Fundamental theorem of calculus Study Guide

📖 Core Concepts Fundamental Theorem of Calculus (FTC) – Connects differentiation (instantaneous rate) with integration (accumulated area). The two operations are inverses. Area (or “integral”) function \(F(x)=\displaystyle\int{a}^{x}f(t)\,dt\). Continuous \(f\) on \([a,b]\) ⇒ \(F\) is continuous on \([a,b]\) and differentiable on \((a,b)\). First Part (FTC‑1): \(F'(x)=f(x)\). Antiderivative – Any function \(G\) with \(G'(x)=f(x)\). Second Part (FTC‑2): If \(G\) is an antiderivative of \(f\) on \([a,b]\), then \(\displaystyle\int{a}^{b} f(t)\,dt = G(b)-G(a)\). Riemann integrable – A function whose upper and lower sums can be made arbitrarily close; the FTC‑2 still works even if \(f\) has a few discontinuities, as long as it’s Riemann integrable. --- 📌 Must Remember FTC‑1: For continuous \(f\), \(\displaystyle\frac{d}{dx}\int{a}^{x}f(t)\,dt = f(x)\). FTC‑2: \(\displaystyle\int{a}^{b} f(t)\,dt = F(b)-F(a)\) for any antiderivative \(F\) of \(f\). Uniform continuity of the area function \(F\) on a closed interval \([a,b]\). Mean Value Theorem for integrals: \(\displaystyle\int{x}^{x+\Delta x} f(t)\,dt = f(c)\,\Delta x\) for some \(c\) in \((x,x+\Delta x)\). When \(f\) is not continuous but Riemann integrable, FTC‑2 still holds. --- 🔄 Key Processes Using FTC‑1 to find a derivative of an integral Define \(F(x)=\int{a}^{x}f(t)\,dt\). Differentiate: \(F'(x)=f(x)\). Evaluating a definite integral with FTC‑2 Find any antiderivative \(G\) of \(f\). Compute \(G(b)-G(a)\). Riemann‑sum proof sketch (FTC‑2) Partition \([a,b]\) into \([x{i-1},xi]\) with widths \(\Delta xi\). Apply mean‑value theorem: \(F(xi)-F(x{i-1}) = f(ci)\Delta xi\). Sum: \(\sum f(ci)\Delta xi = F(b)-F(a)\). Take limit as \(\max\Delta xi \to 0\) → \(\int{a}^{b} f(t)\,dt = F(b)-F(a)\). --- 🔍 Key Comparisons FTC‑1 vs. FTC‑2 FTC‑1: Differentiates an integral → gives the original integrand. FTC‑2: Integrates a function by evaluating an antiderivative at the endpoints. Continuous \(f\) vs. Riemann‑integrable \(f\) Continuous → guarantees both parts of FTC automatically. Riemann‑integrable (may have jump discontinuities) → still valid for FTC‑2, but FTC‑1 requires continuity for the derivative to equal \(f\). --- ⚠️ Common Misunderstandings “The derivative of any integral equals the integrand” – Only true when the upper limit is the variable and the integrand is continuous on the interval (FTC‑1). Confusing antiderivative with indefinite integral – An indefinite integral \(\int f(x)\,dx\) denotes a family of antiderivatives; FTC‑2 uses any single antiderivative \(F\). Assuming FTC works for improper integrals automatically – The theorem requires the function to be (properly) Riemann integrable on the closed interval. --- 🧠 Mental Models / Intuition “Area‑as‑a‑function”: Imagine the graph of \(f\) and a sliding vertical line at \(x\). The area swept from \(a\) to the line is \(F(x)\). Moving the line a tiny bit adds a thin rectangle of height \(f(x)\); the rectangle’s height is exactly the instantaneous rate of change of the accumulated area → \(F'(x)=f(x)\). Inverse machines: Think of integration as a “store” that accumulates tiny pieces, and differentiation as a “scanner” that reads the current piece. The scanner (derivative) tells you exactly what is being stored at that point. --- 🚩 Exceptions & Edge Cases Discontinuous but integrable functions – FTC‑2 still works; however, FTC‑1 may fail at points of discontinuity (the derivative of the area function may not equal \(f\) there). Endpoints – FTC‑1 guarantees differentiability only on the open interval \((a,b)\); at the exact endpoints the derivative may not be defined. --- 📍 When to Use Which Need a quick value of a definite integral? → Find any antiderivative (FTC‑2) and apply \(F(b)-F(a)\). Given an integral with a variable upper limit and asked for its derivative? → Apply FTC‑1 directly: derivative equals the integrand (provided continuity). Function is piecewise or has a few jump points? → Check Riemann integrability; use FTC‑2 for the integral, avoid FTC‑1 at the jumps. --- 👀 Patterns to Recognize Integral of a derivative → \(\displaystyle\int{a}^{b} F'(x)\,dx = F(b)-F(a)\). Derivative of an integral with variable limit → \(\displaystyle\frac{d}{dx}\int{a}^{x} f(t)\,dt = f(x)\). Repeated “+C” in indefinite integrals – any antiderivative works for FTC‑2; the constant cancels in \(F(b)-F(a)\). --- 🗂️ Exam Traps Choosing the wrong limit – Forgetting that FTC‑2 uses the same antiderivative for both limits; mixing different antiderivatives adds extra constants. Assuming continuity automatically – A question may give a function with a removable discontinuity; you must verify continuity on the interval before applying FTC‑1. Confusing \(\inta^b f(t)\,dt\) with \(\inta^x f(t)\,dt\) – The former is a number; the latter is a function of \(x\). Their derivatives behave differently. Mis‑applying the Mean Value Theorem for integrals – The theorem guarantees a point \(c\) inside the subinterval, not at the endpoints; using endpoint values leads to wrong estimates. ---