One-sided limits describe what a function approaches from just one side of a number. The left-hand limit uses x-values less than a, and the right-hand limit uses x-values greater than a. The two-sided limit exists only when both one-sided limits exist and are equal.
A limit is the value a function approaches as x gets closer and closer to a number. We write lim f(x) = L as x → a when f(x) can be made as close to L as we like by taking x close enough to a, from both sides. The function does not even have to be defined at a.
Mathematical induction proves that a statement is true for every positive whole number n. First, show it is true for n = 1 (the base case). Next, assume it is true for some n = k and use that to show it is true for n = k + 1 (the inductive step). Together, these prove it for all n.
The binomial theorem expands a power of a binomial without multiplying it out by hand: (a + b)ⁿ is the sum of C(n, k)aⁿ⁻ᵏbᵏ for k = 0 to n. The coefficients come from combinations or from row n of Pascal’s triangle. The powers of a go down from n to 0 while the powers of b go up from 0 to n.
Permutations and combinations count the ways to choose r items from n. In a permutation, order matters: P(n, r) = n!/(n − r)!. In a combination, order does not matter: C(n, r) = n!/(r!(n − r)!). Choosing a president and a vice president is a permutation. Choosing a two-person team is a combination.
Projectile motion with parametric equations models a thrown or launched object with one equation for horizontal distance and one for height. With launch speed v₀, angle θ and starting height h, in feet, x = (v₀ cos θ)t and y = h + (v₀ sin θ)t − 16t². In meters, replace 16 with 4.9.
Parametric equations describe a curve with two equations, x = f(t) and y = g(t), that share a third variable t, called the parameter. Each value of t gives one point (x, y). Plotting the points in order of t shows the curve and the direction it is traced. Eliminating t gives one equation in x and y.
Vector word problems combine forces, velocities or trips that act in different directions. To solve one, draw a sketch, write each vector in component form, add the components to get the resultant, then turn the resultant back into a magnitude and direction. Work problems use the dot product instead.
The dot product of two vectors multiplies the matching components and adds them: ⟨a, b⟩ · ⟨c, d⟩ = ac + bd. The answer is a number, not a vector. It also equals ‖u‖‖v‖ cos θ, so it gives the angle between two vectors. A dot product of 0 means they are perpendicular.
Adding and subtracting vectors is done one component at a time: ⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩, and subtraction works the same way. On a graph, place the tail of the second vector at the tip of the first. The arrow from start to finish is the sum, called the resultant.