∫ Calculus Intro

Limits · Derivatives · Integrals · Applications

🎯 What Is a Limit?

A limit describes what value a function approaches as the input gets close to a certain point — even if the function isn't defined there.

lim(x→a) f(x) = L

"As x gets closer and closer to a, f(x) gets closer and closer to L."

📈 Visual Explorer

Function:
x approaches: 0
LIMIT VALUE
L = 1
sin(x)/x → 1 as x → 0, even though the function is undefined at x=0

📋 Famous Limits

lim(x→0) sin(x)/x = 1
lim(x→∞) (1 + 1/n)ⁿ = e
lim(x→0) (eˣ - 1)/x = 1
lim(x→∞) 1/x = 0

⚡ L'Hôpital's Rule

When a limit gives 0/0 or ∞/∞, you can take the derivative of numerator and denominator separately:

lim f(x)/g(x) = lim f'(x)/g'(x)

Example: sin(x)/x → cos(x)/1 = cos(0) = 1 ✓

📐 The Derivative — Tangent Line Visualizer

f(x) =
x = 0.5

📊 Results

x =0.5
f(x) =0.25
f'(x) =1.0
f'(x) = slope of the red tangent line

📋 Differentiation Rules

Power: d/dx xⁿ = nxⁿ⁻¹
Sum: (f+g)' = f' + g'
Product:(fg)' = f'g + fg'
Chain: d/dx f(g(x)) = f'(g(x))·g'(x)
sin: d/dx sin(x) = cos(x)
cos: d/dx cos(x) = -sin(x)
eˣ: d/dx eˣ = eˣ
ln(x): d/dx ln(x) = 1/x

💡 What Does f'(x) Mean?

f'(x) > 0 → function increasing
f'(x) < 0 → function decreasing
f'(x) = 0 → local max or min
f''(x) > 0 → concave up
f''(x) < 0 → concave down

📊 Riemann Sum — Area Under a Curve

f(x) =
Method:
n = 6

📊 Results

n =6
Riemann ≈—
Exact =—
Error =—

As n → ∞, the Riemann sum → exact integral

📋 Integration Rules

Power: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
sin: ∫sin(x) dx = −cos(x) + C
cos: ∫cos(x) dx = sin(x) + C
eˣ: ∫eˣ dx = eˣ + C
1/x: ∫1/x dx = ln|x| + C

⭐ Fundamental Theorem

∫ₐᵇ f'(x) dx = f(b) − f(a)

Differentiation and integration are inverse operations. The area under f'(x) equals the change in f(x).

🚗 Velocity & Position

v(t) = dx/dt · a(t) = dv/dt

Position is the integral of velocity. Velocity is the integral of acceleration. The speedometer reading is a derivative — the rate of change of distance with time.

x(t) = x₀ + v₀t + ½at²
v(t) = v₀ + at

📐 Area & Volume

A = ∫ₐᵇ f(x) dx

Integration finds areas under curves, volumes of 3D shapes (disk/shell method), surface areas, and arc lengths of curves.

Volume of sphere = ∫₋ᵣʳ π(r²−x²) dx = 4/3πr³

📈 Optimization

f'(x) = 0 → max/min

Find maximum profit, minimum cost, optimal box dimensions. Set the derivative to zero and solve. Check second derivative to confirm max vs min.

f'(x) = 0 AND f''(x) < 0 → local maximum

🌊 Differential Equations

dy/dx = ky → y = Ce^(kx)

Equations relating a function to its own derivative. Model population growth, radioactive decay, heat transfer, and compound interest.

Exponential growth: P(t) = P₀ e^(rt)

📊 Where Calculus Appears in STEM

Physics
F = ma = m·d²x/dt², Maxwell's equations, quantum mechanics wave functions
Engineering
Beam deflection, fluid flow (Navier-Stokes), control systems, signal processing
Biology
Population models, drug concentration curves, neural firing rates, epidemics
Economics
Marginal cost/revenue, consumer surplus, continuous compounding, Nash equilibria
Computer Science
Machine learning gradients (backpropagation), graphics (Bezier curves), algorithms analysis
Medicine
Pharmacokinetics, MRI reconstruction, blood flow modeling, ECG analysis
The Big Three Ideas

Limit: what a function approaches. Derivative: instantaneous rate of change (slope). Integral: accumulated total (area). They are inverses of each other — the Fundamental Theorem of Calculus.

Key terms
  • Limit — the value a function approaches as x → a
  • Continuity — no holes, jumps or asymptotes at a point
  • Derivative f'(x) — slope of the tangent line; instantaneous rate of change
  • Integral ∫f(x)dx — area under the curve; antiderivative
  • Riemann sum — approximating area with rectangles
  • Definite integral — integral from a to b; gives a number
  • Indefinite integral — general antiderivative; gives a family of functions + C
Newton vs Leibniz

Isaac Newton and Gottfried Leibniz independently invented calculus in the 1660s–1680s. Newton used it to derive the laws of planetary motion. Leibniz developed the notation we still use today: dx/dy and ∫.

🎯 Try this challenge

The derivative of a function at a point is the slope of the tangent line there. Move the slider on the Derivatives tab to x = 0 for f(x) = sin(x). What is the slope? What does that tell you about the function at that point?

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