Question

Find approximate value of f (3.01) where f (x)=x³-2x²-3x+1

Answer 1

This question is based on concept of application of derivatives.

function, y = f(x) = x³ - 2x² - 3x + 1

let x = 3, so that x + ∆x = 3.01

⇒3 + ∆x = 3.01

⇒∆x = 3.01 - 3 = 0.01

now differentiating y = f(x) with respect to x,

dy/dx = d(x³ - 2x² - 3x + 1)/dx

= 3x² - 4x - 3

at x = 3,

dy/dx = 3(3)² - 4(3) - 3 = 27 - 15 = 12

so, ∆y = (dy/dx)∆x

⇒∆y = 12 × 0.01

⇒∆y = 0.12

now applying, f(x + ∆x) = y + ∆y

or, f(3.01) = f(3) + 0.12

= (3)³ - 2(3)² - 3(3) + 1 + 0.12

= 27 - 18 - 9 + 1 + 0.12

= 1.12

hence, f(3.01) = 1.12

Answer 2

Answer:

1.12

Step-by-step explanation:

We have given that f(x)= x³-2x²-3x+1.

We have to find the value of f(3.01).

Now we will calculate it.

It is clear that, f(3.01)= (3.01)³-2(3.01)²-3(3.01)+1

⇒f(3.01)=(3+0.01)³-2(3+0.01)²-3(3.01)+1

⇒f(3.01)={3³+3×3²×0.01+3×3×0.01²+0.01³}-2{3²+2×3×0.01+0.01²)-3(3.01)+1

We have applied the formulas

(a+b)³=a³+3a²b+3ab²+b³ and (a+b)²= a²+2ab+b²

Now, we will ignore the higher order terms of 0.01 because those will be very small in value.

⇒f(3.01)= (3³+3×3²×0.01)-2(3²+2×3×0.01)-3(3.01)+1

⇒f(3.01)= (27+0.27)-2(9+0.06)-3(3.01)+1

⇒f(3.01)= 27.27-2(9.06)-9.03+1

⇒f(3.01)= 27.27-18.12-9.03+1

⇒f(3.01)= 1.12 (Approximate)

(Answer)