BCA Semester 2 – Mathematics II – Unit 3: Application of Differentiation (8Hrs)

Question 1

State Rolle's Theorem. Verify Rolle's Theorem for the function \[ f(x)=x^2-9 \] in the interval $ -3 \le x \le 3 $.

Marks: 5

Year: 2022 Final TU FOHSS

$\textbf{Rolle's Theorem}$ If a function $ f(x) $ satisfies the following conditions on the closed interval $ [a, b] $: 1. $\text{The function } f(x) \text{ is continuous on } [a, b]$ 2. $\text{Th

Question 2

What do you mean by stationary points and inflection points? Using derivatives, find two numbers whose sum is 20 and the sum of whose squares is minimum.

Marks: 10

Year: 2022 Final TU FOHSS

$\textbf{Stationary Points}$ A point $ (a, f(a)) $ on the curve $ y = f(x) $ is called a stationary point if \[ f'(a) = 0 \] At a stationary point, the tangent to the curve is parallel to the \( x

Question 3

Verify Rolle's theorem for the function $ f(x) = \sin x,\quad x \in [0,\pi] $ and find the point on the curve where the tangent is parallel to the $x$-axis.

Marks: 5

Year: 2023 Final TU FOHSS

$\textbf{Step 1: State Rolle's Theorem}$ If a function $ f(x) $ satisfies: 1. $ f(x) $ is continuous on $[a, b]$ 2. $ f(x) $ is differentiable on $(a, b)$ 3. $ f(a) = f(b) $ then there exi

Question 4

Find the maximum and minimum values of the function \[ \quad \quad f(x) = x^3 + 6x^2 + 9x - 2 \] Also, find the point of inflection, if any.

Marks: 5

Year: 2023 Final TU FOHSS

Step 1: Find the derivative \[ f'(x) = \frac{d}{dx} (x^3 + 6x^2 + 9x - 2) = 3x^2 + 12x + 9 \] --- Step 2: Find stationary points Set $ f'(x) = 0 $: \[ 3x^2 + 12x + 9 = 0 \] Divide by 3: \[ x^2 + 4x

Question 5

a. Verify $$\text{Lagrange’s Mean Value Theorem}$$ for the function \[ \quad \quad f(x) = \sqrt{x - 1},\quad x \in [1,3] \] b. Solve the differential equation \[ \quad \quad xy \frac{dy}{dx} =

Marks: 10

Year: 2023 Final TU FOHSS

$\textbf{Unit 3: Application of Differentiation}$ --- (a) Verification of Lagrange’s Mean Value Theorem (LMVT) Step 1: Recall LMVT If $ f(x) $ is continuous on $[a, b]$ and differentiable on \

Question 6

a) Using Simpson’s $ \tfrac{1}{3} $ Rule, evaluate \[ \int0^1 \frac{1}{1+2x^2}\,dx;\quad h = 0.25 \] b) Find the maximum and minimum values of the function \[ f(x) = 4x^3 - 15x^2 + 12x - 1 \]

Marks: 10

Year: 2024 Final TU FOHSS

$\textbf{Unit 4: Integration and Its Applications}$ --- (a) Simpson’s $ \tfrac{1}{3} $ Rule Composite Simpson's rule formula: \[ \inta^b f(x) dx \approx \frac{h}{3} \left[ f(x0) + 4 \sum{i=1,3,\dot