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  1. 15.Math-Review Tuesday 8/15/00

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

  3. Notation: for a function y = f(x), the derivative of f with respect to x can be written as: 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.

  4. y= (x-t)f’(t)+f(t) f(t) f(s) y= (x-s)f’(s)+f(s) s t Differentiation • This graphically: y y=f(x) x

  5. Differentiation • Rules of differentiation: (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

  6. Inverse rule as a special case of this: 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 • Example: f(x) = 3x2 f(x) = 3x3 – 4 x2 + 6x – 20 f(x) = (3–7x)–3

  7. (h) • Inverse rule as a special case of this: Differentiation • More rules of differentiation: (g) f(x) = g(x)h(x) => f’(x) = g’(x)h(x)+ g(x)h’(x) (i) f(x) = g(h(x)) => f’(x) = g’(h(x))h’(x) • 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)

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

  9. 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.

  10. 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.

  11. Differentiation • To differentiate is a trade….

  12. Higher order derivatives are defined analogously. 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: • Example: Second order derivative of f(x) = 3x2-12x +6 f(x) = x3/4-x3/2 +5x

  13. slope=f’(t +) slope=f’(t) y=f(x) t+ t Differentiation • Application of f’’(x) • 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.

  14. 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. Global Maximum Local Maximum Local Maximum 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:

  16. Local Minimum Global Minimum 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:

  17. 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?

  18. a3 a1 a2 a4 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.

  19. 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?

  20. Points of Inflection Points of Inflection 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!!

  21. 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

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

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