Question

Consider f(x)=x2/3, using the following two forward difference methods to evaluate

df(x)/dx at x=3 with h=1 and h=5 and compare with the exact solution to see which one
is more accurate (note that df(x)/dx =1/4x¹/⁴=1/4 when x=1).
(a) df(x)/dx = f(x+h)−f(x)/h
(b) df(x)/dx =−3f(x)+4f(x+h)−f(x+2h)/2h

Answer 1

Using two forward difference methods, df(x)/dx = f(x+h)−f(x)/h and (b) df(x)/dx =−3f(x)+4f(x+h)−f(x+2h)/2h, the derivative of f(x) = x2/3 is evaluated at x=3 for h=1 and h=5. These results are then compared with the exact derivative to determine the accuracy of each method.

Step-by-step explanation:

The student has been asked to use two forward difference methods to evaluate df(x)/dx at x=3 for a given function f(x) = x2/3. The two methods are:

1. df(x)/dx = f(x+h)−f(x)/h
2. (b) df(x)/dx =−3f(x)+4f(x+h)−f(x+2h)/2h

To apply the first method at x=3 with h=1:

f(3+1) - f(3) = (42/3 - 32/3) / 1 = −f(3 + 2)

Using the second method at x=3 with h=1:

−3f(3) + 4f(3+1) = −3(32/3) + 4(42/3) −f(3 + 2*1) = −3(32/3) + 4(42/3) −(52/3)

We repeat these calculations for h=5 and compare the results with the exact solution df(x)/dx = 1/4x1/4, to determine which h value gives a more accurate result.