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ICS (Intermediate in Computer Science) Mathematics (Part II) - Calculus and Analytic Geometry Flashcards
50 question-and-answer cards covering Mathematics (Part II) - Calculus and Analytic Geometry as it is examined in ICS (Intermediate in Computer Science). 24 of them are printed below, taken from across the deck — no signup, no paywall on the preview.
24 sample cards from the Mathematics (Part II) - Calculus and Analytic Geometry deck
Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.
Using first principles, what is the derivative of f(x) = x^n?
By first principles (and the binomial expansion), d/dx (x^n) = n x^(n-1) for any real n. This is the power rule.
State the power rule and the derivative of a constant.
d/dx (x^n) = n x^(n-1). The derivative of any constant c is d/dx (c) = 0.
State the constant-multiple and sum/difference rules of differentiation.
d/dx [c f(x)] = c f'(x); and d/dx [f(x) +/- g(x)] = f'(x) +/- g'(x).
State the product rule of differentiation.
d/dx [f(x) g(x)] = f'(x) g(x) + f(x) g'(x). (First times derivative of second plus second times derivative of first.)
State the quotient rule of differentiation.
d/dx [f(x)/g(x)] = [f'(x) g(x) - f(x) g'(x)] / [g(x)]^2, where g(x) is not 0.
State the chain rule for y = f(g(x)).
If y = f(u) and u = g(x), then dy/dx = (dy/du)(du/dx) = f'(g(x)) g'(x).
What is implicit differentiation and when is it used?
It is differentiating both sides of an equation in x and y with respect to x without first solving for y, applying the chain rule to y-terms (treating y as a function of x), then solving for dy/dx. Used when y is not given explicitly in terms of x.
Differentiate x^2 + y^2 = r^2 implicitly to find dy/dx.
Differentiating: 2x + 2y (dy/dx) = 0, so dy/dx = -x/y.
Give the derivatives of sin x and cos x.
d/dx (sin x) = cos x and d/dx (cos x) = -sin x.
Give the derivatives of tan x and sec x.
d/dx (tan x) = sec^2 x and d/dx (sec x) = sec x tan x.
Give the derivatives of cot x and csc x.
d/dx (cot x) = -csc^2 x and d/dx (csc x) = -csc x cot x.
Give the derivatives of e^x and a^x.
d/dx (e^x) = e^x and d/dx (a^x) = a^x ln a.
Give the derivatives of ln x and log_a x.
d/dx (ln x) = 1/x and d/dx (log_a x) = 1/(x ln a).
State the derivative of the inverse sine function, d/dx (sin^-1 x).
d/dx (sin^-1 x) = 1 / sqrt(1 - x^2), for -1 < x < 1.
What is an antiderivative (indefinite integral) of a function f?
F is an antiderivative of f if F'(x) = f(x). The indefinite integral is the integral of f(x) dx = F(x) + C, where C is an arbitrary constant of integration.
State the power rule for integration.
The integral of x^n dx = x^(n+1)/(n+1) + C, valid for n not equal to -1. For n = -1, the integral of (1/x) dx = ln|x| + C.
Give the integrals of sin x and cos x.
The integral of sin x dx = -cos x + C, and the integral of cos x dx = sin x + C.
Give the integrals of sec^2 x and csc^2 x.
The integral of sec^2 x dx = tan x + C, and the integral of csc^2 x dx = -cot x + C.
Give the integrals of e^x and a^x.
The integral of e^x dx = e^x + C, and the integral of a^x dx = a^x / ln a + C.
State the integration by substitution method.
If u = g(x), then the integral of f(g(x)) g'(x) dx = the integral of f(u) du. You substitute u and du to simplify the integrand into a standard form.
State the formula for integration by parts.
The integral of u dv = u v - the integral of v du. Choose u and dv so that the resulting integral is simpler (a common guide is the LIATE order).
State the Fundamental Theorem of Calculus for evaluating a definite integral.
If F is an antiderivative of f, then the definite integral from a to b of f(x) dx = F(b) - F(a).
State two key properties of definite integrals regarding limits of integration.
The integral from a to b of f(x) dx = -(integral from b to a of f(x) dx); and the integral from a to a of f(x) dx = 0. Also the integral can be split: integral a to b = integral a to c + integral c to b.
How is the area under a curve y = f(x), above the x-axis, between x = a and x = b found?
When f(x) >= 0 on [a, b], the area = the definite integral from a to b of f(x) dx. If the curve is given as x = g(y), the area between the curve and the y-axis from y = c to y = d is the integral from c to d of g(y) dy.
What this deck covers
The Mathematics (Part II) - Calculus and Analytic Geometry deck follows the ICS (Intermediate in Computer Science) Mathematics (Part II) - Calculus and Analytic Geometry syllabus — 7 chapters and 26 topics — so questions land on material that is genuinely examinable rather than trivia around it. That works out to roughly 7.1 cards per chapter.
Answers are written to be recallable, not just readable — averaging about 102 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.
Mathematics (Part II) - Calculus and Analytic Geometry flashcards FAQ
How many Mathematics (Part II) - Calculus and Analytic Geometry flashcards are in this ICS (Intermediate in Computer Science) deck?
50 cards. This page previews 24 of them, sampled evenly across the deck so you can judge the difficulty before installing anything.
Are these ICS (Intermediate in Computer Science) flashcards free?
Yes. The preview here is free to read with no signup, and the full 50-card deck is free inside the Examius app.
What do the Mathematics (Part II) - Calculus and Analytic Geometry cards cover?
They follow the ICS (Intermediate in Computer Science) Mathematics (Part II) - Calculus and Analytic Geometry syllabus — 7 chapters and 26 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.