If $y = e^{\sin^{-1} x}$ and $z = e^{-\cos^{-1} x}$, prove that $\frac{dy}{dz} = e^{\frac{\pi}{2}}$
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Continuity, Differentiability and Differentiation - ISC Class 12 Mathematics Questions with Answers, Page 4
72 past-paper questions on Continuity, Differentiability and Differentiation from ISC Class 12 Mathematics papers (2027-2017), newest first, in full. Questions 61-72 are on this page, 20 to a page. Tap "Show answer" under a question to see its answer.
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Verify Rolle’s Theorem for the following function:
$f(x) = e^x \sin x, x \in [0, \pi]$
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Verify Rolle’s theorem for the following function: $f(x) = e^{-x} \sin x$ on $[0, \pi]$.
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Find the value of constant $k$ so that the function $f(x)$ defined as:
$f(x) = \begin{cases} \frac{x^2 - 2x - 3}{x + 1}, & x \ne -1 \\ k, & x = -1 \end{cases}$
is continuous at $x = -1$.
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Find $\frac{dy}{dx}$, if $x = at^2$ and $y = 2at$.
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Prove that the function $f(x) = |x - 1|, x \in \mathbb{R}$, is continuous at $x = 1$ but not differentiable.
Given: $f(x) = |x - 1|, x \in \mathbb{R}$
To show: $f(x)$ is continuous at $x = 1$ but not differentiable
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If $x = \tan\left(\frac{1}{a} \log y\right)$, prove that $(1 + x^2) \frac{d^2y}{dx^2} + (2x - a) \frac{dy}{dx} = 0$.
Given: $x = \tan\left(\frac{1}{a} \log y\right)$
To show: $(1 + x^2) \frac{d^2y}{dx^2} + (2x - a) \frac{dy}{dx} = 0$
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Show that the function $f(x) = \begin{cases} x^2, & x \le 1 \\ \frac{1}{x}, & x > 1 \end{cases}$ is continuous at $x = 1$ but not differentiable.
Given: $f(x) = \begin{cases} x^2, & x \le 1 \\ \frac{1}{x}, & x > 1 \end{cases}$
To show: $f(x)$ is continuous at $x = 1$ but not differentiable
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If $y = e^{a \cos^{-1} x}$, where $-1 \leq x \leq 1$, then show that:
$(1 - x^2) y_2 - xy_1 - a^2y = 0$
Given: $y = e^{a \cos^{-1} x}$, where $-1 \leq x \leq 1$
To show: $(1 - x^2) y_2 - xy_1 - a^2y = 0$
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Verify Lagrange’s mean value theorem for the function:
$f(x) = x(1 - \log x)$ and find the value of ‘c’ in the interval $[1, 2]$.
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Using L’Hospital’s Rule, evaluate:
$\lim_{x \to \pi/2} \left(x \tan x - \frac{\pi}{2} \sec x\right)$
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If $y = \cos(\sin x)$, show that:
$\frac{d^2y}{dx^2} + \tan x \frac{dy}{dx} + y \cos^2 x = 0$
Given: $y = \cos(\sin x)$
To show: $\frac{d^2y}{dx^2} + \tan x \frac{dy}{dx} + y \cos^2 x = 0$
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