derivative, DefinedDefine the derivative as the limit of secant slopes and compute it for x².Secant slopes are average rates; their limit as the interval shrinks is the tangent slope — the derivative. Applying the definition to x² gives 2x after the crucial cancellation of h, the computation every shortcut rule replaces.
7. Derivative Rules: Power, Constant, SumDifferentiate polynomials with the power, constant, sum, and constant-multiple rules.d/dx xⁿ = n·xⁿ⁻¹, constants vanish, sums split, and coefficients ride along — enough to differentiate any polynomial on sight, e.g. x³ + 4x² − 7x + 9 → 3x² + 8x − 7. Lagrange's primes, Newton's dots, and Leibniz's dy/dx all name the same object.
8. Product and Quotient RulesApply the product and quotient rules and avoid the multiply-the-derivatives trap.(fg)′ = f′g + fg′ — each factor changes while the other holds still — verified cleanly on x²·x³ = x⁵. Even Leibniz first guessed d(uv) = du·dv before correcting himself; the quotient rule (f′g − fg′)/g² completes the pair.
9. The Chain RuleUse the chain rule on composite functions and recognize chained rates in context.When functions nest, rates multiply: dy/dx = f′(g(x))·g′(x) — outside derivative times inside derivative, as in (3x+1)² → 6(3x+1). Real systems chain dependencies constantly, making this the workhorse rule of applied calculus.
10. Derivatives of Sine, Cosine, and ExponentialsKnow the derivatives of sin, cos, eˣ, and ln x and connect them to oscillation and growth.sin → cos and cos → −sin: differentiation shifts the wave a quarter cycle, which is why pendulum acceleration opposes displacement. eˣ is its own derivative — growth rate equal to value — and ln x differentiates to 1/x.
11. Related RatesSolve related-rates problems with the differentiate-then-substitute discipline.Link the variables (V = (4/3)πr³), differentiate the whole equation with respect to time, then substitute the instant's values: dV/dt = 4πr²·dr/dt explains precisely why an inflating balloon's radius growth slows — quartering when the radius doubles.
12. Maxima and MinimaFind critical points and classify them with the first-derivative test.Smooth extremes hide where f′ = 0, turning "find the best" into an equation — the vertex of x² − x − 6 falls at x = ½. The sign of f′ on either side classifies each critical point as a max, min, or mere pause.
13. Reading Curves: f, f′, and f″Interpret f, f′, and f″ together: slope, concavity, and inflection.f′ gives direction and steepness; f″ gives the bend — concave up cups like a bowl, concave down like a dome, and inflection points mark the flip. In motion language the trio is position, velocity, acceleration, narrating anything from a downhill run to a skydiver approaching terminal velocity.
14. Optimization in the Real WorldRun the full optimization workflow: model, differentiate, solve, classify, answer.The fence problem — A(x) = x(50−x), A′ = 0 at x = 25, square wins with 625 m² — is the complete template. Nature runs the same mathematics: soap films minimize area, and the hexagonal honeycomb was proven minimal by Hales in 1999.
15. The Bridge to IntegrationPreview integration as accumulation and state the Fundamental Theorem.Integration accumulates a rate into a total — area under the rate curve, computed by Archimedes-style slicing — the inverse of differentiation, as the Fundamental Theorem guarantees. Rocket guidance integrating acceleration into velocity and position shows the direction Calculus II travels.

Questions this course answers

Grounded in trusted sources

• OpenStax, Calculus Volume 1 (Rice University)
• James Stewart, Calculus: Early Transcendentals (Cengage)
• MIT OpenCourseWare 18.01 — Single Variable Calculus
• Carl B. Boyer, The History of the Calculus and Its Conceptual Development (Dover)

Every Wunder lesson is built from real, reputable sources — never invented.

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