Question

Find the approximate value of f(x) = x³ + 5x² − 2x + 3 at x = 1.98.

Answer 1

f(x) = x³ + 5x² − 2x + 3

∴ f'(x) = `d/dx (x^3 + 5x^2 - 2x + 3)`

= 3x² + 5 × 2x − 2 × 1 + 0

= 3x² + 10x − 2

Take a = 2, h = −0.02

Then f(a) = f(2) = (2)³ + 5(2)² − 2(2) + 3

= 8 + 20 − 4 + 3

= 27

and f′(a) = f′(2) = 3(2)² + 10(2) − 2

= 12 + 20 − 2

= 30

The formula for approximation is

f(a + h) ≑ f(a) + h.f'(a)

∴ f(1.98) = f(2 − 0.02) ≑ f(2) − (0.02) f′(2)

∴ f(1.98) ≑ 27 − 0.02 × 30

∴ f(1.98) ≑ 27 − 0.6

∴ f(1.98) ≑ 26.4