What does the set of real numbers R contain?
All numbers used in this course and its subsets.
What defines an interval I of real numbers R?
It contains every real x between any two numbers a and b in I.
Which endpoints does the interval [a;b] include?
Both endpoints a and b.
Which endpoints does the interval ]a;b[ exclude?
Both endpoints a and b.
Which endpoint does the interval [a;b[ include and exclude?
Includes a but excludes b.
Which endpoint does the interval ]a;b] exclude and include?
Excludes a but includes b.
What is the set R* in terms of real numbers?
R without zero, R \ {0}.
What intervals define R− and R*−?
R− is ]−∞;0], R*− is ]−∞;0[.
What does F ⊂ E mean for sets F and E?
Every element of F belongs to E.
What does the symbol ∃ mean in quantifiers?
There exists at least one.
What is an example of a quantified algebraic assertion involving real x?
For every real x ≠ −1 and 0, 1/(x−1)−1/(x+1)=1/(x²+x).
When is an integer divisible by 2 based on its units digit?
When its units digit is even.
When is an integer divisible by 5 based on its units digit?
When its units digit is 0 or 5.
When is an integer divisible by 4 based on its last two digits?
When its last two digits form a number divisible by 4.
When is an integer divisible by 11?
When its alternating sum of digits is divisible by 11.
What is the approximate value of √2?
Approximately 1.4.
What is the approximate value of ln 3?
Approximately 1.1.
How do you multiply a number by 5 using multiplication and division?
Multiply by 10 and divide by 2.
How do you multiply a number by 9 using multiplication and subtraction?
Multiply by 10 and subtract the original number.
What defines a prime number?
An integer ≥ 2 divisible only by 1 and itself.
What is a fraction a/b when b ≠ 0?
The result of dividing numerator a by nonzero denominator b.
What is the formula for adding fractions a/b and c/d?
(ad + bc) / bd
What is the formula for multiplying fractions a/b and c/d?
(ac) / (bd)
What is the formula for dividing fractions a/b by c/d with nonzero denominators?
(ad) / (bc)
What is the square root √a for a ≥ 0?
The unique nonnegative real number whose square equals a.
What is √(a²) equal to for any real a?
The absolute value of a, |a|.
What is the rule for multiplying powers aⁿ and aᵐ with nonzero a?
aⁿ × aᵐ = aⁿ⁺ᵐ
What are three methods to prove A = B in algebra?
Transform A into B, transform both into the same expression C, or show A − B = 0.
Why must domains be checked before proving two expressions in x are equal?
Because the equality may hold only on their common domain.
How is the equality proved for x ≠ 1?
By putting the right-hand side over the denominator x − 1.
What does a function f defined on E associate with each real x in E?
At most one real value f(x).
In a function f, what is f(x) called?
The image of x.
In a function f, what is x called relative to f(x)?
An antecedent of f(x).
What conditions define the domain of a quotient A(x)/B(x)?
A and B must be defined and B(x) ≠ 0.
What additional condition restricts the domain of √A(x)?
A(x) must be greater than or equal to zero.
What points make up the graph Cf of a function f?
Points (x,y) where x is in the domain of f and y = f(x).
When is a function considered even?
When its domain is symmetric about zero and f(−x) = f(x) for all x.
What does it mean for a function to be increasing?
For x0 < x1, f(x0) ≤ f(x1).
What does it mean for a function to be decreasing?
For x0 < x1, f(x0) ≥ f(x1).
What does a nonnegative derivative imply about a function?
The function is increasing on that interval.
What does a nonpositive derivative imply about a function?
The function is decreasing on that interval.
What does a zero derivative everywhere imply about a function?
The function is constant on that interval.
When is the composition g ◦ f defined?
When f's image is contained in g's domain.
How is the composition g ◦ f defined?
By (g ◦ f)(x) = g(f(x)).
Why is the square-root function the inverse of the square function on R+?
Because every nonnegative real has one unique nonnegative square root.
How is the rational power a^(p/q) defined for a > 0?
By (a^p)^(1/q) = (a^(1/q))^p.
What is the product rule for rational powers of a positive base a?
a^r times a^{r'} equals a^{r+r'}.
What is the quotient rule for rational powers of a positive base a?
a^r divided by a^{r'} equals a^{r-r'}.
What is the unique positive solution to the equation x³ = 125?
x = 125^{1/3} = 5.
How is a vector represented in R² using components?
As a pair (a,b) representing a i + b j.
What are the components of vector AB for points A and B?
(xB - xA, yB - yA).
How do you add two vectors (x,y) and (x′,y′)?
By summing components to get (x+x′, y+y′).
How do you multiply vector (x,y) by scalar k?
By scaling components to get (kx, ky).
What is the formula for the midpoint I of points A and B?
I = ((xA + xB)/2, (yA + yB)/2).
How is the norm of vector (x,y) calculated in an orthonormal system?
As the square root of x² + y².
What is the distance formula between points A and B?
The square root of (xB - xA)² + (yB - yA)².
When are two vectors considered collinear?
When one is zero or a nonzero scalar multiple of the other.
What condition on the determinant shows vectors (x,y) and (x′,y′) are collinear?
Their determinant xy′ - x′y equals zero.
How is the scalar product of vectors (x,y) and (x′,y′) calculated?
It is xx′ + yy′.
When are two vectors orthogonal based on their scalar product?
When their scalar product is zero.
Which vector is orthogonal to every vector?
The zero vector.
What is a direction vector of line (AB) for distinct points A and B?
Vector AB is a direction vector of line (AB).
What characterizes every other direction vector of line (AB)?
They are nonzero vectors collinear with AB.
What are the parametric equations of a line through A(xA,yA) with direction vector (α,β)?
x = xA + tα and y = yA + tβ for t in ℝ.
What is the normal vector if (α,β) is a direction vector of a line?
The vector (−β, α) is a normal vector.
What is the Cartesian equation of a line with direction vector (α,β) through A(xA,yA)?
β(x − xA) − α(y − yA) = 0.
What is the general Cartesian equation of any line?
ax + by + c = 0.
What are the normal and direction vectors from the line equation ax + by + c = 0?
Normal vector is (a,b) and direction vector is (−b,a).
When are two nonvertical lines y = mx + p parallel or perpendicular?
Parallel if slopes are equal; perpendicular if slope product is −1.
What is the formula defining an affine function on R?
f(x) = mx + p for real numbers m and p.
When is an affine function strictly increasing on R?
When m > 0.
When is an affine function constant on R?
When m = 0.
How is the slope m of an affine function through two points calculated?
m = (y2 - y1) / (x2 - x1).
What is the equation of the affine function through points (x1,y1) and (x2,y2)?
f(x) = m(x - x1) + y1.
How is the absolute value |x| defined for x ≥ 0?
|x| = x.
What does the absolute value represent?
The distance from zero.
How do you find the zero of ax + b when solving inequalities?
The zero is −b/a.
Test your knowledge with 50 questions on Mathematics Foundations for Economics.
1. Which interval contains every real number between its two endpoints whenever both endpoints belong to the interval?
2. Which notation represents an interval that includes its left endpoint but excludes its right endpoint?
Review the complete course in the revision sheet for Mathematics Foundations for Economics.
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