Calculus 1: The key for Science, Engineering and Economics.

Examine functions as domain to codomain mappings, use the vertical line test, and explore image, preimage, and explicit or implicit forms.

Classify functions as even or odd and identify increasing or decreasing behavior, including increasing or decreasing cases, for power, polynomial, rational, exponential, and logarithmic functions with base e and ln.

Study periodic functions, unit-circle definitions of sine and cosine, the sine squared plus cosine squared equals one identity, then review tangent, cotangent, and composite functions with domain rules.

Explore the core function types: surjective, injective, and bijective, and their inverses, with examples using x^2, x^3+2x^2, and the square root of x; learn when inverse functions exist.

Explore transformations of functions, including rigid translations and reflections, and the non-rigid dilation, using constants to shift, reflect, or stretch the graph of y = f(x).

Explore limits of numerical sequences, including convergence, divergence, and monotonic behavior. Define limits using epsilon, and illustrate with a_n = n/(n+1) and the Euler number e.

Study limits of functions with real domain, including finite and infinite limits, via epsilon-delta definitions and neighborhoods, and apply sum, difference, product, and quotient rules.

Explore continuity and discontinuity through limits at x0 via epsilon-delta definitions, left- and right-hand limits, and types like jump, infinite (vertical asymptote), removable, and endpoint discontinuities.

Explore theorems on limits, including the uniqueness of the limit and epsilon-neighborhood reasoning. Apply the squeeze theorem, comparison theorems, and limits at infinity, such as sin x over x.

Explore key limit theorems for continuous functions, including Bolzano’s theorem, the intermediate value theorem, and the extreme value theorem, with examples on polynomials, rational limits, and special limits.

Explore big-O notation to compare function behavior near a point, using f and g to define big-O and little-o relations and apply them to limits and continuity.

Explore big-O notation and Landau symbols to analyze special limits, compare x^n and x^m as x→0 or ∞, apply little-o rules, and solve limits like sin x over x.

Define infinitesimal and infinite functions and analyze vertical, horizontal, and oblique asymptotes through limits, right- and left-hand approaches, and example calculations.

Learn how derivatives analyze functions and real life applications through the tangent slope, with f'(x) or df/dx defined as a limit.

Master derivative rules for calculus, including linearity with constants, product and quotient rules, and the chain rule for composite functions, with examples of sine, cosine, exponential, and logarithmic functions.

Identify non differentiable points, corners, vertical tangents, and cusps, and locate critical points where the derivative equals zero to determine function maxima and minima.

Fermat's theorem links local extrema to critical points where the derivative vanishes, while Rolle's theorem, the mean value theorem (Legrange's theorem), and L'Hôpital's rule connect differentiability to limits.

Explore how monotony follows the derivative's sign to locate local maxima and minima, using critical points and non-differentiability. Learn higher-order derivatives, second derivative, concavity, convexity, and inflection points.

Determine the domain from the square root and denominator; identify horizontal asymptotes y = 1 and y = −1, no vertical asymptotes, and analyze sign, roots, and critical points.

Study functions by determining the domain (excluding x = −1), finding horizontal and vertical asymptotes, analyzing sign, and using derivatives to locate maxima and minima and sketch qualitative graphs.

Learn indefinite integrals by identifying primitive functions, with a constant c. Explore rules for x^α, 1, 1/x, sin, cos, and e^x and how boundary conditions determine a single primitive.

This lecture presents the linearity of the indefinite integral, showing term-by-term integration and constant extraction, then covers integration by parts and substitution with practical x e^x and x e^{x^2} examples.

Explore the rules of indefinite integration for rational functions, using substitution, polynomial division, and decomposition to express integrals as natural logarithms and polynomials plus constants.

Explore definite integrals: compute precise values as areas under the curve using Riemann sums, understand integrability conditions, and apply the fundamental theorem of calculus to evaluate with antiderivatives.

Explore definite integrals through integration by parts and substitution, using the fundamental theorem of calculus to compute areas, with even/odd function considerations and applications in economics and physics.

Explore complex numbers, the imaginary unit i, real and imaginary parts in rectangular form on the complex plane, and polar and exponential forms via Euler's formula for multiplication and division.

Explore Taylor and Maclaurin series to approximate functions by polynomials in a neighborhood of a point x0, improving accuracy by increasing polynomial degree.

Explore commonly used Maclaurin series for e^x, ln(1+x), sine, and cosine, and learn how increasing polynomial degree improves approximations and simplifies limits.