List of practice Questions

Let \( f : \mathbb{R} \to \mathbb{R} \) be such that \( |f(x) - f(y)| \leq (x - y)^2 \) for all \( x, y \in \mathbb{R} \). Then \[ f(1) - f(0) = \ \underline{\hspace{2cm}} \quad \text{(Answer in integer)} \]

Let \( f : \mathbb{R} \to \mathbb{R} \) be such that \( |f(x) - f(y)| \leq (x - y)^2 \) for all \( x, y \in \mathbb{R} \). Then \[ f(1) - f(0) = \ \underline{\hspace{2cm}} \quad \text{(Answer in integer)} \]

Consider the given function \( f(x) \). \[ f(x) = \begin{cases} ax + b & \text{for } x < 1 \\ x^3 + x^2 + 1 & \text{for } x \geq 1 \end{cases} \] If the function is differentiable everywhere, the value of \( b \) must be __________ (rounded off to one decimal place).

For the function \( f(x) = e^x |\sin x|, \; x \in \mathbb{R}, \) which of the following statements is/are TRUE?}

The value of the following limit is \(\underline{\hspace{1cm}}\). \[ \lim_{x \to 0^+} \frac{\sqrt{x}}{1 - e^{2\sqrt{x}}} \]

Consider the following expression.
\[ \lim_{x \to -3} \frac{\sqrt{2x + 22} - 4}{x + 3} \] The value of the above expression (rounded to 2 decimal places) is \(\underline{\hspace{2cm}}\).