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Praxis (Praxis Core and Praxis Subject Assessments) Praxis Subject Assessment: Mathematics: Content Knowledge (5165) Flashcards
52 question-and-answer cards covering Praxis Subject Assessment: Mathematics: Content Knowledge (5165) as it is examined in Praxis (Praxis Core and Praxis Subject Assessments). 24 of them are printed below, taken from across the deck — no signup, no paywall on the preview.
24 sample cards from the Praxis Subject Assessment: Mathematics: Content Knowledge (5165) deck
Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.
What is the formal idea of the limit of f(x) as x approaches a?
It is the single value L that f(x) gets arbitrarily close to as x gets arbitrarily close to a (from both sides), regardless of the value f(a) itself.
What three conditions must hold for a function f to be continuous at x = a?
f(a) is defined, the limit of f(x) as x→a exists, and that limit equals f(a).
How can L'Hôpital's Rule be used to evaluate a limit of indeterminate form 0/0 or ∞/∞?
Replace the limit of f(x)/g(x) with the limit of f′(x)/g′(x), provided that new limit exists; repeat if still indeterminate.
What is the limit definition of the derivative f′(x)?
f′(x) = lim_(h→0) [f(x + h) − f(x)] / h.
State the power rule for derivatives.
d/dx [xⁿ] = n·x^(n−1).
State the product rule and quotient rule for derivatives.
Product: (fg)′ = f′g + fg′. Quotient: (f/g)′ = (f′g − fg′)/g².
State the chain rule for the derivative of a composite function f(g(x)).
d/dx f(g(x)) = f′(g(x)) · g′(x).
How do you use the first derivative to find and classify relative extrema?
Find critical points where f′(x) = 0 or is undefined; if f′ changes from positive to negative there is a relative maximum, and from negative to positive a relative minimum (first derivative test).
What does the second derivative tell you about concavity and inflection points?
f″ > 0 means concave up, f″ < 0 means concave down; an inflection point occurs where concavity changes (f″ changes sign).
State the power rule for integration of xⁿ (n ≠ −1).
∫ xⁿ dx = x^(n+1)/(n+1) + C.
State the Fundamental Theorem of Calculus (evaluation part) for a definite integral.
If F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a).
How is the definite integral ∫ₐᵇ f(x) dx interpreted geometrically?
It is the net (signed) area between the curve y = f(x) and the x-axis from x = a to x = b; area above the axis is positive, below is negative.
How many edges does a complete graph Kₙ have, and what is the degree of each vertex?
Kₙ has n(n−1)/2 edges, and every vertex has degree n − 1.
What distinguishes a permutation from a combination, and what are their formulas?
Permutations count ordered arrangements: nPr = n!/(n−r)!. Combinations count unordered selections: nCr = n!/[r!(n−r)!]. Order matters for permutations, not combinations.
State the Pythagorean Theorem and its converse.
In a right triangle with legs a, b and hypotenuse c: a² + b² = c². Converse: if a² + b² = c² for a triangle's sides, then the triangle is right-angled.
What are the triangle congruence criteria?
SSS, SAS, ASA, AAS, and HL (hypotenuse-leg for right triangles). SSA and AAA are not valid congruence criteria.
What are the formulas for the distance and midpoint between points (x₁,y₁) and (x₂,y₂)?
Distance = √[(x₂−x₁)² + (y₂−y₁)²]; Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).
What is the relationship between the slopes of parallel lines and of perpendicular lines?
Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (their product is −1).
Which rigid transformations (isometries) preserve distance, and which transformation does not?
Translations, rotations, and reflections are isometries that preserve distance and shape; a dilation is not an isometry (it changes size but preserves shape).
State the law of sines and the law of cosines for a triangle.
Law of Sines: a/sin A = b/sin B = c/sin C. Law of Cosines: c² = a² + b² − 2ab·cos C.
What is the addition rule for the probability of A or B, and how does it simplify for mutually exclusive events?
P(A or B) = P(A) + P(B) − P(A and B). If A and B are mutually exclusive, P(A and B) = 0, so P(A or B) = P(A) + P(B).
What is the formula for conditional probability P(A | B), and when are two events independent?
P(A | B) = P(A and B)/P(B). Events are independent when P(A and B) = P(A)·P(B), equivalently P(A | B) = P(A).
How are mean, median, and mode each affected by an extreme outlier in a data set?
The mean is strongly pulled toward the outlier; the median is resistant and changes little; the mode is unaffected unless the outlier creates a new most-frequent value.
In a normal distribution, what does the 68-95-99.7 (empirical) rule state?
About 68% of data lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3 standard deviations.
What this deck covers
The Praxis Subject Assessment: Mathematics: Content Knowledge (5165) deck follows the Praxis (Praxis Core and Praxis Subject Assessments) Praxis Subject Assessment: Mathematics: Content Knowledge (5165) syllabus — 4 chapters and 15 topics — so questions land on material that is genuinely examinable rather than trivia around it. That works out to roughly 13.0 cards per chapter.
Answers are written to be recallable, not just readable — averaging about 104 characters, which is long enough to carry the reasoning and short enough to say out loud.
A deck like this earns its keep on the second and third pass. Read the syllabus first so you know the shape of the subject, then use the cards to find the specific facts that have not stuck.
Praxis Subject Assessment: Mathematics: Content Knowledge (5165) flashcards FAQ
How many Praxis Subject Assessment: Mathematics: Content Knowledge (5165) flashcards are in this Praxis (Praxis Core and Praxis Subject Assessments) deck?
52 cards. This page previews 24 of them, sampled evenly across the deck so you can judge the difficulty before installing anything.
Are these Praxis (Praxis Core and Praxis Subject Assessments) flashcards free?
Yes. The preview here is free to read with no signup, and the full 52-card deck is free inside the Examius app.
What do the Praxis Subject Assessment: Mathematics: Content Knowledge (5165) cards cover?
They follow the Praxis (Praxis Core and Praxis Subject Assessments) Praxis Subject Assessment: Mathematics: Content Knowledge (5165) syllabus — 4 chapters and 15 topics — so the questions track what is actually examinable.
How should I use these flashcards?
Read the syllabus first so you know the shape of the subject, then drill the deck. Examius schedules each card with spaced repetition, so cards you keep missing come back sooner and ones you know drift further apart.