Calculus Essentials

Cheatsheet Content

Calculus Cheat Sheet Topic: Fundamental Concepts & Applications Key Formulas Derivative Definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$ Product Rule: $(fg)' = f'g + fg'$ Quotient Rule: $(\frac{f}{g})' = \frac{f'g - fg'}{g^2}$ Chain Rule: $(f(g(x)))' = f'(g(x))g'(x)$ Definite Integral: $\int_a^b f(x) dx = F(b) - F(a)$ Integration by Parts: $\int u \, dv = uv - \int v \, du$ Geometric Series: $\sum_{n=0}^\infty ar^n = \frac{a}{1-r}$ for $|r| Taylor Series: $f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$ Common Pitfalls Algebra Errors: Especially with fractions, exponents, and signs. Limit Evaluation: Forgetting L'Hôpital's Rule or applying it incorrectly. Chain Rule Mistakes: Not applying it, or applying it multiple times when not needed. Integration Constant: Forgetting the "+C" for indefinite integrals. Bounds for U-Substitution: Not changing integral bounds when using u-sub in definite integrals. Series Convergence Tests: Misapplying conditions for Integral, Ratio, Root, or Comparison Tests. Domain Issues: Not considering where functions are defined or differentiable. Definitions Limit: The value a function approaches as the input approaches some value. Continuity: A function is continuous if its graph can be drawn without lifting the pen (no breaks, jumps, or holes). Derivative: The instantaneous rate of change of a function; the slope of the tangent line. Antiderivative: The reverse process of differentiation. Definite Integral: Represents the net signed area under a curve. Sequence: An ordered list of numbers. Series: The sum of the terms of a sequence. Convergence: When a sequence or series approaches a finite value. Divergence: When a sequence or series does not approach a finite value. Critical Point: Where $f'(x)=0$ or $f'(x)$ is undefined. Potential local max/min. Inflection Point: Where the concavity of a function changes. Units & Constants While Calculus itself doesn't have fundamental physical constants like physics, here are some related mathematical constants and their relevance: Constant Value / Description $e$ Euler's number, approx. $2.71828$. Base of natural logarithm. $\pi$ Pi, approx. $3.14159$. Ratio of a circle's circumference to its diameter. $C$ Constant of integration (arbitrary real number). $dx, dy$ Differentials, representing infinitesimally small changes in $x, y$. $\infty$ Infinity, representing an unbounded quantity. Diagrams (Conceptual) Visualizing concepts is key in Calculus. Here are some conceptual diagrams to aid understanding: Tangent Line: A line that touches a curve at a single point and has the same slope as the curve at that point ($f'(x)$). Area Under Curve: The region bounded by a function's graph, the x-axis, and vertical lines, calculated by definite integral. Concavity: Concave Up (U-shape): $f''(x) > 0$ Concave Down (n-shape): $f''(x) For instance, a diagram for a tangent line might look like this: Tangent Point Function $f(x)$ And a simple representation of Area Under Curve: Area $f(x)$ x-axis