Functions in calculus

Constant function f(x) = c The constant function f(x) = c: its horizontal-line graph, why its derivative is 0, its integral cx + C, limits, and common mistakes with constants. Linear function f(x) = mx + b The linear function f(x) = mx + b: slope and intercept, its straight-line graph, why its derivative is the slope m, its integral and worked examples. Quadratic function f(x) = ax^2 + bx + c, \quad a \ne 0 The quadratic function f(x) = ax² + bx + c: parabola graph, vertex formula, roots, derivative 2ax + b, integral and worked calculus examples. Power functions f(x) = x^n Power functions f(x) = xⁿ for whole, negative and fractional n: graphs, domains, the derivative nxⁿ⁻¹, the integral and the n = -1 exception. Polynomial functions f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 Polynomial functions explained: degree, leading term, end behavior, roots and turning points, plus how to differentiate and integrate any polynomial. Rational functions f(x) = \frac{P(x)}{Q(x)}, \quad Q(x) \ne 0 Rational functions P(x)/Q(x): domain, vertical and horizontal asymptotes, holes, derivatives with the quotient rule and integrals by partial fractions. Reciprocal function f(x) = \frac{1}{x} The reciprocal function 1/x: its hyperbola graph and asymptotes, the derivative -1/x², why its integral is ln|x| + C, and its one-sided limits. Square root function f(x) = \sqrt{x} = x^{1/2} The square root function f(x) = √x: its domain and graph, the derivative 1/(2√x), the integral (2/3)x^(3/2) + C, and the vertical tangent at 0. Absolute value function f(x) = |x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases} The absolute value function |x|: its V-shaped graph, why it is continuous but not differentiable at 0, its derivative x/|x|, and its integral.

Natural exponential function f(x) = e^x The natural exponential function e^x: what e is, its graph, why it is its own derivative and integral, key limits, series and worked examples. Exponential functions f(x) = b^x, \quad b > 0,\ b \ne 1 Exponential functions f(x) = b^x: growth and decay graphs, properties, the derivative b^x ln b, the integral b^x / ln b + C and worked examples. Natural logarithm f(x) = \ln x The natural logarithm ln x: its graph, log rules, why its derivative is 1/x, the integral x ln x - x + C, and worked calculus examples. Logarithmic functions f(x) = \log_b x, \quad b > 0,\ b \ne 1 Logarithmic functions log_b x: definition, graphs, log rules, change of base, the derivative 1/(x ln b), the integral and worked examples.

Sine function f(x) = \sin x The sine function sin(x): its graph, domain, range, period, key values, derivative cos(x), integral and the mistakes students make with it. Cosine function f(x) = \cos x The cosine function cos(x): its graph, domain, range, period, key values, derivative −sin(x), integral sin(x) + C and common mistakes. Tangent function f(x) = \tan x = \frac{\sin x}{\cos x} The tangent function tan(x) = sin(x)/cos(x): its graph, vertical asymptotes, period π, derivative sec²(x), integral ln|sec x| + C and examples. Secant, cosecant and cotangent \sec x = \frac{1}{\cos x}, \quad \csc x = \frac{1}{\sin x}, \quad \cot x = \frac{\cos x}{\sin x} The reciprocal trig functions sec x, csc x and cot x: definitions, graphs, asymptotes, all three derivatives and integrals, with worked examples.

Arcsine f(x) = \arcsin x = \sin^{-1} x The arcsine function arcsin(x), the inverse of sine: domain [−1, 1], range [−π/2, π/2], graph, derivative 1/√(1−x²), integral and examples. Arccosine f(x) = \arccos x = \cos^{-1} x The arccosine function arccos(x), the inverse of cosine: domain [−1, 1], range [0, π], graph, derivative −1/√(1−x²), integral and examples. Arctangent f(x) = \arctan x = \tan^{-1} x The arctangent function arctan(x), the inverse of tangent: domain all reals, range (−π/2, π/2), graph, derivative 1/(1+x²), integral and examples.

Hyperbolic functions \sinh x = \frac{e^x - e^{-x}}{2}, \quad \cosh x = \frac{e^x + e^{-x}}{2}, \quad \tanh x = \frac{\sinh x}{\cosh x} The hyperbolic functions sinh x, cosh x and tanh x: exponential definitions, graphs, identities, derivatives, integrals and worked examples.

Piecewise functions f(x) = \begin{cases} f_1(x) & x < a \\ f_2(x) & x \ge a \end{cases} Piecewise functions explained: how to read and graph them, find one-sided limits, check continuity and differentiability, and integrate across pieces. Floor and ceiling functions \lfloor x \rfloor = \max\{n \in \mathbb{Z} : n \le x\}, \quad \lceil x \rceil = \min\{n \in \mathbb{Z} : n \ge x\} The floor function ⌊x⌋ and ceiling function ⌈x⌉: definitions, step graphs, one-sided limits, continuity, derivative and how to integrate them.