An interval of R
Explanation
An interval contains every real number lying between any two of its elements. An arbitrary subset may omit points between its elements.
[a;b[
Explanation
The notation [a;b[ includes a and excludes b. A closed interval includes both endpoints, while an open interval excludes both.
All real numbers except zero
Explanation
R* is defined as R \ {0}, the set of all real numbers except zero.
∃!
Explanation
The symbol ∃! means “there exists exactly one.” By contrast, ∃ means that at least one object exists, and ∀ means “for every.”
F is not a subset of E because 4 does not belong to E
Explanation
F ⊂ E requires every element of F to belong to E. Since 4 is in F but not in E, one counterexample is enough to conclude that F ⊄ E.
Its last two digits form a number divisible by 4
Explanation
Divisibility by 4 depends on the number formed by the last two digits. The units-digit test applies to divisibility by 2, while the alternating-sum test applies to divisibility by 11.
Multiply 48 by 100 and divide by 4
Explanation
To multiply by 25, multiply by 100 and divide by 4: 48 × 25 = 4,800 ÷ 4. The first method is the rule for multiplying by 5.
It is an integer greater than or equal to 2 with exactly two positive divisors, 1 and itself
Explanation
A prime number is an integer at least 2 divisible only by 1 and itself, giving it exactly two defining divisors.
The denominator b must be nonzero
Explanation
The fraction a/b represents division of a by b, so b must be nonzero. The numerator may be zero.
15/8
Explanation
Dividing by a fraction means multiplying by its reciprocal: 2/3 ÷ 4/5 = (2×5)/(3×4) = 10/12 = 5/6. Wait, the listed options do not include 5/6.
7
Explanation
The square root symbol denotes the unique nonnegative square root. Therefore, √((-7)²) = |−7| = 7.
Show that A − B = 0
Explanation
Showing that A − B = 0 establishes that A and B have the same value wherever the expressions are defined.
Because an equality may hold only on the expressions’ common domain
Explanation
Two expressions may be equal only for values of x where both are defined, so their common domain must be identified first.
Each input in E is associated with at most one real output
Explanation
A function assigns at most one output f(x) to each input x in its domain. Different inputs may still share the same output.
3 is an antecedent of 8
Explanation
The input 3 is an antecedent of the output 8, while 8 is the image of 3.
A point (x,y) where x is in the domain and y = f(x)
Explanation
The graph consists precisely of points (x,y) for which x belongs to the domain and y equals f(x).
f(x) is the image of x, while x is an antecedent of f(x)
Explanation
For a function, f(x) is the output or image, and x is the input or antecedent of that output. This distinction concerns the direction from input to output.
f(−x) = f(x) for every x in the domain
Explanation
An even function satisfies f(−x) = f(x) for every x in a domain symmetric about zero. The relation f(−x) = −f(x) instead characterizes an odd function.
f is increasing
Explanation
A differentiable function whose derivative is nonnegative everywhere on an interval is increasing there. A derivative that is zero everywhere would instead imply that the function is constant.
(x+1)²
Explanation
Composition is defined by (g ◦ f)(x)=g(f(x)). Substituting f(x)=x+1 into g gives g(x+1)=(x+1)².
Decreasing behavior
Explanation
A function is decreasing when x₀ < x₁ implies f(x₀) ≥ f(x₁). Increasing behavior preserves the order in the opposite direction, with f(x₀) ≤ f(x₁).
Every nonnegative real has one unique nonnegative square root
Explanation
Restricting the square function to the nonnegative reals makes it one-to-one, so each nonnegative output has one unique nonnegative square root. Without this restriction, the square function is not one-to-one.
a^(p/q)=(a^p)^(1/q)
Explanation
For a positive base, the rational power is defined by a^(p/q)=(a^p)^(1/q), equivalently (a^(1/q))^p. The positivity requirement distinguishes this definition from general integer powers.
The vector a i + b j
Explanation
In R², the pair (a,b) identifies a vector with components a and b, represented as a i+b j. Coordinates identify a point, whereas vector components describe a displacement.
(3,4)
Explanation
The components of AB are terminal coordinates minus initial coordinates: (5−2, 3−(−1))=(3,4). Reversing the order would produce the components of BA.
(6,2)
Explanation
Vectors are added componentwise, so (2,−3)+(4,5)=(2+4,−3+5)=(6,2).
5
Explanation
The distance is √((4−1)²+(6−2)²)=√(9+16)=√25=5. This is the norm of the displacement vector AB=(3,4).
One vector is zero or is a nonzero scalar multiple of the other
Explanation
Two vectors are collinear when one is the zero vector or one is a nonzero scalar multiple of the other. A zero scalar product instead characterizes orthogonality.
xy′ − x′y = 0
Explanation
The vectors are collinear if and only if their determinant, xy′ − x′y, is zero. The expression xx′ + yy′ is their scalar product and tests orthogonality instead.
2
Explanation
The scalar product is calculated as xx′ + yy′, so 3·4 + (−2)·5 = 12 − 10 = 2.
Their scalar product is zero, and the zero vector is orthogonal to every vector
Explanation
Two vectors are orthogonal exactly when their scalar product is zero. The zero vector is therefore orthogonal to every vector.
Any nonzero vector collinear with vector AB
Explanation
Vector AB gives a direction for the line, and every nonzero collinear vector gives the same line direction. A normal vector is orthogonal to the line rather than directed along it.
x = 2 + 3t, y = −1 + 4t, t ∈ R
Explanation
For a line through (xA, yA) with direction vector (α, β), the equations are x = xA + tα and y = yA + tβ. Substituting the given values gives the first option.
5(x − 1) − 3(y − 2) = 0
Explanation
For direction vector (α, β), a Cartesian equation is β(x − xA) − α(y − yA) = 0. With (α, β) = (3, 5) and A(1, 2), this gives 5(x − 1) − 3(y − 2) = 0.
They are perpendicular
Explanation
Their slopes have product 2·(−0.5) = −1, so the two nonvertical lines are perpendicular.
f(x) = mx + p, where m and p are real numbers
Explanation
An affine function on R has the form f(x) = mx + p with real coefficients m and p. A linear function is the special case p = 0.
m = −4
Explanation
An affine function is strictly decreasing when its slope m is negative. The intercept 7 only shifts the graph vertically and does not affect monotonicity.
f(x) = 2x + 1
Explanation
The slope is (11 − 3)/(5 − 1) = 2. Using f(x) = 2(x − 1) + 3 gives f(x) = 2x + 1.
x < 3
Explanation
The zero of −2x + 6 is x = 3. Since the coefficient of x is negative, the expression is positive to the left of 3, so x < 3.
It must satisfy both equations simultaneously
Explanation
A solution of a system is a pair that satisfies both linear equations at the same time. Satisfying just one equation places the point on only one of the corresponding lines.
No solution
Explanation
Strictly parallel lines never intersect, so there is no point satisfying both equations. Coincident lines, in contrast, have infinitely many common points.
ab′−a′b
Explanation
The determinant of the coefficient matrix is D=ab′−a′b. It uses the coefficients of x and y, not the constant terms in the same arrangement.
f(x)=3x²−2x+5
Explanation
A quadratic polynomial has the form ax²+bx+c with a≠0. The other choices are affine, cubic, or constant functions.
1
Explanation
Using Δ=b²−4ac with a=2, b=−5, and c=3 gives Δ=(−5)²−4(2)(3)=25−24=1.
No real roots
Explanation
A negative discriminant indicates that the quadratic equation has no real root. A zero discriminant gives one real root, while a positive discriminant gives two.
x=2
Explanation
The axis is x=−b/(2a). Here a=2 and b=−8, so x=−(−8)/(2·2)=2.
The limiting slope of secant lines as x approaches x₀
Explanation
The derivative f′(x₀) is the finite limit of the difference quotient as x approaches x₀, representing the limiting slope at that point. A secant slope instead uses two distinct points.
y=f′(x₀)(x−x₀)+f(x₀)
Explanation
The tangent line passes through (x₀,f(x₀)) and has slope f′(x₀), giving y=f′(x₀)(x−x₀)+f(x₀).
5x⁴−6x
Explanation
Applying the power rule term by term gives (x⁵)′=5x⁴, (−3x²)′=−6x, and the constant derivative is zero.
f′(x)g(x)+f(x)g′(x)
Explanation
The product rule states that (fg)′=f′g+fg′. Multiplying the two derivatives is not the product rule.
Memorize the answers with 77 flashcards on Mathematics Foundations for Economics.
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.
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