# Introduction to Bayesian statistics - Bolstad M.

ISBN 0-471-27020-2

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F(a) = lim F(x) and F(b) = lim F(x)

x^-Ð° x^-b

provided those limits exist. Then we define the definite integral with the same formula as before

f f(x) = F(b) â€” F(a)

Ð°

For example, let f (x) = x-1/2. This function is defined over the half-open interval (0,1]. It is not defined over the closed interval [0,1] because it is not defined at the endpoint x = 0. This curve is shown in Figure A.9. We see the curve has a

290 INTRODUCTION TO CALCULUS

vertical asymptote at x = 0. We will define

F(0) = lim F(x)

X^0

= lim 2x1/2

x^-0 = 0.

Then

f1 x-1/2 = 2x1/2 0

1

= 2.

0

Theorem 9 Integration by parts. Let F(x) and G(x) be differentiable functions defined on an interval [a, b]. Then

bb / F'(x) X G(x)dx = F(x) X G(x)ia â€” / F(x) X G'(x)dx.

aa

Proof: Integration by parts is the inverse of finding the derivative of the product

F( x) X G(x) :

â€” [F(x) X G(x)] = F(x) X G'(x) + F(x) X G'(x). dx

Integrating both sides, we see that

/* b /* b

F(b) X G(b) â€” F(a) X G(a) = F(x) X G'(x)dx + / F'(x) X G(x)dx .

aa

Theorem 10 Change of variable formula. Let x = g(y) be a differentiable function on the interval [a, b]. Then

rb /-s(b)

/ f (g(y))g'(y)dy = / f (y)dy

Ja Jg(a)

The change of variable formula is the inverse of the chain rule for differentiation. The derivative of the function of a function F (g(y)) is

d-[F(g(y)] = F'(g(y)) x g'(y). dx

Integrating both sides from y = a to y = b gives

F(g(b)) â€” F(g(a)) = fÐ F'(g(y)) X g'(y)dy .

a

The left-hand-side equals F'(y)dy. Let f (x) = F'(x), and the theorem is

proved.

INTRODUCTION TO CALCULUS 291

MULTIVARIATE CALCULUS

Partial Derivatives

In this section we consider the calculus of two or more variables. Suppose we have a function of two variables f (x, y). The function is continuous at the point (a, b) if and only if

lim f (x, y) = f (a, b).

(x,y)^(a,b)

The first partial derivatives at the point (a, b) are defined to be

df (x, y)

and

dx

df (x, y)

dy

(a,b)

(a,b)

lim

h^0

lim

h^0

f (a + h, b) â€” f (a, b)

f (a, b + h) â€” f (a, b) h

provided these limits exist. In practice, the first partial derivative in the x-direction is found by treating y as a constant and differentiating the function with respect to x, and vice versa, to find the first partial derivative in the y-direction.

If the function f (x, y) has first partial derivatives for all points (x, y) in a continuous two-dimensional region, then the first partial derivative function with respect to x is the function that has value at point (x, y) equal to the partial derivative of f (x, y) with respect to x at that point. It is denoted

fx(x,y) =

df (x, y)

dx

(x,y)

The first partial derivative function with respect to y is defined similarly. The first derivative functions fx(x, y) and fy (x, y) give the instantaneous rate of change of the function in the x-direction and y-direction, respectively.

The second partial derivatives at the point (a, b) are defined to be

d2f(x,y)

and

dx2

d2f (x,y)

lim fx(x + h,y) â€” fx(x,y)

dy2

(a,b)

(a,b)

h^0

lim fy (x,y + h) â€” fy (x,y)

h^0

The second cross partial derivatives at (a, b) are

d2f(x,y)

and

dxdy d2 f (x,y)

. fy(x + h,y) â€” fy (x,y)

dydx

(a,b)

(a,b)

lim y

h^0

h

lim fx(x,y + h) â€” fx(x,y)

h^0

h

292 INTRODUCTION TO CALCULUS

For all the functions that we consider, the cross partial derivatives are equal, so it doesnâ€™t matter which order we differentiate.

If the function f (x, y) has second partial derivatives (including cross partial derivatives) for all points (x, y) in a continuous two-dimensional region, then the second partial derivative function with respect to x is the function that has value at point (x, y) equal to the second partial derivative of f (x, y) with respect to x at that point. It is denoted

dfx(x,y)

fxx(x,y) =

dx

(x,y)

The second partial derivative function with respect to y is defined similarly. The second cross partial derivative functions are

and

fxy (x, y)

fyx (x, y)

dfx(x,y)

dy

dfy (x,y)

dx

(x,y)

(x,y)

The two cross partial derivative functions are equal.

Partial derivatives of functions having more than 2 variables are defined in a similar manner.

Finding Minima and Maxima of a Multivariate Function

A univariate functions with a continuous derivative achieves minimum or maximum at an interior point x only at points where the derivative function f '(x) = 0. However, not all such points were minimum or maximum. We had to check either the first derivative test, or the second derivative test to see whether the critical point was minimum, maximum, or point of inflection.

The situation is more complicated in two dimensions. Suppose a continuous differentiable function f (x, y) is defined on a two dimensional rectangle. It is not enough that both fx(x, y) = 0 and fy (x, y) = 0.

The directional derivative of the function f (x, y) in direction Ð² at a point measures the rate of change of the function in the direction of the line through the point that has angle Ð² with the positive x-axis. It is given by

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