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15.Math-Review Tuesday 8/15/00 Convexity and Concavity Consider the function f(x)=x 2 over the interval [-1,1]. Is this function convex or concave? Prove it. Notation: for a function y = f(x), the derivative of f with respect to x can be written as: Differentiation The derivative

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15 math review l.jpg

15.Math-Review

Tuesday 8/15/00


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Convexity and Concavity

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


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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.


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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


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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


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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


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(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)


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  • 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


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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.


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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.


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Differentiation

  • To differentiate is a trade….


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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


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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.


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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


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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:


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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:


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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?


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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.


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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?


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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!!


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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


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Tough examples to kill time

  • Application of derivative: L’Hopital rule.


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  • 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.


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