Calculus 1, also known as differential and integral calculus, is a branch of mathematics that deals with the study of rates of change (derivatives) and accumulation (integrals) of functions, where functions can be represented by algebraic expressions like f(x) = ax^n and the key concepts include limits, such as lim_(x->c) f(x) = L, which describe the behavior of a function as the input approaches a particular value; continuity, which requires that lim_(x->c) f(x) = f(c) for a function to be continuous at x = c; derivatives, denoted as f'(x) or df/dx, which quantify the instantaneous rate of change of a function with respect to its input; differentiation rules, like the sum rule, f'(x) + g'(x) = (f+g)'(x), product rule, (fg)'(x) = f'(x)g(x) + f(x)g'(x), quotient rule, (f/g)'(x) = (f'(x)g(x) – f(x)g'(x))/g^2(x), and chain rule, (f(g(x)))’ = f'(g(x))g'(x); basic derivative formulas, such as d/dx(x^n) = nx^(n-1), d/dx(sin(x)) = cos(x), d/dx(cos(x)) = -sin(x), and d/dx(e^x) = e^x; higher-order derivatives, like f”(x), which represent the rate of change of the first derivative; applications of derivatives, including finding local extrema (maxima and minima), where f'(x) = 0 or is undefined, and determining concavity and inflection points, which involve analyzing the second derivative; indefinite integrals, or antiderivatives, denoted by ∫f(x)dx = F(x) + C, which represent the inverse operation of differentiation; integration rules, such as the reverse power rule, ∫x^n dx = (x^(n+1))/(n+1) + C, and substitution, ∫f(g(x))g'(x)dx = ∫f(u)du, where u = g(x); definite integrals, denoted by ∫[a,b] f(x)dx, which represent the accumulated value of a function over an interval [a, b]; the Fundamental Theorem of Calculus, which states that if F'(x) = f(x), then ∫[a,b] f(x)dx = F(b) – F(a), connecting the concepts of derivatives and integrals; and applications of integrals, such as finding areas under curves and solving real-world problems related to motion, growth, and other phenomena that involve rates of change or accumulation, while advanced integration methods take place in here, they are used to evaluate more complex integrals, include integration by parts, which is based on the product rule for differentiation and can be expressed as ∫u dv = uv – ∫v du, where u and v are functions of x, and choosing appropriate functions u and dv can simplify the integral; partial fractions, a technique applicable to rational functions (fractions with polynomials in the numerator and denominator), where the goal is to decompose the given function into a sum of simpler fractions, with expressions like (Ax + B)/(x^2 + Cx + D) decomposed into (A/(x – r1)) + (B/(x – r2)), where r1 and r2 are roots of the denominator, and then integrate each simpler fraction; trigonometric substitution, which involves substituting a trigonometric function for a variable to simplify integrals with expressions like √(a^2 – x^2), √(a^2 + x^2), or √(x^2 – a^2) by using substitutions such as x = a sin(θ), x = a tan(θ), or x = a sec(θ), respectively, and then transforming the integral into a trigonometric integral; integration by trigonometric identities, which relies on applying various trigonometric identities like sin^2(x) + cos^2(x) = 1, 1 + tan^2(x) = sec^2(x), or sin(2x) = 2sin(x)cos(x) to rewrite and simplify the given integral, making it easier to evaluate; and numerical integration methods, such as the midpoint rule, trapezoidal rule, or Simpson’s rule, which involve approximating the definite integral of a function by dividing the interval [a, b] into n subintervals and summing the areas of geometric shapes (rectangles, trapezoids, or parabolas, respectively) constructed using the function values at specific points within each subinterval, with formulas like ∫[a,b] f(x)dx ≈ (b-a)/n * [f(x1) + f(x2) + … + f(xn)], where xi are the chosen points within each subinterval, and n is the number of subintervals, to provide an estimate of the integral’s value when an exact antiderivative is difficult or impossible to find.