On the unit circle (radius r = 1), how are cos(θ) and sin(θ) defined in terms of Cartesian coordinates?

cos(θ) is the x-coordinate and sin(θ) is the y-coordinate of the terminal point on the circle: (x, y) = (cos θ, sin θ).

Under what condition does an infinite geometric series converge, and what is its sum formula?

Converges if and only if |r| < 1; sum formula is S_∞ = a_1 / (1 - r).

How is radian measure defined, and what conversion factor converts degrees to radians?

One radian is the central angle subtending an arc length equal to the circle's radius (s = r); conversion factor is θ_rad = θ_deg * (π / 180°).

State the Addition Rule of Probability for any two events A and B.

P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

What are the standard equation of an ellipse centered at (h, k) and the relationship between semi-axes a, b and focal distance c?

(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1; focal relationship is c^2 = a^2 - b^2 (assuming horizontal major axis a > b).

What are the formulas for the n-th term (a_n) and sum of first n terms (S_n) of an arithmetic sequence?

a_n = a_1 + (n - 1)*d; S_n = (n / 2) * (a_1 + a_n) = (n / 2) * [2*a_1 + (n - 1)*d].

What is the statistical definition and practical interpretation of a p-value?

The probability of obtaining test results at least as extreme as observed, assuming the null hypothesis H0 is true; if p <= α, reject H0.

What are the formulas for the n-th term (a_n) and sum of first n terms (S_n) of a geometric sequence with common ratio r != 1?

a_n = a_1 * r^(n - 1); S_n = a_1 * (1 - r^n) / (1 - r).

What mathematical test establishes that two events A and B are statistically independent?

P(A ∩ B) = P(A) * P(B), or equivalently P(A | B) = P(A).

What are the domains and principal ranges of the inverse trigonometric functions arcsin(x) and arccos(x)?

arcsin(x): domain [-1, 1], range [-π/2, π/2]; arccos(x): domain [-1, 1], range [0, π].

How do permutations and combinations differ in combinatorics, and what are their computational formulas?

Permutations account for order: P(n, k) = n! / (n - k)!; Combinations ignore order: C(n, k) = (n choose k) = n! / (k! * (n - k)!).

What is the standard error of the mean (SEM), and how does increasing sample size n affect it?

SEM = σ / √n; increasing sample size decreases standard error inversely proportional to the square root of n, increasing estimate precision.

State the double-angle formulas for sin(2θ) and the three standard forms for cos(2θ).

sin(2θ) = 2sin(θ)cos(θ); cos(2θ) = cos^2(θ) - sin^2(θ) = 2cos^2(θ) - 1 = 1 - 2sin^2(θ).

What percentages of data fall within 1, 2, and 3 standard deviations of the mean in a Normal distribution (Empirical Rule)?

Approximately 68.27% within μ ± 1σ, 95.45% within μ ± 2σ, and 99.73% within μ ± 3σ.

What are the trigonometric addition formulas for sin(α + β) and cos(α + β)?

sin(α + β) = sin(α)cos(β) + cos(α)sin(β); cos(α + β) = cos(α)cos(β) - sin(α)sin(β).

How is a data point x converted to a standard normal z-score, and what does the z-score represent?

z = (x - μ) / σ; it represents the signed number of standard deviations the value x lies above or below the mean.

What is the formula for the conditional probability of event A given event B has occurred, P(A | B)?

P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.

How is the expected value E[X] of a discrete random variable X calculated?

E[X] = μ = Σ [x_i * P(X = x_i)] over all possible values x_i.

What is the computational formula for the variance Var(X) of a random variable X?

Var(X) = σ^2 = E[X^2] - (E[X])^2.

What are the period, amplitude, and horizontal phase shift of y = A*sin(Bx - C) with B > 0?

Amplitude = |A|; Period T = 2π / B; Phase shift = C / B (to the right if C/B > 0).

How do you convert polar coordinates (r, θ) to Cartesian coordinates (x, y) and vice versa?

Polar to Cartesian: x = r*cos(θ), y = r*sin(θ); Cartesian to polar: r = √(x^2 + y^2), θ = atan2(y, x).

What does the Central Limit Theorem (CLT) state regarding sample means from any population distribution?

For independent, identically distributed samples of sufficiently large size (n >= 30), the sample mean X_bar is approximately normally distributed: X_bar ~ N(μ, σ^2 / n).

State Bayes' Theorem for revising the probability of hypothesis A given observed evidence B.

P(A | B) = [P(B | A) * P(A)] / P(B) = [P(B | A) * P(A)] / [P(B | A)*P(A) + P(B | ¬A)*P(¬A)].

Distinguish between a Type I error (α) and a Type II error (β) in statistical hypothesis testing.

Type I error (false positive) rejects the null hypothesis H0 when H0 is true; Type II error (false negative) fails to reject H0 when H0 is false.

State De Moivre's Theorem for raising a complex number in polar form to the integer power n.

[r*(cos θ + i*sin θ)]^n = r^n * [cos(nθ) + i*sin(nθ)] = r^n * e^(i*n*θ).

State the Law of Sines and the Law of Cosines for a triangle with sides a, b, c and opposite angles A, B, C.

Law of Sines: a / sin(A) = b / sin(B) = c / sin(C); Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos(C).

State the Binomial Theorem for expanding (x + y)^n for a positive integer n.

(x + y)^n = Σ [k=0 to n] (n choose k) * x^(n - k) * y^k, where (n choose k) = n! / (k! * (n - k)!).

What are the formulas for arc length (s) and area of a circular sector (A) subtended by central angle θ in radians?

Arc length: s = r * θ; Sector area: A = (1/2) * r^2 * θ.

What is statistical power, and how is it related to the Type II error rate (β)?

Power = 1 - β; it is the probability of correctly rejecting a false null hypothesis.

What are the probability mass function, mean, and variance of a Binomial distribution B(n, p)?

P(X = k) = (n choose k) * p^k * (1 - p)^(n - k); Mean μ = n*p; Variance σ^2 = n*p*(1 - p).

State the fundamental Pythagorean trigonometric identity and its two derived identities involving tan(θ) and cot(θ).

1. sin^2(θ) + cos^2(θ) = 1 2. 1 + tan^2(θ) = sec^2(θ) 3. 1 + cot^2(θ) = csc^2(θ).

Why is (n - 1) used as the divisor instead of n in the sample variance formula s^2 = Σ(x_i - x_bar)^2 / (n - 1)?

Bessel's correction: dividing by (n - 1) corrects downward bias, providing an unbiased estimator of population variance σ^2.

Differential Equations (Svenska) — Medelnivå (B1/B2) 2: Trigonometry, Precalculus, Probability & Statistics

[Svenska] Differential Equations (Svenska): Unit circle, trigonometric identities, limits, sequences, permutations, combinations, normal distribution, and hypothesis testing basics

Level: intermediate
Languages: sv → sv
Cards: 32
Markdown: Differential Equations (Svenska) — Medelnivå (B1/B2) 2: Trigonometry, Precalculus, Probability & Statistics (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
1	On the unit circle (radius r = 1), how are cos(θ) and sin(θ) defined in terms of Cartesian coordinates?	cos(θ) is the x-coordinate and sin(θ) is the y-coordinate of the terminal point on the circle: (x, y) = (cos θ, sin θ).
2	Under what condition does an infinite geometric series converge, and what is its sum formula?	Converges if and only if \|r\| < 1; sum formula is S_∞ = a_1 / (1 - r).
3	How is radian measure defined, and what conversion factor converts degrees to radians?	One radian is the central angle subtending an arc length equal to the circle's radius (s = r); conversion factor is θ_rad = θ_deg * (π / 180°).
4	State the Addition Rule of Probability for any two events A and B.	P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
5	What are the standard equation of an ellipse centered at (h, k) and the relationship between semi-axes a, b and focal distance c?	(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1; focal relationship is c^2 = a^2 - b^2 (assuming horizontal major axis a > b).
6	What are the formulas for the n-th term (a_n) and sum of first n terms (S_n) of an arithmetic sequence?	a_n = a_1 + (n - 1)d; S_n = (n / 2) (a_1 + a_n) = (n / 2) * [2a_1 + (n - 1)d].
7	What is the statistical definition and practical interpretation of a p-value?	The probability of obtaining test results at least as extreme as observed, assuming the null hypothesis H0 is true; if p <= α, reject H0.
8	What are the formulas for the n-th term (a_n) and sum of first n terms (S_n) of a geometric sequence with common ratio r != 1?	a_n = a_1 * r^(n - 1); S_n = a_1 * (1 - r^n) / (1 - r).
9	What mathematical test establishes that two events A and B are statistically independent?	P(A ∩ B) = P(A) * P(B), or equivalently P(A \| B) = P(A).
10	What are the domains and principal ranges of the inverse trigonometric functions arcsin(x) and arccos(x)?	arcsin(x): domain [-1, 1], range [-π/2, π/2]; arccos(x): domain [-1, 1], range [0, π].
11	How do permutations and combinations differ in combinatorics, and what are their computational formulas?	Permutations account for order: P(n, k) = n! / (n - k)!; Combinations ignore order: C(n, k) = (n choose k) = n! / (k! * (n - k)!).
12	What is the standard error of the mean (SEM), and how does increasing sample size n affect it?	SEM = σ / √n; increasing sample size decreases standard error inversely proportional to the square root of n, increasing estimate precision.
13	State the double-angle formulas for sin(2θ) and the three standard forms for cos(2θ).	sin(2θ) = 2sin(θ)cos(θ); cos(2θ) = cos^2(θ) - sin^2(θ) = 2cos^2(θ) - 1 = 1 - 2sin^2(θ).
14	What percentages of data fall within 1, 2, and 3 standard deviations of the mean in a Normal distribution (Empirical Rule)?	Approximately 68.27% within μ ± 1σ, 95.45% within μ ± 2σ, and 99.73% within μ ± 3σ.
15	What are the trigonometric addition formulas for sin(α + β) and cos(α + β)?	sin(α + β) = sin(α)cos(β) + cos(α)sin(β); cos(α + β) = cos(α)cos(β) - sin(α)sin(β).
16	How is a data point x converted to a standard normal z-score, and what does the z-score represent?	z = (x - μ) / σ; it represents the signed number of standard deviations the value x lies above or below the mean.
17	What is the formula for the conditional probability of event A given event B has occurred, P(A \| B)?	P(A \| B) = P(A ∩ B) / P(B), provided P(B) > 0.
18	How is the expected value E[X] of a discrete random variable X calculated?	E[X] = μ = Σ [x_i * P(X = x_i)] over all possible values x_i.
19	What is the computational formula for the variance Var(X) of a random variable X?	Var(X) = σ^2 = E[X^2] - (E[X])^2.
20	What are the period, amplitude, and horizontal phase shift of y = A*sin(Bx - C) with B > 0?	Amplitude = \|A\|; Period T = 2π / B; Phase shift = C / B (to the right if C/B > 0).
21	How do you convert polar coordinates (r, θ) to Cartesian coordinates (x, y) and vice versa?	Polar to Cartesian: x = rcos(θ), y = rsin(θ); Cartesian to polar: r = √(x^2 + y^2), θ = atan2(y, x).
22	What does the Central Limit Theorem (CLT) state regarding sample means from any population distribution?	For independent, identically distributed samples of sufficiently large size (n >= 30), the sample mean X_bar is approximately normally distributed: X_bar ~ N(μ, σ^2 / n).
23	State Bayes' Theorem for revising the probability of hypothesis A given observed evidence B.	P(A \| B) = [P(B \| A) * P(A)] / P(B) = [P(B \| A) * P(A)] / [P(B \| A)P(A) + P(B \| ¬A)P(¬A)].
24	Distinguish between a Type I error (α) and a Type II error (β) in statistical hypothesis testing.	Type I error (false positive) rejects the null hypothesis H0 when H0 is true; Type II error (false negative) fails to reject H0 when H0 is false.
25	State De Moivre's Theorem for raising a complex number in polar form to the integer power n.	[r(cos θ + isin θ)]^n = r^n * [cos(nθ) + isin(nθ)] = r^n e^(inθ).
26	State the Law of Sines and the Law of Cosines for a triangle with sides a, b, c and opposite angles A, B, C.	Law of Sines: a / sin(A) = b / sin(B) = c / sin(C); Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos(C).
27	State the Binomial Theorem for expanding (x + y)^n for a positive integer n.	(x + y)^n = Σ [k=0 to n] (n choose k) * x^(n - k) * y^k, where (n choose k) = n! / (k! * (n - k)!).
28	What are the formulas for arc length (s) and area of a circular sector (A) subtended by central angle θ in radians?	Arc length: s = r * θ; Sector area: A = (1/2) * r^2 * θ.
29	What is statistical power, and how is it related to the Type II error rate (β)?	Power = 1 - β; it is the probability of correctly rejecting a false null hypothesis.
30	What are the probability mass function, mean, and variance of a Binomial distribution B(n, p)?	P(X = k) = (n choose k) * p^k * (1 - p)^(n - k); Mean μ = np; Variance σ^2 = np*(1 - p).
31	State the fundamental Pythagorean trigonometric identity and its two derived identities involving tan(θ) and cot(θ).	1. sin^2(θ) + cos^2(θ) = 1 2. 1 + tan^2(θ) = sec^2(θ) 3. 1 + cot^2(θ) = csc^2(θ).
32	Why is (n - 1) used as the divisor instead of n in the sample variance formula s^2 = Σ(x_i - x_bar)^2 / (n - 1)?	Bessel's correction: dividing by (n - 1) corrects downward bias, providing an unbiased estimator of population variance σ^2.