Prove that is an increasing function in

OR

If the radius of a sphere is measured as 9 cm with an error of 0.03 cm, then find the approximate error in calculating its surface area.

Given,

Key point: A function f(x) is said to be increasing over an interval [a, b] if f’(x) > 0 ∀ x ∈ [a, b]

∴ To check whether y is increasing or not, we will differentiate it with respect to θ and will check its sign over the interval [0, π/2]

⇒

Applying the quotient rule of differentiation –

⇒

⇒

⇒

⇒ {∵ sin²θ + cos²θ = 1}

⇒

⇒

⇒

∵ for θ ∈ [0, π/2] ; cos θ ∈ [0, 1]

∴

Thus, we can say that y is increasing function in [0, π/2].

OR

Let, r = radius of sphere = 9 cm

Error in measurement = dr = 0.03

We know that the surface area of the sphere is given by 4πr²

Let A represents the area of the sphere.

∴ A = 4πr²

Differentiating both sides w.r.t r –

⇒

⇒

⇒

⇒ {putting the values given in question}

⇒ dA = 2.16π cm² = 6.786 cm² {taking π = 3.1416}

∴ Approximate error in surface area = 6.786 cm²