Calculus – Limits & Derivatives Quiz - ICSE Class 11 Mathematics

Free practice quiz: Calculus – Limits & Derivatives (ICSE Class 11 Mathematics). 38 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This ICSE Class 11 Mathematics practice quiz has 38 multiple-choice questions on Calculus – Limits & Derivatives, and 38 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

Sample questions

1. Evaluate the limit: $\lim_{x \to 0} \frac{\sin(ax)}{bx}$, where $a, b \neq 0$.

   1. $\frac{a}{b}$
   2. $ab$
   3. $1$
   4. $0$
   Show answer

   Answer: A. $\frac{a}{b}$

   This is a standard limit. We can rewrite the expression as $\frac{a}{b} \cdot \frac{\sin(ax)}{ax}$. As $x \to 0$, $ax o 0$. We know that $\lim_{y \to 0} \frac{\sin(y)}{y} = 1$. Therefore, the limit is $\frac{a}{b} \cdot 1 = \frac{a}{b}$.

2. What is $\frac{d}{dx}(c)$ where $c$ is a constant?

   1. $0$
   2. $c$
   3. $1$
   4. $x$
   Show answer

   Answer: A. $0$

   The derivative of a constant is always zero. This is because the rate of change of a constant value is zero.

3. Find the derivative of $f(x) = \sin^{-1}(x)$.

   1. $\frac{1}{\sqrt{1-x^2}}$
   2. $\frac{1}{1+x^2}$
   3. $\frac{1}{\sqrt{x^2-1}}$
   4. $-\frac{1}{\sqrt{1-x^2}}$
   Show answer

   Answer: A. $\frac{1}{\sqrt{1-x^2}}$

   The derivative of the inverse sine function is a standard result: $\frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1-x^2}}$ for $|x| < 1$.

4. If $f(x) = \sec(x)$, find $f'(x)$.

   1. $\sec(x)\tan(x)$
   2. $\sec^2(x)$
   3. $-\csc(x)\cot(x)$
   4. $-\sec(x)\tan(x)$
   Show answer

   Answer: A. $\sec(x)\tan(x)$

   The derivative of the secant function is a standard result: $\frac{d}{dx}(\sec(x)) = \sec(x)\tan(x)$. This can be derived using the quotient rule on $\frac{1}{\cos(x)}$.

5. Find the derivative of $f(x) = x^a$, where $a$ is a constant.

   1. $ax^{a-1}$
   2. $a x^a$
   3. $a x^{a-1}$
   4. $x^{a-1}$
   Show answer

   Answer: A. $ax^{a-1}$

   This is the power rule for differentiation: $\frac{d}{dx}(x^n) = nx^{n-1}$. Here, the exponent is $a$, so the derivative is $a x^{a-1}$.

More questions in this quiz

Answers and explanations are in the full quiz.

1. Evaluate $\lim_{x \to \infty} \frac{3x^2 + 5x - 2}{2x^2 - 7x + 1}$.
2. What is the derivative of $\ln(x)$?
3. Evaluate $\lim_{x \to 0} \frac{\tan(x)}{x}$.
4. If $f(x) = x^3 - 2x^2 + 5x - 7$, find $f'(x)$.
5. Evaluate $\lim_{x \to 0} \frac{\sin(3x)}{\sin(5x)}$.
6. Evaluate $\lim_{x \to 0} \frac{1 - \cos(x)}{x^2}$.
7. Evaluate $\lim_{x \to 0} \frac{\sin(ax) - \sin(bx)}{x}$ where $a, b \neq 0$.
8. Evaluate $\lim_{x \to 0} \frac{e^{x} - 1}{x}$.
9. Evaluate $\lim_{x \to \infty} \frac{\sin(x)}{x}$.
10. If $y = \tan(x)$, find $\frac{dy}{dx}$.
11. Evaluate $\lim_{x \to 0} \frac{\tan(ax)}{bx}$ where $a, b \neq 0$.
12. If $f(x) = x^x$, find $f'(x)$.
13. Evaluate $\lim_{x \to 0} \frac{\sqrt{x+a} - \sqrt{a}}{x}$, where $a > 0$.
14. If $f(x) = e^{ax+b}$, find $f'(x)$.
15. Evaluate $\lim_{x \to 0} \frac{1 - \cos(x)}{x}$.
16. Find the derivative of $f(x) = \ln(kx)$ where $k eq 0$.
17. Evaluate $\lim_{x \to 1} \frac{x^n - 1}{x^m - 1}$, where $n, m$ are integers and $m eq 0$.
18. Find the derivative of $f(x) = \frac{u(x)}{v(x)}$.
19. Evaluate $\lim_{x \to 0} \frac{\sin(x)}{x}$ using the definition of derivative of $\sin(x)$ at $x=0$.
20. Evaluate $\lim_{x \to 0} \frac{\sinh(x)}{x}$.
21. Evaluate $\lim_{x \to 0} \frac{a^x - 1}{x}$ where $a > 0, a eq 1$.
22. If $f(x) = \cot(x)$, find $f'(x)$.
23. Evaluate $\lim_{x \to 0} \frac{x}{\sin(x)}$.
24. Find the derivative of $f(x) = e^{x^2}$.
25. Evaluate $\lim_{x \to \infty} (1 + \frac{1}{x})^x$.
26. If $f(x) = \tan^{-1}(x)$, find $f'(x)$.
27. Evaluate $\lim_{x \to 0} \frac{\sin^{-1}(x)}{x}$.
28. If $f(x) = \ln(\cos(x))$, find $f'(x)$.
29. Evaluate $\lim_{x \to 0} \frac{\tan(x) - x}{x^3}$.
30. If $f(x) = \sin(x) \cos(x)$, find $f'(x)$.
31. Find the derivative of $f(x) = \tan^{-1}(\frac{x}{a})$.
32. Evaluate $\lim_{x \to 0} \frac{\ln(1+x)}{x}$.
33. What is the limit of $\frac{x^2-4}{x-2}$ as $x$ approaches 2?

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