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Notation: x in A means x belongs to A. A is a subset of B if every element of A is in B; a proper subset if A differs from B. Operations on two sets A and B - Union: elements in A or B. - Intersection: elements in both. - Difference A minus B: elements in A not in B. - Cartesian product A x B: all ordered pairs (x, y) with x in A and y in B. Number sets: natural numbers N = {1, 2, 3, ...}, integers Z, rationals Q (fractions), reals R, and complex numbers C. Each set contains the previous one. Quantifiers: for all (universal) and there exists (existential); there exists a unique is written with an exclamation mark.

A function f: X -> Y assigns to each x in X exactly one f(x) in Y. The graph of f is the set of pairs (x, f(x)). Types - Injective (one-to-one): different inputs give different outputs. - Surjective (onto): every element of Y is an output. - Bijective: both. Only bijective functions have an inverse f^-1 defined on the whole of Y: for every y in Y there is a unique x with f(x) = y. Composition: (g o f)(x) = g(f(x)), defined when the range of f lies in the domain of g. Exam trap: the square function on all reals is neither injective nor surjective onto the reals. Restricted to the non-negative reals it is a bijection, and its inverse is the square root. Applying the square root to both sides of an equation needs care with signs.

Even function: f(-x) = f(x) for all x in the domain (graph symmetric about the y-axis), for example x squared and cos x. Odd function: f(-x) = -f(x) (graph symmetric about the origin), for example x cubed and sin x. Elementary functions: polynomials, rational functions, powers and roots, exponential and logarithm, trigonometric functions and their inverses. The exponential function is increasing for base greater than 1, and the logarithm is its inverse. Also useful: a function is monotonic if it only increases or only decreases, and it is periodic with period T if f(x + T) = f(x).

1. Define union, intersection and Cartesian product. 2. When does a function have an inverse? 3. Is f(x) = x squared on the real numbers injective? Why? 4. Classify sin x, cos x and x cubed as even or odd. 5. What is needed to compose g with f?

1. Union: in A or B. Intersection: in both. Cartesian product: ordered pairs (a, b) with a in A, b in B. 2. When it is bijective. 3. No. f(2) = f(-2) = 4. 4. sin x odd, cos x even, x cubed odd. 5. The range of f must be contained in the domain of g.

An m x n matrix has m rows and n columns. The entry in row i and column j is a_ij. Square matrices have m = n; the diagonal runs from top left to bottom right. Operations - Sum and multiplication by a scalar: entry by entry (same size required). - Product A B: defined only if the number of columns of A equals the number of rows of B. The entry (i, j) is the row i of A multiplied by column j of B. - Transpose: rows and columns exchanged. - Trace: sum of the diagonal entries of a square matrix. Exam trap: matrix multiplication is not commutative (A B is generally not B A), and the product can be defined in one order only. Special matrices: identity (1 on the diagonal, acts like the number 1), zero, diagonal, symmetric (equal to its transpose).

A linear system A x = b has coefficient matrix A, unknowns x and known terms b. The augmented matrix [A | b] holds all the information. Elementary row operations (do not change the solution set) 1. Swap two rows. 2. Multiply a row by a non-zero number. 3. Add a multiple of one row to another. Gaussian elimination reduces the matrix to row echelon form (leading entry of each row to the right of the one above). Continuing to make every leading entry 1 with zeros above it gives the reduced row echelon form. Number of solutions - No solution if a row reads 0 = non-zero (inconsistent). - A unique solution if the number of leading entries equals the number of unknowns. - Infinitely many solutions if there are free variables: n minus r free parameters, where r is the rank. Worked idea: for a system with a parameter k, reduce the matrix and find the values of k that change the rank; these give the cases with no solution, one solution or many.

A homogeneous system has zero right-hand side. It is always consistent because x = 0 is a solution. If the rank r is less than the number of unknowns n, there are n minus r free variables and infinitely many solutions; the solution set is a subspace. For a square system, non-trivial solutions exist exactly when the determinant of A is zero.

1. When is the product of two matrices defined? 2. List the elementary row operations. 3. How can a linear system have no solution? 4. If the rank is 2 and there are 4 unknowns, how many free variables are there? 5. Does a homogeneous system always have a solution?

1. When the number of columns of the first equals the number of rows of the second. 2. Swap rows, multiply a row by a non-zero scalar, add a multiple of a row to another. 3. If row reduction gives a row 0 = non-zero number. 4. Two. 5. Yes, the trivial solution.

2 x 2: det of [a b; c d] is a d minus b c. Larger matrices: expand along a row or column with cofactors (minors with alternating signs). Choose the row or column with the most zeros. For a triangular matrix the determinant is the product of the diagonal entries. Properties - Swapping two rows changes the sign. - Multiplying a row by k multiplies the determinant by k. - Adding a multiple of one row to another leaves it unchanged. - det(A B) = det(A) det(B); det(A transpose) = det(A). - A row of zeros or two equal rows gives determinant 0. A matrix is singular if its determinant is 0. It is then not invertible.

The inverse A^-1 satisfies A A^-1 = A^-1 A = I. It exists if and only if det(A) is not 0. Methods: the adjugate formula A^-1 = (1/det A) times the transposed matrix of cofactors, or Gauss-Jordan elimination on [A | I]. Use: if A is invertible, A x = b has the unique solution x = A^-1 b. Also (A B)^-1 = B^-1 A^-1. For a matrix with a parameter k, find the values of k for which det = 0 to know when the inverse does not exist.

1. Compute det [2 1; 3 4]. 2. What happens to the determinant if two rows are swapped? 3. When is a matrix invertible? 4. What is (A B)^-1?

1. 2 x 4 minus 1 x 3 = 5. 2. It changes sign. 3. When its determinant is non-zero. 4. B^-1 A^-1.

A vector in R^n is an ordered list of n numbers. It can be added component by component and multiplied by a scalar. Dot (scalar) product: u . v = u1 v1 + ... + un vn, a number. Length (norm): the length of u is the square root of u . u. Angle: cos(theta) = (u . v) divided by (length of u times length of v). Orthogonal vectors have u . v = 0. Distance between points u and v: the length of u minus v. Exam trap: the dot product is defined for two vectors only. An expression like (u . v) . w is not defined because u . v is a number.

1. Compute (1, 2, 3) . (2, 0, -1). 2. When are two vectors orthogonal? 3. Write the formula for the angle between two vectors.

1. 2 + 0 - 3 = -1. 2. When their dot product is zero. 3. cos(theta) = u . v / (|u| |v|).

A sequence is a function from the natural numbers to the reals: a_1, a_2, a_3, ... Properties: increasing, decreasing, bounded above or below, positive, alternating in sign. Limit: the sequence converges to L if the terms get arbitrarily close to L for large n; it diverges to plus or minus infinity, or has no limit (for example cos n, which oscillates). Theorem: a monotone sequence that is bounded converges. An increasing sequence not bounded above diverges to plus infinity. Useful limits: a^n tends to 0 if |a| < 1, and to infinity if a > 1; (1 + 1/n)^n tends to e; n-th root of n tends to 1.

A series is the sum of the terms a_n. It converges if the sequence of partial sums S_n converges to a finite limit. Necessary condition: if the series converges then a_n tends to 0. The converse is false (the harmonic series 1 + 1/2 + 1/3 + ... diverges). Geometric series: the sum of q^n for n = 0, 1, ... converges to 1/(1 - q) if |q| < 1 and diverges otherwise. Tests for positive terms: comparison test (compare with a known series), ratio test (limit of a_(n+1)/a_n below 1 gives convergence), root test (limit of the n-th root of a_n below 1 gives convergence). Telescoping series collapse and can be summed exactly. Generalised harmonic series: the sum of 1/n^p converges if p > 1.

1. State the necessary condition for series convergence and give a counter-example to the converse. 2. When does a geometric series converge and what is its sum? 3. Which theorem guarantees convergence of an increasing bounded sequence? 4. Apply the ratio test to the sum of 1/n!.

1. The terms must tend to 0; the harmonic series has terms tending to 0 but diverges. 2. When |q| < 1, with sum 1/(1 - q) starting from n = 0. 3. The monotone convergence theorem. 4. a_(n+1)/a_n = 1/(n+1) tends to 0 < 1, so it converges.

The limit of f(x) as x approaches a is L if f(x) is close to L whenever x is close to a (x different from a). Limits can be one-sided and can be infinite. Rules: the limit of a sum, product and quotient equals the sum, product and quotient of the limits (denominator not 0). Indeterminate forms: 0/0, infinity/infinity, infinity minus infinity, 0 times infinity, 1^infinity, 0^0, infinity^0. They need manipulation (factoring, rationalising, dividing by the highest power). Remarkable limits - sin x / x tends to 1 as x tends to 0. - (1 - cos x) / x^2 tends to 1/2. - (1 + 1/x)^x tends to e as x tends to infinity. - (e^x - 1)/x tends to 1 and ln(1 + x)/x tends to 1. Comparing orders: as x tends to infinity, powers are dominated by exponentials, and logarithms grow slower than any power.

A function is infinitesimal at a if it tends to 0. Two infinitesimals f and g are comparable if f/g has a finite non-zero limit; f is of higher order if f/g tends to 0. Continuity: f is continuous at a if the limit of f(x) as x tends to a equals f(a). Graphically this means the graph can be drawn without lifting the pencil. Elementary functions are continuous on their domains. Types of discontinuity: removable (the limit exists but differs from the value), jump (different one-sided limits) and infinite.

1. Calculate the limit of sin(3x)/x as x tends to 0. 2. List the indeterminate forms. 3. Define continuity at a point. 4. What is a jump discontinuity?

1. Write it as 3 sin(3x)/(3x), which tends to 3. 2. 0/0, infinity/infinity, infinity minus infinity, 0 times infinity, 1^infinity, 0^0, infinity^0. 3. The limit at the point exists and equals the value of the function there. 4. The one-sided limits exist but are different.

Definition: f'(a) is the limit of (f(a + h) - f(a))/h as h tends to 0. Geometrically it is the slope of the tangent line at a. Rules - Sum: (f + g)' = f' + g'. - Product: (f g)' = f' g + f g'. - Quotient: (f/g)' = (f' g - f g')/g^2. - Chain rule: (f(g(x)))' = f'(g(x)) g'(x). - Reciprocal (inverse function) rule: the derivative of the inverse is 1/f'. Elementary derivatives: x^n gives n x^(n-1); e^x gives e^x; ln x gives 1/x; sin x gives cos x; cos x gives -sin x. A differentiable function is continuous, but a continuous function need not be differentiable (for example |x| at 0).

If f'(x) > 0 on an interval then f is increasing there; if f'(x) < 0 it is decreasing. Stationary points are where f'(x) = 0. A local maximum or minimum is found by the sign change of f' or by the second derivative: f'' > 0 gives a minimum, f'' < 0 a maximum. Procedure for studying a function: domain, symmetry, limits at the boundaries (asymptotes), first derivative (monotonicity and extrema), second derivative (concavity), then sketch the graph. Higher-order derivatives f'', f''' are obtained by repeated differentiation.

1. Write the definition of the derivative. 2. Differentiate x^2 sin x. 3. Differentiate ln(3x + 1). 4. How does the second derivative classify a stationary point? 5. Is |x| differentiable at 0?

1. f'(a) = limit as h tends to 0 of (f(a + h) - f(a))/h. 2. 2x sin x + x^2 cos x. 3. 3/(3x + 1). 4. f'' > 0 gives a minimum and f'' < 0 a maximum. 5. No, the left and right slopes differ (-1 and +1).