ICS (Intermediate in Computer Science) Mathematics (Part II) - Calculus and Analytic Geometry Syllabus

State the three conditions for a function f to be continuous at x = a.

1) f(a) is defined; 2) lim(x->a) f(x) exists; 3) lim(x->a) f(x) = f(a). All three must hold.

What is a removable discontinuity?

A removable discontinuity occurs at x = a when lim(x->a) f(x) exists but either f(a) is undefined or not equal to the limit. It can be 'removed' by redefining f(a) to equal the limit.

Distinguish a jump (non-removable) discontinuity from a removable one.

In a jump discontinuity the left and right limits exist but are unequal, so the overall limit does not exist; it cannot be removed by redefining a single value. A removable discontinuity has an existing limit that simply differs from (or replaces) f(a).

Is a polynomial function continuous? Where are rational functions discontinuous?

Every polynomial is continuous for all real x. A rational function p(x)/q(x) is continuous everywhere except where the denominator q(x) = 0.

Define the derivative of f(x) by the first principle (limit definition).

f'(x) = lim(h->0) [f(x + h) - f(x)] / h, provided the limit exists. This is also called differentiation from first principles or ab-initio.

What is the geometric interpretation of the derivative f'(a)?

f'(a) is the slope of the tangent line to the curve y = f(x) at the point (a, f(a)). It is the instantaneous rate of change of f at a.

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.