Calculus Study Guide

📖 Core Concepts Calculus – study of continuous change; two linked branches: differential (instantaneous rates, slopes) and integral (accumulation, areas). Limit – describes a function’s behavior as the input approaches a point; foundation for rigor via the $\varepsilon$–$\delta$ definition. Derivative $f'(a)=\displaystyle\lim{h\to0}\frac{f(a+h)-f(a)}{h}$ – instantaneous rate of change; geometric slope of the tangent line; linear operator on functions. Leibniz notation $\frac{dy}{dx}$ – read “dy over dx”; shorthand for the limit definition; $d/dx$ acts on a function to produce its derivative. Indefinite integral $\displaystyle\int f(x)\,dx$ – family of antiderivatives $F$ such that $F'=f$; includes a constant $C$. Definite integral $\displaystyle\int{a}^{b} f(x)\,dx=\lim{\| \Delta\|\to0}\sum{i=1}^{n} f(xi^)\,\Delta xi$ – limit of Riemann sums, gives area‑type quantities. Fundamental Theorem of Calculus – differentiation and integration are inverses: if $F'=f$ on $(a,b)$, then $\displaystyle\int{a}^{b} f(x)\,dx = F(b)-F(a)$. 📌 Must Remember Derivative definition: $f'(a)=\lim{h\to0}\frac{f(a+h)-f(a)}{h}$. Indefinite integral: $\int f(x)\,dx = F(x)+C$ where $F'=f$. Definite integral as Riemann sum limit. FTC formula: $\int{a}^{b} f(x)\,dx = F(b)-F(a)$. Newton’s second law (physics): $F = m\,a = m\,\dfrac{d^{2}x}{dt^{2}}$. Radioactive decay solution: $N(t)=N{0}e^{-\lambda t}$ from $\dfrac{dN}{dt}=-\lambda N$. Simple population growth: $N(t)=N{0}e^{rt}$ from $\dfrac{dN}{dt}=rN$. Logistic growth differential equation: $\displaystyle\dfrac{dN}{dt}=rN\!\left(1-\dfrac{N}{K}\right)$. Marginal cost/revenue: $MC=\dfrac{dC}{dQ}$, $MR=\dfrac{dR}{dQ}$; profit max when $MC=MR$. 🔄 Key Processes Differentiation (Finding $f'(x)$) Write the limit definition. Simplify algebraically; cancel $h$ where possible. Take the limit $h\to0$. Integration (Finding $\int f(x)\,dx$) Identify antiderivative patterns (power rule, trig, exponential, etc.). Add constant $C$ for indefinite integrals. Evaluating Definite Integrals via FTC Find any antiderivative $F$ of $f$. Compute $F(b)-F(a)$. Solving First‑Order ODEs (e.g., decay, growth) Separate variables: $\dfrac{dN}{N}= -\lambda\,dt$ (or $r\,dt$). Integrate both sides. Exponentiate to solve for $N(t)$. 🔍 Key Comparisons Derivative vs. Integral Derivative: local rate, slope of tangent, units of “output per input”. Integral: global accumulation, area under curve, units of “output × input”. Leibniz $\frac{dy}{dx}$ vs. Lagrange $f'$ $\frac{dy}{dx}$ emphasizes the variable with respect to which we differentiate; useful in chain rule and implicit differentiation. $f'$ is concise, ideal for writing higher‑order derivatives ($f'', f^{(n)}$). Indefinite vs. Definite Integral Indefinite: produces a family of functions + $C$. Definite: yields a number (area, net change) by applying limits $a$ and $b$. ⚠️ Common Misunderstandings “Infinitesimals are real numbers.” – Modern calculus replaces them with limits; they are not actual numbers. Treating $\frac{dy}{dx}$ as a simple fraction. It is a notation for a limit; however, the fraction view works in the chain rule and separable ODEs because of the underlying limit properties. Forgetting the constant $C$ in indefinite integrals. Leads to missing solutions, especially in differential equations. Assuming the derivative always exists if a function is drawn smoothly. Differentiability requires a limit to exist; corners, cusps, or vertical tangents break it. 🧠 Mental Models / Intuition Slope‑as‑instantaneous‑rate – Picture a tiny “zoomed‑in” straight line touching the curve; the steeper the line, the larger the derivative. Area‑as‑stacked‑rectangles – Riemann sums: think of piling thin rectangles under the curve; as they get thinner, the total height approaches the true area. FTC as “undo” – Integration “adds up” infinitesimal changes; differentiation “undoes” that addition. 🚩 Exceptions & Edge Cases Discontinuous functions – Derivative does not exist at jump discontinuities. Improper integrals – When $a$ or $b$ is infinite or $f$ has an infinite discontinuity, the limit definition must be used to test convergence. Functions with vertical tangents – Derivative may be infinite ($\pm\infty$), still a valid limit in extended real numbers. 📍 When to Use Which Use derivative when asked for instantaneous speed, slope, marginal cost/revenue, or to find extrema (set $f'=0$). Use definite integral when the problem asks for total distance, area, accumulated quantity, or net change over an interval. Apply FTC when a definite integral can be evaluated by finding an antiderivative (most “nice” functions). Use separation of variables for first‑order ODEs that can be written as $g(y)dy = h(x)dx$. 👀 Patterns to Recognize $f'(x)=0$ → potential maxima/minima (follow with second‑derivative test). Integrand of the form $f'(x)\,g(f(x))$ → suggests substitution $u = f(x)$. Growth/decay ODEs have the pattern $\dfrac{dy}{dt}=ky$ → solution $y(t)=y0e^{kt}$. Logistic equation always includes a factor $(1-\frac{N}{K})$. 🗂️ Exam Traps Choosing the wrong sign in FTC: $\int{a}^{b} f(x)\,dx = F(b)-F(a)$, not $F(a)-F(b)$. Dropping the $C$ in indefinite integrals, especially when solving differential equations – leads to missing families of solutions. Confusing marginal cost (derivative of cost) with average cost – marginal is instantaneous, average is $C/Q$. Assuming all continuous functions are differentiable – remember corners and cusps break differentiability. Misreading $\frac{dy}{dx}$ as “dy divided by dx” in contexts where it must be treated as a limit operator (e.g., implicit differentiation). --- Study tip: Flash these bullets, then write one quick example for each process (differentiate $x^3$, integrate $e^x$, solve $dN/dt=rN$). Master the patterns, and the traps disappear!