Calculus – Continuity, Differentiation & Integration Quiz - ICSE Class 12 Mathematics

Free practice quiz: Calculus – Continuity, Differentiation & Integration (ICSE Class 12 Mathematics). 26 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This ICSE Class 12 Mathematics practice quiz has 26 multiple-choice questions on Calculus – Continuity, Differentiation & Integration, and 26 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. If $f(x) = \frac{\sin(x)}{x}$ for $x \neq 0$ and $f(0) = k$, for what value of $k$ is $f(x)$ continuous at $x=0$?

   1. 1
   2. 0
   3. undefined
   4. $\cos(0)$
   Show answer

   Answer: A. 1

   For $f(x)$ to be continuous at $x=0$, we must have $\lim_{x \to 0} f(x) = f(0)$. We know that $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$. Therefore, $k$ must be 1.

2. For the function $f(x) = \begin{cases} x+1 & \text{if } x \le 1 \ 3-ax^2 & \text{if } x > 1 \end{cases}$, find the value of $a$ that makes $f(x)$ continuous at $x=1$.

   1. 2
   2. 1/2
   3. 1
   4. 0
   Show answer

   Answer: C. 1

   For continuity at $x=1$, we need $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1)$. $f(1) = 1+1=2$. $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x+1) = 1+1=2$. $\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (3-ax^2) = 3-a(1)^2 = 3-a$. For continuity, $2 = 3-a$, which gives $a=1$.

3. Consider the function $f(x) = \frac{x^2-9}{x-3}$ for $x \neq 3$. Is $f(x)$ continuous at $x=3$ if we define $f(3)=6$?

   1. Yes
   2. No
   3. Depends on the definition of f(3)
   4. Cannot be determined
   Show answer

   Answer: A. Yes

   For continuity at $x=3$, we need $\lim_{x \to 3} f(x) = f(3)$. $\lim_{x \to 3} \frac{x^2-9}{x-3} = \lim_{x \to 3} \frac{(x-3)(x+3)}{x-3} = \lim_{x \to 3} (x+3) = 3+3=6$. Since $f(3)$ is defined as 6, which is equal to the limit, the function is continuous at $x=3$.

4. For the function $f(x) = x^n$, when is it continuous?

   1. Always
   2. Only for positive integers n
   3. Only for negative integers n
   4. Never
   Show answer

   Answer: A. Always

   Polynomials are continuous everywhere. $f(x) = x^n$ is a polynomial for any real number $n$ (though the domain might be restricted for non-integer $n$). For the context of calculus up to Grade 12, $x^n$ is considered continuous over its domain.

5. Find the derivative of $f(x) = \sinh(x)$.

   1. $\cosh(x)$
   2. $\sinh(x)$
   3. $-\cosh(x)$
   4. $-\sinh(x)$
   Show answer

   Answer: A. $\cosh(x)$

   The derivative of the hyperbolic sine function, $\sinh(x)$, is the hyperbolic cosine function, $\cosh(x)$.

More questions in this quiz

Answers and explanations are in the full quiz.

1. Find the derivative of $f(x) = \tan^{-1}(x) + \cot^{-1}(x)$.
2. Evaluate $\int_0^1 x e^x dx$.
3. If $f(x) = |x-2|$, is $f(x)$ continuous at $x=2$?
4. Find the derivative of $f(x) = \sin^2(x)$.
5. Find the derivative of $f(x) = x^3 \ln(x)$.
6. Evaluate $\int_0^{\pi/2} \sin(x) dx$.
7. Let $f(x) = \begin{cases} x^2 & \text{if } x < 0 \ x & \text{if } x \ge 0 \end{cases}$. Is $f(x)$ continuous at $x=0$?
8. Find the derivative of $f(x) = e^{2x+1}$.
9. If $f(x) = \sqrt{x}$ for $x \ge 0$, is $f(x)$ differentiable at $x=0$?
10. Let $f(x) = \begin{cases} ax+b & \text{if } x \le 1 \ x^2 & \text{if } x > 1 \end{cases}$. If $f(x)$ is differentiable at $x=1$, find the relationship between $a$ and $b$.
11. For the function $f(x) = x^3$, is it differentiable at $x=0$?
12. If $f(x) = \begin{cases} \frac{\sin(x)}{x} & \text{if } x \neq 0 \ 1 & \text{if } x = 0 \end{cases}$, is $f(x)$ continuous at $x=0$?
13. Find the second derivative of $f(x) = x^4$.
14. If $f(x) = |x^2 - 4|$, is $f(x)$ differentiable at $x=2$?
15. If $f(x) = \begin{cases} x^2-1 & \text{if } x < 1 \ 2x-2 & \text{if } x \ge 1 \end{cases}$, is $f(x)$ continuous at $x=1$?
16. Is the function $f(x) = \frac{1}{x}$ continuous at $x=0$?
17. If $f(x) = |x-1| + |x+1|$, is $f(x)$ differentiable at $x=0$?
18. Find the value of $a$ for which the function $f(x) = \begin{cases} x^2 & \text{if } x < 2 \ ax+1 & \text{if } x \ge 2 \end{cases}$ is continuous at $x=2$.
19. If $f(x) = |x^2 - 1|$, is $f(x)$ differentiable at $x=1$?
20. If $f(x) = egin{cases} \frac{x^2-4}{x-2} & ext{if } x e 2 \ 4 & ext{if } x = 2 \end{cases}$, is $f(x)$ continuous at $x=2$?
21. Find the derivative of $f(x) = \ln(\sin(x))$.

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