Study sheet: Mathematics Foundations for Economics

📋 Course Outline

1. Real Numbers and Intervals
2. Sets and Quantifiers
3. Mental Calculation Rules
4. Fractions, Radicals, and Powers
5. Proving Algebraic Equalities
6. Functions and Domains
7. Symmetry, Variation, and Composition
8. Inverse and Rational Powers
9. Vectors and Distances
10. Collinearity and Orthogonality
11. Lines and Their Equations
12. Affine Functions and Inequalities
13. Systems of Two Equations
14. Quadratic Polynomials
15. Derivatives and Tangents

📖 1. Real Numbers and Intervals

🔑 Key Concepts & Definitions

• Real numbers : The set of numbers used throughout this course and its subsets.
• Interval : An interval I of R contains every real x between any two real numbers a and b belonging to I.

★ Must-know

📌 The interval [a;b] includes both endpoints, ]a;b[ excludes both, [a;b[ includes a but excludes b, and ]a;b] excludes a but includes b.

Further detail

• The standard subsets are:
  • R* = R \ {0}
  • R− = ]−∞;0]
  • R*− = ]−∞;0[
  • R+ = [0;+∞[
  • R*+ = ]0;+∞[

💡 Memory Hook

Closed intervals include endpoints, whereas open intervals exclude them.

📖 2. Sets and Quantifiers

🔑 Key Concepts & Definitions

• Set inclusion : For sets F and E, F ⊂ E means that every element of F belongs to E, while F ⊄ E means that at least one element of F does not belong to E.
• Quantifiers : The symbols ∃, ∃!, and ∀ mean respectively “there exists at least one,” “there exists exactly one,” and “for every.”

📝 Essential Points

• For every real x different from −1 and 0, $\frac{1}{x-1}-\frac{1}{x+1}=\frac{1}{x^2+x}$ is an example of a quantified algebraic assertion.

📖 3. Mental Calculation Rules

★ Must-know

📌 An integer is divisible by 2, 5, or 4 when respectively its units digit is even, its units digit is 0 or 5, or its last two digits form a number divisible by 4.

Further detail

📌 An integer is divisible by 11 when its alternating sum of digits is divisible by 11.

• Useful approximate values are:

  • √2 ≈ 1.4
  • √3 ≈ 1.7
  • √5 ≈ 2.2
  • ln 2 ≈ 0.7
  • ln 3 ≈ 1.1
  • ln 5 ≈ 1.6

• To multiply by 5, multiply by 10 and divide by 2; to multiply by 25, multiply by 100 and divide by 4; to multiply by 9, multiply by 10 and subtract the original number.

📖 4. Fractions, Radicals, and Powers

🔑 Key Concepts & Definitions

• Prime number : An integer greater than or equal to 2 divisible only by 1 and itself.
• Fraction : The result of dividing the numerator a by the nonzero denominator b.
• Square root : The unique nonnegative real number whose square equals a, for a ≥ 0.

📝 Essential Points

📐 Formula — For nonzero denominators, $\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$ and $\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}$.

📐 Formula — For nonzero denominators and a nonzero divisor, $\frac{a}{b}\div\frac{c}{d}=\frac{ad}{bc}$.

📌 For a ≥ 0, √(a²) = a when a ≥ 0 and √(a²) = −a when a ≤ 0, so √(a²) = |a|.

📐 Formula — For nonzero a and integers n and m, $a^n a^m=a^{n+m}$, $\frac{a^n}{a^m}=a^{n-m}$, and $(a^n)^m=a^{nm}$.

📖 5. Proving Algebraic Equalities

★ Must-know

• 🔄 Three methods can prove an equality:
  1. Transform A into B
  2. Transform A and B into the same expression C
  3. Show that A − B = 0

📌 Before proving that two expressions in x are equal, their domains must be checked because the equality may hold only on their common domain.

Further detail

• For x ≠ 1, $\frac{2x^2-5x-1}{x-1}=2x-3-\frac{4}{x-1}$ is proved by putting the right-hand side over the denominator x − 1.

💡 Memory Hook

Transform one side, transform both sides, or show the difference is zero.

📖 6. Functions and Domains

🔑 Key Concepts & Definitions

• Function : A function f defined on E associates at most one real f(x) with each real x belonging to E.
• Graph of a function : The graph Cf consists of points (x,y) such that x belongs to the domain of f and y = f(x).

📝 Essential Points

📌 For a function f, f(x) is the image of x and x is an antecedent of f(x).

📌 The domain of a quotient A(x)/B(x) requires A and B to be defined and B(x) ≠ 0, while the domain of √A(x) additionally requires A(x) ≥ 0.

📖 7. Symmetry, Variation, and Composition

🔑 Key Concepts & Definitions

• Even function : A function is even when its domain is symmetric about zero and f(−x) = f(x) for every x in its domain.
• Monotonicity : A function is increasing when x0 < x1 implies f(x0) ≤ f(x1), and decreasing when x0 < x1 implies f(x0) ≥ f(x1).
• Composition : If f(Df) ⊂ Dg, the composition g ◦ f is defined by (g ◦ f)(x) = g(f(x)).

📝 Essential Points

📌 If f is differentiable on an interval and f′ is nonnegative everywhere, f is increasing; if f′ is nonpositive everywhere, f is decreasing; and if f′ is zero everywhere, f is constant.

💡 Memory Hook

Even functions mirror across the y-axis, whereas odd functions mirror through the origin.

📖 8. Inverse and Rational Powers

🔑 Key Concepts & Definitions

• Inverse function : The square-root function is the inverse of the square function restricted from R to R+ because every nonnegative real has one unique nonnegative square root.
• Rational power : For a > 0, p ∈ Z, and q ∈ N*, the rational power a^(p/q) is defined by (a^p)^(1/q) = (a^(1/q))^p.

📝 Essential Points

📐 Formula — For positive a and b and rational r and r′, $a^r a^{r′}=a^{r+r′}$ and $\frac{a^r}{a^{r′}}=a^{r-r′}$.

• The equation x³ = 125 has the unique positive solution x = 125^(1/3) = 5.

💡 Memory Hook

Power, restrict to positive inputs, then invert with the corresponding root.

📖 9. Vectors and Distances

🔑 Key Concepts & Definitions

• Vector components : A pair of real components representing a i + b j.

★ Must-know

• For A(xA,yA) and B(xB,yB), the vector AB has components $(x_B-x_A,y_B-y_A)$.

• The sum of (x,y) and (x′,y′) is (x+x′,y+y′), and multiplication of (x,y) by k gives (kx,ky).

📐 Formula — In an orthonormal coordinate system, the norm of (x,y) is $\|(x,y)\|=\sqrt{x^2+y^2}$ and the distance AB is $\sqrt{(x_B-x_A)^2+(y_B-y_A)^2}$.

Further detail

📐 Formula — The midpoint I of A(xA,yA) and B(xB,yB) has coordinates $I=\left(\frac{x_A+x_B}{2},\frac{y_A+y_B}{2}\right)$.

📖 10. Collinearity and Orthogonality

🔑 Key Concepts & Definitions

• Collinear vectors : Two vectors are collinear when one is zero or one is a nonzero scalar multiple of the other.
• Scalar product : The scalar product of (x,y) and (x′,y′) is xx′ + yy′.

📝 Essential Points

• Vectors (x,y) and (x′,y′) are collinear if and only if their determinant $xy′-x′y$ is zero.

📌 Two vectors are orthogonal if and only if their scalar product is zero, and the zero vector is orthogonal to every vector.

💡 Memory Hook

A zero determinant means collinearity, whereas a zero dot product means orthogonality.

📖 11. Lines and Their Equations

🔑 Key Concepts & Definitions

• Direction vector : For distinct points A and B, AB is a direction vector of the line (AB), and every other direction vector is a nonzero collinear vector.

★ Must-know

📐 Formula — A line through A(xA,yA) with direction vector (α,β) has the parametric equations $x=x_A+t\alpha,\quad y=y_A+t\beta,\quad t\in\mathbb R$.

📐 Formula — If (α,β) is a direction vector, then (−β,α) is a normal vector and a Cartesian equation of the line is $\beta(x-x_A)-\alpha(y-y_A)=0$.

📌 For y = mx + p, m is the slope and p is the y-intercept; two nonvertical lines are parallel when their slopes are equal and perpendicular when their slope product is −1.

Further detail

• Every line has a Cartesian equation ax + by + c = 0 with normal vector (a,b) and direction vector (−b,a).

💡 Memory Hook

Point and direction produce a parametric equation; direction and normal produce a Cartesian equation.

📖 12. Affine Functions and Inequalities

🔑 Key Concepts & Definitions

• Affine function : An affine function is defined on R by f(x) = mx + p for real numbers m and p.
• Absolute value : The absolute value is defined by |x| = x when x ≥ 0 and |x| = −x when x ≤ 0, and represents the distance from zero.

📝 Essential Points

📌 An affine function is strictly increasing on R when m > 0, strictly decreasing when m < 0, and constant when m = 0.

📐 Formula — The affine function through two distinct points (x1,y1) and (x2,y2) has slope $m=\frac{y_2-y_1}{x_2-x_1}$ and equation $f(x)=m(x-x_1)+y_1$.

📌 To solve an inequality involving ax+b, one determines its zero −b/a and uses the sign of a on the two resulting intervals.

📖 13. Systems of Two Equations

🔑 Key Concepts & Definitions

• System of two equations : A 2×2 linear system consists of two linear equations in two unknowns x and y, and its solutions are the pairs satisfying both equations.

★ Must-know

📌 Graphically, a system has one solution when its lines intersect once, no solution when they are strictly parallel, and infinitely many solutions when they coincide.

📐 Formula — For the system ax+by=c and a′x+b′y=c′, the determinant is $D=ab′-a′b$.

Further detail

• When D ≠ 0, a system can be solved by elementary row operations that preserve its solution set until each unknown is isolated.

💡 Memory Hook

A nonzero determinant gives one solution; a zero determinant gives none or infinitely many.

📖 14. Quadratic Polynomials

🔑 Key Concepts & Definitions

• Quadratic polynomial : A quadratic polynomial function has the form f(x)=ax²+bx+c with a ≠ 0 and is defined on R.
• Discriminant : For ax²+bx+c=0, the discriminant is Δ=b²−4ac.

📝 Essential Points

📌 If Δ>0 there are two roots x1=(−b−√Δ)/(2a) and x2=(−b+√Δ)/(2a); if Δ=0 there is one root x0=−b/(2a); and if Δ<0 there is no real root.

📐 Formula — The canonical form of f(x)=ax²+bx+c is $f(x)=a\left(\left(x+\frac{b}{2a}\right)^2-\frac{\Delta}{4a^2}\right)$.

📌 The parabola has axis x=−b/(2a), vertex S(−b/(2a),−Δ/(4a)), a minimum when a>0, and a maximum when a<0.

📖 15. Derivatives and Tangents

🔑 Key Concepts & Definitions

• Derivative : A function is differentiable at x0 when [f(x)−f(x0)]/(x−x0) has a finite limit as x tends to x0, and this limit is f′(x0).

★ Must-know

📐 Formula — The tangent to the graph of f at (x0,f(x0)) has equation $y=f′(x_0)(x-x_0)+f(x_0)$.

• The basic derivatives include (ax+b)′=a, (x^n)′=nx^(n−1), (√x)′=1/(2√x), (1/x)′=−1/x², (e^x)′=e^x, and (ln x)′=1/x on their domains.

• For differentiable f and g, (f+g)′=f′+g′, (fg)′=f′g+fg′, and (f/g)′=(f′g−fg′)/g² when g is nonzero.

📐 Formula — For a differentiable composition g ◦ f, the chain rule is $(g\circ f)′(x)=f′(x)g′(f(x))$.

Further detail

📌 When a function depends on a variable and parameters, differentiation is performed with respect to the chosen variable while the parameters are treated as constants.

💡 Memory Hook

Secant slope approaches derivative, which determines the tangent slope.

📊 Synthesis Tables

Line Representations

Representation	Key data	Equation
Parametric	Point A and direction (α,β)	x=xA+tα; y=yA+tβ
Cartesian	Point A and normal (a,b)	ax+by+c=0
Reduced	Slope m and intercept p	y=mx+p