AMUEEE Mathematics - Calculus, Geometry And Statistics Flashcards

24 sample cards from the Mathematics - Calculus, Geometry And Statistics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. State the definition of an indefinite integral and the power rule for integration.

   ∫f(x)dx = F(x) + C, where F′(x) = f(x) and C is the constant of integration. Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1.

2. Give the integrals of 1/x, eˣ, sin x and cos x.

   ∫(1/x)dx = ln|x| + C; ∫eˣ dx = eˣ + C; ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C.

3. State the integration by parts formula.

   ∫u dv = uv − ∫v du. The choice of u is commonly guided by the ILATE/LIATE order (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential).

4. State the Fundamental Theorem of Calculus (the evaluation part for definite integrals).

   If F is an antiderivative of f on [a, b], then ∫ₐᵇ f(x)dx = F(b) − F(a).

5. How do you compute the area between a curve y = f(x) and the x-axis from x = a to x = b?

   Area = ∫ₐᵇ |f(x)| dx. If the curve lies below the x-axis on part of the interval, that portion's integral is negative, so absolute value (or splitting the interval) gives the geometric area.

6. State two key properties of definite integrals involving limit interchange and additivity.

   ∫ₐᵇ f(x)dx = −∫ᵇₐ f(x)dx (swapping limits changes sign), and ∫ₐᵇ f(x)dx = ∫ₐᶜ f(x)dx + ∫꜀ᵇ f(x)dx (additivity over a subdivision point c).

7. State the property ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a − x)dx and why it is useful.

   This 'king property' lets you replace x with (a − x) in the integrand over [0, a]; it often simplifies symmetric integrals, especially those involving sin x and cos x over [0, π/2].

8. What is the order and the degree of a differential equation?

   Order = the order of the highest derivative present. Degree = the power of the highest-order derivative, after the equation is made free of radicals and fractions in derivatives (degree is defined only then).

9. Describe the method of solving a variable-separable differential equation.

   Write the equation as dy/dx = g(x)h(y), rearrange to dy/h(y) = g(x)dx, then integrate both sides: ∫dy/h(y) = ∫g(x)dx + C.

10. What is the standard form and integrating factor of a linear first-order differential equation?

    Standard form: dy/dx + P(x)y = Q(x). The integrating factor is IF = e^∫P(x)dx, and the solution is y·(IF) = ∫Q(x)·(IF)dx + C.

11. What is a homogeneous differential equation and how is it solved?

    It has the form dy/dx = f(y/x), a function of y/x only. Substitute y = vx, so dy/dx = v + x(dv/dx); this converts it into a variable-separable equation in v and x.

12. What is the difference between a scalar and a vector quantity?

    A scalar has only magnitude (e.g. mass, temperature, distance). A vector has both magnitude and direction (e.g. velocity, force, displacement).

13. Define the dot (scalar) product of two vectors and give its geometric formula.

    a·b = |a||b|cos θ, where θ is the angle between them. In components, a·b = a1b1 + a2b2 + a3b3. It yields a scalar and is zero when the vectors are perpendicular.

14. Define the cross (vector) product of two vectors, its magnitude and direction.

    a × b is a vector with magnitude |a||b|sin θ, directed perpendicular to both a and b (by the right-hand rule). It is zero when the vectors are parallel; its magnitude equals the area of the parallelogram formed by a and b.

15. How do you find a unit vector in the direction of a given vector a?

    Divide the vector by its magnitude: â = a / |a|, where |a| = √(a1² + a2² + a3²). The result has magnitude 1 in the same direction.

16. State the scalar triple product and what its value represents geometrically.

    [a b c] = a·(b × c). Its absolute value equals the volume of the parallelepiped formed by a, b and c. If it is zero, the three vectors are coplanar.

17. State the classical (theoretical) definition of the probability of an event.

    P(E) = (number of favourable outcomes) / (total number of equally likely outcomes). It satisfies 0 ≤ P(E) ≤ 1, and P(sure event) = 1, P(impossible event) = 0.

18. State the addition theorem of probability for two events A and B.

    P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

19. Define conditional probability and the multiplication theorem.

    P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. Multiplication theorem: P(A ∩ B) = P(B)·P(A|B) = P(A)·P(B|A). For independent events, P(A ∩ B) = P(A)·P(B).

20. State Bayes' Theorem.

    P(Aᵢ|B) = [P(Aᵢ)·P(B|Aᵢ)] / Σⱼ P(Aⱼ)·P(B|Aⱼ), where the Aᵢ form a partition of the sample space. It updates prior probabilities to posterior probabilities given evidence B.

21. What is a Bernoulli trial, and what is the binomial probability formula for r successes in n trials?

    A Bernoulli trial has exactly two outcomes (success/failure) with fixed probability p of success. P(X = r) = ⁿCᵣ · pʳ · (1 − p)ⁿ⁻ʳ, for r = 0, 1, ..., n.

22. What is the difference between mean, median and mode as measures of central tendency?

    Mean is the arithmetic average (sum ÷ count). Median is the middle value when data is ordered. Mode is the most frequently occurring value. The empirical relation is Mode ≈ 3·Median − 2·Mean.

23. Define variance and standard deviation for a data set.

    Variance σ² = (1/n)Σ(xᵢ − x̄)², the mean of squared deviations from the mean. Standard deviation σ = √(variance); it measures the spread of data in the original units.

24. What does the coefficient of variation measure and how is it computed?

    It measures relative variability, enabling comparison of dispersion between data sets with different units or means. CV = (standard deviation / mean) × 100%. A lower CV indicates more consistent (less variable) data.

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