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.
15.Math-Review

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

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.
15.Math-Review

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.
15.Math-Review

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

12. 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

13. 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

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.Math-Review

15. 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

16. 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

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?
15.Math-Review

18. 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

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?
15.Math-Review

20. 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

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
15.Math-Review

22. 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

23. 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