Calculus 12

A piecewise function is one function with a rulebook: which formula applies depends on which x-interval you're in.

The seams (boundary points) are where interesting things happen — evaluate with the piece whose interval CONTAINS the point (watch ≤ vs <).

The parent library (polynomial, rational, radical, exponential, log, trig) supplies the graphs whose limits and slopes calculus will interrogate.

sin⁻¹, cos⁻¹, tan⁻¹ answer 'what angle produced this ratio?' — but periodic functions need RESTRICTED ranges to make the answer unique.

The conventions: sin⁻¹ returns [-π/2, π/2]; cos⁻¹ returns [0, π]; tan⁻¹ returns (-π/2, π/2).

The limit is the value f(x) approaches as x approaches a — the destination, whether or not f(a) exists or agrees.

A two-sided limit exists only when the left and right limits agree; jumps make them disagree.

Algebraic evaluation: substitute; if 0/0, factor and cancel (or rationalize), then substitute again.

Limits at infinity read end behaviour: divide by the highest power — rational functions settle to 0, a leading-coefficient ratio, or ±∞.

Continuity at a: f(a) exists, lim(x→a) f(x) exists, and they're EQUAL — three checks, one failure breaks it.

Average rate = secant slope over an interval; instantaneous rate = tangent slope at a point — the limit of secants as the interval shrinks.

f'(a) = lim(h→0) [f(a+h) - f(a)]/h — the difference quotient's limit IS the derivative.

Notations name the same object: f'(x), dy/dx, d/dx[f(x)] — Newton's and Leibniz's rival notations both survived.

Power rule: d/dx[xⁿ] = nxⁿ⁻¹. Product: (fg)' = f'g + fg'. Quotient: (f/g)' = (f'g - fg')/g². Chain: outside' times inside'.

The transcendental facts: d/dx[eˣ] = eˣ, d/dx[ln x] = 1/x, d/dx[sin x] = cos x, d/dx[cos x] = -sin x.

A Riemann sum adds rectangle areas f(xᵢ)·Δx; left, right, and midpoint choices bracket the truth, trapezoids split the difference.

The definite integral ∫ₐᵇ f(x)dx is the limit of Riemann sums as slices shrink — signed area under the curve.

The indefinite integral ∫f(x)dx is the antiderivative FAMILY — hence the +C.

FTC: ∫ₐᵇ f(x)dx = F(b) - F(a) where F' = f — area computed by antiderivative, no rectangles required.

Substitution reverses the chain rule: set u = inner function, convert dx to du completely, integrate in u.