15.Math-Review Tuesday 8/15/00

Convexity and Concavity Consider the function f(x)=x2 over the interval [-1,1]. Is this function convex or concave? Prove it. 15.Math-Review

Differentiation The derivative The derivative of a function at a point is the instantaneous slope of the function at that point. This is, the slope of the tangent line to the function at that point. Notation: for a function y = f(x), the derivative of f with respect to x can be written as: 15.Math-Review

Differentiation This graphically: y y= (x-t)f’(t)+f(t) f(t) y=f(x) f(s) y= (x-s)f’(s)+f(s) y=f(x) x 15.Math-Review

Differentiation Rules of differentiation: Example: (a) f(x) = k => f’(x) = 0 (b) f(x) = ax => f’(x) = a (c) f(x) = xn => f’(x) = nxn–1 Example: f(x) = x f(x) = x5 f(x) = x2/3 f(x) = x–2/5 15.Math-Review

Differentiation Rules of differentiation: (d) f(x) = g(x) + h(x) => f’(x) = g’(x) + h’(x) (e) f(x) = kg(x) => f’(x) = kg’(x) (f) f(x) = g(x)n => f’(x) = n g’(x)g(x)n–1 Inverse rule as a special case of this: Example: f(x) = 3x2 f(x) = 3x3 – 4 x2 + 6x – 20 f(x) = (3–7x)–3 15.Math-Review

Differentiation More rules of differentiation: (g) f(x) = g(x)h(x) => f’(x) = g’(x)h(x)+ g(x)h’(x) (h) (i) f(x) = g(h(x)) => f’(x) = g’(h(x))h’(x) Inverse rule as a special case of this: Example: product, quotient and chain for the following: g(x) = x+2, h(x) = 3x2 g(x) = 3x2 + 2, h(x) = 2x – 5 g(x) = 6x2, h(x) = 2x + 1 g(x) = 3x, h(x) = 7x2 – 10 g(x) = 3x + 6, h(x) = (2x2 + 5).(3x – 2) 15.Math-Review

Differentiation Even more rules of differentiation: Example: (j) f(x) = ax => f’(x) = ln(a)ax (k) f(x) = ln(x) => f’(x) = 1/x Example: f(x) = ex f(x) = ln(3x3 + 2x+6) f(x) = ln(x-3) 15.Math-Review

Differentiation Example: logs, rates and ratios: For the following examples we will consider y a function of x, ( y(x) ). Compute: For this last example find an expression in terms of rates of changes of x and y. 15.Math-Review

Differentiation A non-linear model of the demand for door knobs, relating the quantity Q to the sales price P was estimated by our sales team as Q = e9.1 P-0.10 Derive an expression for the rate of change in quantity to the rate of change in price. 15.Math-Review

Differentiation To differentiate is a trade…. 15.Math-Review

Differentiation Higher order derivatives: The second derivative of f(x) is the derivative of f’(x). It is the rate of change of function f’(x). Notation, for a function y=f(x), the second order derivative with respect to x can be written as: Higher order derivatives are defined analogously. Example: Second order derivative of f(x) = 3x2-12x +6 f(x) = x3/4-x3/2 +5x 15.Math-Review

Differentiation Application of f’’(x) Therefore: We have that f’(t)  f’(t+) This means that the rate of change of f’(x) around t is negative. f’’(t)  0 We also note that around t, f is a concave function. Therefore: f’’(t) 0 is equivalent to f a concave function around t. f’’(t)  0 is equivalent to f a convex function around t. y=f(x) slope=f’(t +) t slope=f’(t) t+ 15.Math-Review

Differentiation Partial derivatives: For functions of more than one variable, f(x,y), the rate of change with respect to one variable is given by the partial derivative. The derivative with respect to x is noted: The derivative with respect to y is noted: Example: Compute partial derivatives w/r to x and y. f(x,y) = 2x + 4y2 + 3xy f(x,y) = (3x – 7)(4x2 – 3y3) f(x,y) = exy 15.Math-Review

Stationary Points Maximum A point x is a local maximum of f, if for every point y ‘close enough’ to x, f(x) > f(y). A point x is a global maximum of f, if f(x) > f(y) for any point y in the domain. In general, if x is a local maximum, we have that: f’(x)=0, and f’’(x)<0. Graphically: Global Maximum Local Maximum 15.Math-Review

Stationary Points Minimum A point x is a local minimum of f, if for every point y ‘close enough’ to x, f(x) < f(y). A point x is a global minimum of f, if f(x) < f(y) for any point y in the domain. In general, if x is a local minimum, we have that: f’(x)=0, and f’’(x)>0. Graphically: Local Minimum Global Minimum 15.Math-Review

Stationary Points Example: Consider the function defined over all x>0, f(x) = x - ln(x). Find any local or global minimum or maximum points. What type are they? 15.Math-Review

Stationary Points Consider the following example: The function is only defined in [a1, a4]. Points a1 and a3 are maximums. Points a2 and a4 are minimums. And we have: f’(a1) < 0 and f’’ (a1) ? 0 f’(a2) = 0 and f’’ (a2)  0 f’(a3) = 0 and f’’ (a3)  0 f’(a4) < 0 and f’’ (a4) ? 0 The problem arises in points that are in the boundary of the domain. a1 a2 a3 a4 15.Math-Review

Stationary Points Example: Consider the function defined over all x[-3,3], f(x) = x3-3x+2. Find any local or global minimum or maximum points. What type are they? 15.Math-Review

Stationary Points Points of Inflection. Is where the slope of f shifts from increasing to decreasing or vice versa. Or where the function changes from convex to concave or v.v. In other words f’’(x) = 0!! Points of Inflection 15.Math-Review

Stationary Points Finding Stationary Points Given f(x), find f’(x) and f”(x). Solve for x in f’(x) = 0. Substitute the solution(s) into f”(x). If f”(x)  0, x is a local minimum. If f”(x)  0, x is a local maximum. If f”(x) = 0, x is likely a point of inflection. Example: f(x) = x2 – 8x + 26 f(x) = x3 + 4x2 + 4x f(x) = 2/3 x3 – 10 x2 + 42x – 3 15.Math-Review

Tough examples to kill time Application of derivative: L’Hopital rule. Use this rule to find a limit for f(x)=g(x)/h(x): 15.Math-Review

Tough examples to kill time Let us consider the function Obtain a sketch of this function using all the information about stationary points you can obtain. Sketch the function Hint: for this we will need to know that the ex ‘beats’ any polynomial for very large and very small x. 15.Math-Review