3) find the smallest \( n \) such that the error estimate from the error formula is the approx. of the integral \( \int_{0}^{2} \sqrt{x+10} d x \) is less than 0.00001 using the trape zindal rule

1. State whether the following statements are True or False. Justify your answer with the help of a short proof or a counter-example. No marks are awarded for a question without justification : \[ 2 \times 5=10 \] (i) The function \( f(x, y)=x y^{2} \quad \) satisfies Lipschitz condition on : \[ a \leq x \leq b,-\infty<y<\infty \]

\( f ( x ) = \frac { x + 10 } { x - 4 } \quad D f = ? \)

8. \( \lim _{x \rightarrow 4} \frac{\sqrt{3 x-3}-3}{2-\sqrt{x}} \) Ans.: -2

Calculate the maximum value of \( S \) if \( S=\frac{6}{x^{2}+2} \)

Consider the differential equation \( \frac{dy}{dx} = 4x^2 \) with the initial condition \( y(1) = 5 \). Determine the specific solution.