3 Second derivatives

An approach which has been found to work well for second derivatives involves applying the notion of a central difference three times. We begin with

f ( a ) f ( a + 1 2 h ) f ( a 1 2 h ) h .

Next we approximate the two derivatives in the numerator of this expression using central differences as follows:

f ( a + 1 2 h ) f ( a + h ) f ( a ) h and f ( a 1 2 h ) f ( a ) f ( a h ) h .

Combining these three results gives

f ( a ) f ( a + 1 2 h ) f ( a 1 2 h ) h 1 h f ( a + h ) f ( a ) h f ( a ) f ( a h ) h = f ( a + h ) 2 f ( a ) + f ( a h ) h 2
Key Point 12

Second Derivative Approximation

A central difference approximation to the second derivative f ( a ) is



f ( a ) f ( a + h ) 2 f ( a ) + f ( a h ) h 2

Example 21

The distance x of a runner from a fixed point is measured (in metres) at intervals of half a second. The data obtained are

t 0.0 0.5 1.0 1.5 2.0 ̲ ̲ ̲ ̲ ̲ ̲ x 0.00 3.65 6.80 9.90 12.15

Use a central difference to approximate the runner’s acceleration at t = 1.5 s.

Solution

Our aim here is to approximate x ( t ) .

Using data with t = 1.5 s at its centre we obtain

x ( 1.5 ) x ( 2.0 ) 2 x ( 1.5 ) + x ( 1.0 ) 0 . 5 2 = 3.40 m s 2 ,

from which we see that the runner is slowing down.

Exercises
  1. Let f ( x ) = cosh ( x ) and a = 2 . Let h = 0.01 and approximate f ( a ) using forward, backward and central differences. Work to 8 decimal places and compare your answers with the exact result, which is sinh ( 2 ) .
  2. The distance x , measured in metres, of a downhill skier from a fixed point is measured at intervals of 0.25 s. The data gathered are

    t 0 0.25 0.5 0.75 1 1.25 1.5 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ x 0 4.3 10.2 17.2 26.2 33.1 39.1

    Use a central difference to approximate the skier’s velocity and acceleration at the times

    t = 0.25 s, 0.75 s and 1.25 s. Give your answers to 1 decimal place.

  1. Forward: f ( a ) cosh ( a + h ) cosh ( a ) h = 3.79865301 3.76219569 0.01 = 3.64573199



    Backward: f ( a ) cosh ( a ) cosh ( a h ) h = 3.76219569 3.72611459 0.01 = 3.60810972



    Central: f ( a ) cosh ( a + h ) cosh ( a h ) 2 h = 3.79865301 3.72611459 0.02 = 3.62692086



    The accurate result is sinh ( 2 ) = 3.62686041 .
  2. Velocities at the given times approximated by a central difference are:

    20.4 m s 1 , 32.0 m s 1 and 25.8 m s 1 .

    Accelerations at these times approximated by a central difference are:

    25.6 m s 2 , 32.0 m s 2 and 14.4 m s 2 .