Single-variable Differential Calculus - Calculus and Optimization - Data Science & Artificial Intelligence Previous Year Questions

1

2024 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science & Artificial Intelligence (DA) 2024

For any twice differentiable function \( f: \mathbb{R} \to \mathbb{R} \), if at some \( x^* \in \mathbb{R} \), \( f'(x^*) = 0 \) and \( f''(x^*) > 0 \), then the function \( f \) necessarily has a ______ at \( x = x^* \).
Note: \( \mathbb{R} \) denotes the set of real numbers.

A

local minimum

B

global minimum

C

local maximum

D

global maximum

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2

2024 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science & Artificial Intelligence (DA) 2024

Let f: ℝ → ℝ be the function f(x) = 1 / (1 + e^{-x}).
The value of the derivative of f at x where f(x) = 0.4 is ______ (rounded off to two decimal places).
Note: ℝ denotes the set of real numbers.

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3

2024 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science & Artificial Intelligence (DA) 2024

Let \(f : \mathbb{R} \to \mathbb{R}\) be a function. Note: \(\mathbb{R}\) denotes the set of real numbers.
\(f(x) = \begin{cases} -x, & \text{if } x < -2 \\ ax^2 + bx + c, & \text{if } x \in [-2, 2] \\ x, & \text{if } x > 2 \end{cases}\)
Which ONE of the following choices gives the values of \(a, b, c\) that make the function \(f\) continuous and differentiable?

A

\(a = \frac{1}{4}, b = 0, c = 1\)

B

\(a = \frac{1}{2}, b = 0, c = 0\)

C

\(a = 0, b = 0, c = 0\)

D

\(a = 1, b = 1, c = -4\)

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4

2024 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science & Artificial Intelligence (DA) 2024

Consider the function \(f: \mathbb{R} \to \mathbb{R}\) where \(\mathbb{R}\) is the set of all real numbers.
\(f(x) = \frac{x^4}{4} - \frac{2x^3}{3} - \frac{3x^2}{2} + 1\)
Which of the following statements is/are TRUE?

A

\(x = 0\) is a local maximum of \(f\)

B

\(x = 3\) is a local minimum of \(f\)

C

\(x = -1\) is a local maximum of \(f\)

D

\(x = 0\) is a local minimum of \(f\)

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5

2024 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science & Artificial Intelligence (DA) 2024

Evaluate the following limit:
\[ \lim_{x \to 0} \frac{\ln(x^2+1)\cos x}{x^2} = \] ______.

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6

2025 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science & Artificial Intelligence (DA) 2025

Let \(f(x) = \frac{e^x - e^{-x}}{2}\), \(x \in \mathbb{R}\). Let \(f^{(k)}(a)\) denote the \(k^{th}\) derivative of \(f\) evaluated at \(a\). What is the value of \(f^{(10)}(0)\)? (Note: ! denotes factorial)

A

0

B

1

C

\(\frac{1}{10!}\)

D

\(\frac{2}{10!}\)

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7

2025 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science & Artificial Intelligence (DA) 2025

Consider two functions \(f : \mathbb{R} \to \mathbb{R}\) and \(g : \mathbb{R} \to (1, \infty)\). Both functions are differentiable at a point \(c\). Which of the following functions is/are ALWAYS differentiable at \(c\)? The symbol \(\cdot\) denotes product and the symbol \(\circ\) denotes composition of functions.

A

\(f \pm g\)

B

\(f \cdot g\)

C

\(\frac{f}{g}\)

D

\(f \circ g + g \circ f\)

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8

2025 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science & Artificial Intelligence (DA) 2025

\[\lim_{t \to +\infty} \sqrt{t^2 + t} - t = \] ______________ (Round off to one decimal place)

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9

2025 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science & Artificial Intelligence (DA) 2025

Consider the function \(f(x) = \frac{x^3}{3} + \frac{7}{2}x^2 + 10x + \frac{133}{2}\), \(x \in [-8, 0]\). Which of the following statements is/are correct?

A

The maximum value of \(f\) is attained at \(x = -5\)

B

The minimum value of \(f\) is attained at \(x = -2\)

C

The maximum value of \(f\) is \(\frac{133}{2}\)

D

The minimum value of the derivative of \(f\) is attained at \(x = -\frac{7}{2}\)

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10

2025 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science & Artificial Intelligence (DA) 2025

Let \(f : \mathbb{R} \to \mathbb{R}\) be a twice-differentiable function and suppose its second derivative satisfies \(f''(x) > 0\) for all \(x \in \mathbb{R}\). Which of the following statements is/are ALWAYS correct?

A

\(f\) has a local minima

B

There does not exist \(x\) and \(y\), \(x \neq y\), such that \(f'(x) = f'(y) = 0\)

C

\(f\) has at most one global minimum

D

\(f\) has at most one local minimum

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11

2025 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science & Artificial Intelligence (DA) 2025

Let \(f : \mathbb{R} \rightarrow \mathbb{R}\) be such that \(|f(x) - f(y)| \leq (x - y)^2\) for all \(x, y \in \mathbb{R}\). Then \(f(1) - f(0) = \)__________ (Answer in integer)

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12

2026 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science and Artificial Intelligence (DA) 2026

Let \( f(x) = x^3 - 3x^2 + 2 \) be a function defined on \( (-1, 3] \).
Which of the following statements is/are correct?

A

\(f(x) \) has exactly two roots in \([-0.9, 0] \).

B

\(f(x) \) has a minimum at 2 only.

C

\(f(x) \) has maximum at 0 only.

D

\(f(x) \) has a root at 1.

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