NTS NAT-IE Mathematics Syllabus

What does the sign of the first derivative f'(x) tell you about a function?

If f'(x) > 0 the function is increasing; if f'(x) < 0 it is decreasing; if f'(x) = 0 there is a critical point (possible max, min, or inflection).

State the second-derivative test for classifying a critical point where f'(c) = 0.

If f''(c) > 0, f has a local minimum at c (concave up); if f''(c) < 0, a local maximum (concave down); if f''(c) = 0, the test is inconclusive.

What is a point of inflection in terms of the second derivative?

A point where the concavity of the curve changes; f''(x) = 0 (or undefined) there and changes sign. The curve switches between concave up and concave down.

How do you find the equation of the tangent line to y = f(x) at x = a?

Slope m = f'(a); tangent: y - f(a) = f'(a)(x - a). The normal line has slope -1/f'(a).

In a related-rates problem, what is the general strategy?

Write an equation relating the quantities, differentiate both sides with respect to time t (implicit differentiation), then substitute known values and rates to solve for the unknown rate.

What is an antiderivative (indefinite integral) of a function f?

A function F such that F'(x) = f(x). The indefinite integral is the family F(x) + C, where C is an arbitrary constant of integration.

State the power rule for integration.

Integral of x^n dx = x^(n+1)/(n+1) + C, for n not equal to -1. For n = -1: integral of (1/x) dx = ln|x| + C.

Give the integrals of sin x and cos x.

Integral of sin x dx = -cos x + C; Integral of cos x dx = sin x + C.

Give the integrals of e^x and 1/x.

Integral of e^x dx = e^x + C; Integral of (1/x) dx = ln|x| + C.

Give the integral of sec^2 x and of 1/(1 + x^2).

Integral of sec^2 x dx = tan x + C; Integral of 1/(1 + x^2) dx = arctan x + C.

State the integration by parts formula.

Integral of u dv = u*v - Integral of v du. Choose u (to differentiate) and dv (to integrate); the LIATE rule helps pick u.

What is the method of integration by substitution (u-substitution)?

Let u = g(x) so du = g'(x) dx, converting Integral of f(g(x))g'(x) dx into Integral of f(u) du. It reverses the chain rule.

State the Fundamental Theorem of Calculus (evaluation part).

If F is an antiderivative of f on [a, b], then the definite integral from a to b of f(x) dx = F(b) - F(a).

What does a definite integral represent geometrically?

The net signed area between the curve y = f(x) and the x-axis from x = a to x = b: area above the axis counts positive, area below counts negative.

State three key properties of definite integrals.

1) Integral from a to a = 0; 2) Integral a to b = -(Integral b to a); 3) Integral a to b = Integral a to c + Integral c to b (additivity over intervals).

How do you find the area between a curve y = f(x) and the x-axis when the curve crosses the axis on [a, b]?

Split the interval at the x-intercepts, integrate over each piece, and add the absolute values: Area = integral of |f(x)| dx so areas below the axis count as positive.

How do you find the area between two curves y = f(x) (upper) and y = g(x) (lower) on [a, b]?

Area = Integral from a to b of [f(x) - g(x)] dx, with f the upper curve. Find intersection points to determine the limits.

State the three Pythagorean trigonometric identities.

sin^2 x + cos^2 x = 1; 1 + tan^2 x = sec^2 x; 1 + cot^2 x = cosec^2 x.

State the sine and cosine addition formulas for (A + B).

sin(A + B) = sin A cos B + cos A sin B; cos(A + B) = cos A cos B - sin A sin B.

State the double-angle formulas for sin 2x and cos 2x.

sin 2x = 2 sin x cos x; cos 2x = cos^2 x - sin^2 x = 2cos^2 x - 1 = 1 - 2sin^2 x.

State the tangent addition formula tan(A + B).

tan(A + B) = (tan A + tan B) / (1 - tan A tan B).

State the half-angle identities for sin^2 x and cos^2 x (power-reduction).

sin^2 x = (1 - cos 2x)/2; cos^2 x = (1 + cos 2x)/2.

What is the general solution of sin x = 0, and of cos x = 0?

sin x = 0 gives x = n*pi; cos x = 0 gives x = (2n + 1)*pi/2, where n is any integer.

What is the general solution of sin x = sin a?

x = n*pi + (-1)^n * a, where n is any integer (a is the principal value).

What is the general solution of cos x = cos a, and of tan x = tan a?

cos x = cos a: x = 2n*pi ± a. tan x = tan a: x = n*pi + a, where n is any integer.

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

a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius. Used when given two angles and a side (AAS/ASA) or two sides and a non-included angle (SSA).

State the Law of Cosines.

c^2 = a^2 + b^2 - 2ab*cos C (and cyclic permutations). Used for SAS (two sides and included angle) or SSS (three sides).

Give two formulas for the area of a triangle used in solution of triangles.

Area = (1/2)ab sin C (two sides and included angle); and Heron's formula: Area = sqrt[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2.

For an arithmetic progression with first term a and common difference d, give the nth term and the sum of n terms.

nth term: a_n = a + (n - 1)d. Sum: S_n = (n/2)[2a + (n - 1)d] = (n/2)(a + l), where l is the last term.

For a geometric progression with first term a and common ratio r, give the nth term and the sum of n terms.

nth term: a_n = a*r^(n-1). Sum of n terms: S_n = a(1 - r^n)/(1 - r) for r not equal to 1.

What is the sum of an infinite geometric series, and when does it converge?

S(infinity) = a/(1 - r), valid only when |r| < 1. If |r| >= 1 the series diverges. Also: arithmetic mean = (a+b)/2; geometric mean = sqrt(ab).