Week 1 - Before Calculus

Question 1

Calculus was invented more or less simultaneously and independently by

Answer 1

Newton (1665-1667) and Leibniz (1672-1674)

Question 2

State the fundamental theorem of calculus.

Answer 2

d/dt xa∫ f(t) dt = f(x)

Question 3

For two real numbers (ℝ) exactly one of the following holds.

Answer 3

a<b>b “a is more than b”</b>

a=b “a equals b”</b>

Question 4

Define the notation

:=

Answer 4

By Definition

Question 5

Define the absolute value of the real number a.

Answer 5

|a| := { a if a≥0

{ a if a<0

Question 6

How can we think of |a|.

Answer 6

As the size of a.

Question 7

Define the triangle inequality for real numbers.

Answer 7

For every a and b,

|a + b| ≤ |a| +|b|

Question 8

Definite the notation [a, b], for a < b.

Answer 8

:= {x∈ℝ : a ≤ x and x ≤ b}

Question 9

Definite the notation (a, b], for a < b.

Answer 9

:= {x∈ℝ : a < x and x ≤ b}

Question 10

Definite the notation [a, b), for a < b.

Answer 10

:= {x∈ℝ : a ≤ x and x < b}

Question 11

Definite the notation (a, b), for a < b.

Answer 11

:= {x∈ℝ : a < x and x < b}

Question 12

Express the set
A := {a∈ℝ : |a| ≤ 1}
as an interval.

Answer 12

A = [-1, 1]

Question 13

A function is a collection of 3 ingredients, what are they?

Answer 13

1) A is a set of numbers called the domain of the function
2) B is a set of numbers called the codomain of the function
3) f is a “mapping” between A and B, mapping assigns a unique element y = f(x) ∈B

Question 14

How do we indicate a function?

Answer 14

We write

f : A → B

Question 15

What is the maximal domain of f?

Answer 15

The largest set of real numbers A that can serve as the domain.