Section 14.7Second-Order Partial Derivatives

Old Stuff• Let y = f(x), then

• Now the first derivative (at a point) gives us the slope of the tangent, the instantaneous rate of change, and whether or not a function is increasing

• What does the second derivative give us?

• If f’’(x) > 0 on an interval, f is concave up on that interval

• If f’’(x) < 0 on an interval, f is concave down on that interval

)()]([2

2

xfdx

ydxf

dx

d

New Stuff

• Suppose z = f(x,y)

• Now we’ve looked at and – What do they give us?

• For z we have 4 second-order partial derivatives

xf yf

yxxy

yyxx

fdy

dz

dx

d

dxdy

zdf

dx

dz

dy

d

dydx

zd

fdy

dz

dy

d

dy

zdf

dx

dz

dx

d

dx

zd

22

2

2

2

2

Let’s practice a little• Find the first and second order partial

derivatives of the following functions

• What do you notice about the mixed partials?

xy

xy

eyxyxh

yxxyxg

eyxf

23),(

ln),(

),(

2

2

yxxy ffei and..

Interpretations of second order partial derivatives

• If then f is concave up in the x direction or the rate of change is increasing at an increasing rate in the x direction (similar for )

• When we are looking at the mixed partials, we are looking at how a partial in one variable is changing in the direction of the other

• For example, tells us how the rate of change of f in the x direction is changing as we move in the y direction

0xxf

yyf

xyf

x

y

40302010

50

60

P●

)(Pf x )(Pf y )(Pf xx )(Pf yy )(Pf xy )(Pf yx

Use the following contour plot to determine the sign of the following partial derivatives at the point, P.

We can use these in Taylor approximations

• Recall we can approximate a function using a 1st degree polynomial

• This polynomial is the tangent line approximation

• The tangent line and the curve we are approximating have the same slope at x = a

• The tangent line approximation is generally more accurate at (or around) x = a

))((')()( axafafxf

• To get a more accurate approximation we use a quadratic function instead of a linear function– In order to make this more accurate, we require

that our approximation have the same value, same slope, and same second derivative at a

• The Taylor Polynomial of Degree 2 approximating f(x) for x near a is

22 2

)('')(')()()( x

afxafafxPxf

Now let z = f(x,y) have continuous first and second partial derivatives at (a,b)

• Now we know from 14.3 that

• Note that our approximation has the same function value and the partial derivatives have the same value as f at (a,b)

• Therefore our quadratic Taylor approximation is

))(,())(,(),( aybafaxbafbafz yx

2 2

( , ) ( , )( ) ( , )( )

( , )( , )( ) ( )( ) ( )

2! 2!

x y

yyxxxy

z f a b f a b x a f a b y b

f a bf a bx a f x a y b y b

• From

we can see that our approximation matches the function value at (a,b) as well as all the partials at (a,b)

• Let’s take a look at this with Maple

2 2

( , ) ( , )( ) ( , )( )

( , )( , )( ) ( )( ) ( )

2! 2!

x y

yyxxxy

z f a b f a b x a f a b y b

f a bf a bx a f x a y b y b