University of Nottingham

School of Economics University of Nottingham Pre-sessional Mathematics Masters (MSc) Dr Maria Montero Dr Alex Possajennikov Topic 1 Univariate Calculus Pre-sessional Mathematics Topic 1

Univariate Calculus Functions f : X  Y gives for each element x  X one element y Y y = f(x) Real functions f : R  R, where R is the set of real numbers Example: y(x) = x2 In economics: cost function, production function, demand function, ... Let y = f(x) x is independent variable y is dependent variable Let f(x) = a - bx a,b are parameters Pre-sessional Mathematics Topic 1

Example: f(x) = is defined on non-negative numbers: D = {x: x  0} Univariate Calculus DomainD of a function: the set of interest on which it is defined Composite function: f(g(x)) Example: (x+1)2is composed of f(y) = y2and g(x) = x + 1 Inverse function: f(g(x)) = x and g(f(y)) = y Inverse function is denoted by f -1 Example: Demand function q = f(p) = 100 - 2p Inverse demand function p = f -1(q) Pre-sessional Mathematics Topic 1

Univariate Calculus Limits When x is close to a, f(x) is close to b Definition: limx  a f(x) = b when for any  > 0, there exists  > 0 such that |f(x) - b| <  for any x such that |x - a| <  Example: limx  0 1/x2 Example: limx  0 1/x Continuity: f(x) is continuous at x0 if limx  x0 f(x) = f(x0). f(x) is continuous on D if it is continuous at every point in D Intermediate Value Theorem: f(x) is continuous on [a,b], f(a)  f(b). Then for any y between f(a) and f(b) there exists c between aand b such that f(c) = y. Pre-sessional Mathematics Topic 1

5 4 3 2 slope=2 1 0 -2 2 x Univariate Calculus Differentiation Consider function f(x). Fix x0. The derivative of f(x) at x0 is When the limit exists, f(x) is differentiable at x0. Example: f(x) = x2at x0 = 1 Approximation by derivative Close to x0, f(x)  f(x0) + f’(x0)(x - x0) In economics: marginal cost, marginal product, marginal revenue Pre-sessional Mathematics Topic 1

Univariate Calculus Derivative is itself a function: for each x Examples: f(x) = xk, f’(x) = kxk-1 f(x) = ln x, f’(x) = 1/x = x -1 Rules of differentiation Sum-difference: (f  g)’ = f ’  g’ Product: (f ·g)’ = f ·g’+ f ’ ·g Quotient: Chain: (f(g(x))’ = f ’ ·g’ Example: [(2x + 1)2]’ Pre-sessional Mathematics Topic 1

Univariate Calculus Differentiable functions are continuous Not all continuous functions are differentiable Example: |x| is not differentiable at 0 If f’(x) is continuous, then f(x) is continuously differentiable, f  C1 The derivative of f’(x) is the second derivative of f(x), f’’(x) Further derivatives are constructed in a similar fashion If all derivatives of f(x) are continuous, f  C Pre-sessional Mathematics Topic 1

f(x2) f(x2) f((1-)x1+x2) (1-)f(x1)+f(x2) (1-)f(x1)+f(x2) f((1-)x1+x2) f(x1) f(x1) x1 x2 0 x1 x2 0 Univariate Calculus Concavity and convexity f(x) is convex if f((1 - )x1 + x2)  (1 - )f(x1) + f(x2) f(x) is concave if f((1 - )x1 + x2)  (1 - )f(x1) + f(x2) If f’’(x)  0then f(x) is convex If f(x) is C2: If f’’(x)  0then f(x) is concave Pre-sessional Mathematics Topic 1

Anti-derivative is also denoted by Univariate Calculus Integration Anti-derivative of f(x) is such function F(x) that F’(x) = f(x) Anti-derivative is determined up to a constant Example: f(x) = xk, Integration by parts (f ·g)’ = f ·g’+ f ’ ·g Example: Pre-sessional Mathematics Topic 1

f(xi) f(x) a=x0 xi b=xn Univariate Calculus Definite Integral f(x1)(x1 - x0) + … + f(xn)(xn - xn-1) =  f(xi)xi The Fundamental Theorem of Calculus (Newton-Leibniz formula) In economics: consumer surplus, measures of inequality, probability, stock and flow variables Pre-sessional Mathematics Topic 1

f’= 0 10 8 f’ increasing 8 6 f’< 0 f’’> 0 f’> 0 6 4 f’< 0 f’> 0 f’ decreasing f’’< 0 4 2 f’= 0 2 0 -1 1 2 3 4 5 -1 0 1 2 3 4 5 x x Univariate Calculus Optimisation x0 is the global maximum of f(x) if f(x0)  f(x) for all x in D x0 is the global minimum of f(x) if f(x0)  f(x) for all x in D x0 is the local maximum of f(x) if f(x0)  f(x) for all x in a small interval around x0 Theoremf  C1. If x0 is an interior local maximum or minimum, then f’(x0) = 0 Pre-sessional Mathematics Topic 1

Univariate Calculus Points wheref’(x) = 0 (along with points where f’(x) does not exist) are critical points of f(x). Theorem f  C2. f’(x0) = 0. If f’’(x0) < 0, then x0 is a local maximum. If f’’(x0) > 0, then x0 is a local minimum. Example: f(x) = x3 - 3x2 + 4 Pre-sessional Mathematics Topic 1

Univariate Calculus Theoremf  C1. f’(x0) = 0. x0is the unique solution of f’(x) = 0. If x0 is local maximum, then x0 is global maximum Example: f(x) = x3 - 3x2 - 9x + 5on x  0 Theoremf  C2. f’(x0) = 0. If f’’(x) < 0 for all x(f(x) is concave), then x0 is global maximum Example: f(x) = - x2 + 2x - 3 Pre-sessional Mathematics Topic 1

Univariate Calculus General approach to maximisation of C1 function on an interval 1. Find all critical points of f(x) by solving f’(x) = 0 2. Find values of f(x) at these points 3. Find values of f(x) at the ends of the interval If ends are not included, find lim f(x) as x approaches the ends 4. Choose the highest value from 2 and 3 If the highest value is at the interval end that is not included, there is no maximum If the ends are included in the interval, then maximum (and minimum) exists Example: f(x) = x2 - 4x + 5on [0,3) Pre-sessional Mathematics Topic 1

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