Math Study Notes

Cards · 20

Pythagorean Theorem

a^2 + b^2 = c^2, where a and b are the lengths of the legs of a right triangle, and c is the length of the hypotenuse.

Quadratic Formula

x = (-b +/- sqrt(b^2 - 4ac)) / (2a), used to find the roots of ax^2 + bx + c = 0, where a != 0.

Slope of a Line

m = (y2 - y1) / (x2 - x1), representing the rate of vertical change over horizontal change between two points.

Slope-Intercept Form

y = mx + b, where m represents the slope of the line and b represents the y-intercept.

Area of a Circle

A = pi * r^2, where A is the area and r is the radius of the circle.

Circumference of a Circle

C = 2 * pi * r, where C is the perimeter of the circle and r is its radius.

Distance Formula

d = sqrt((x2 - x1)^2 + (y2 - y1)^2), giving the straight-line distance between two points in a Cartesian plane.

Euler's Formula (Polyhedra)

V - E + F = 2, where V is the number of vertices, E is edges, and F is faces in a convex polyhedron.

Fundamental Theorem of Calculus (Part 1)

d/dx integral from a to x of f(t) dt = f(x), linking differentiation and integration for a continuous function f.

Definition of the Derivative

f'(x) = lim (h -> 0) [f(x + h) - f(x)] / h, measuring the instantaneous rate of change of f at x.

Product Rule for Differentiation

(f * g)' = f' * g + f * g', the formula for computing the derivative of the product of two functions.

Quotient Rule for Differentiation

(f / g)' = (f' * g - f * g') / g^2, the formula for differentiating the ratio of two functions, with g != 0.

Chain Rule

(f(g(x)))' = f'(g(x)) * g'(x), the rule for differentiating composite functions.

Law of Sines

a / sin(A) = b / sin(B) = c / sin(C), relating the side lengths to the sines of their opposite angles in any triangle.

Law of Cosines

c^2 = a^2 + b^2 - 2ab * cos(C), generalizing the Pythagorean theorem to any triangle with angle C.

Arithmetic Sequence N-th Term

a_n = a_1 + (n - 1)d, where a_1 is the first term, d is the common difference, and n is the term index.

Geometric Sequence N-th Term

a_n = a_1 * r^(n - 1), where a_1 is the initial term, r is the common ratio, and n is the term index.

Permutation Formula

P(n, k) = n! / (n - k)!, counting the number of ordered arrangements of k elements from a set of n elements.

Combination Formula

C(n, k) = n! / (k! * (n - k)!), calculating the number of unordered selections of k items chosen from n items.

Euler's Identity

e^(i * pi) + 1 = 0, connecting five fundamental mathematical constants: e, i, pi, 1, and 0.