Calculus (Midterm 2024)

Question 1

Describe and explain the formal definition of a limit

Answer 1

The formal definition of a limit states that lim(x → a) f(x) = L if, for every ε > 0, there exists a δ > 0 such that whenever 0 < |x - a| < δ, it follows that |f(x) - L| < ε.

Question 2

State the formal definition of injectivity

Answer 2

A function f: A → B is injective if, for every x₁ and x₂ in A, whenever f(x₁) = f(x₂), it follows that x₁ = x₂.

Question 3

What is a function?

Answer 3

A function is from a set of x to a set of y (f(x) x —> y), and is a rule that assigns a unique value of y to each value of x. At most, one value of y to each value of x (vertical line test)

Question 4

What is domain?

Answer 4

Subset of f where f is defined.

Question 5

What is range?

Answer 5

Subset of values where f reaches.

Question 6

What is the inverse of a function?

Answer 6

The inverse of a function f is a function, denoted f⁻¹, that reverses the effect of f. Formally, f⁻¹ satisfies the following properties for every element y in the codomain of f and every element x in the domain of f:

1.	f(f⁻¹(y)) = y for all y in the codomain of f.
2.	f⁻¹(f(x)) = x for all x in the domain of f.