List of top Mathematics Questions on Fundamental Theorem of Calculus

Let $ x_0 $ be the real number such that $ e^{x_0} + x_0 = 0 $. For a given real number $ \alpha $, define $$ g(x) = \frac{3xe^x + 3x - \alpha e^x - \alpha x}{3(e^x + 1)} $$ for all real numbers $ x $. Then which one of the following statements is TRUE?

Let $\mathbb{R}$ denote the set of all real numbers. Let $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to (0,4)$ be functions defined by $$ f(x) = \log_e (x^2 + 2x + 4), \quad \text{and} \quad g(x) = \frac{4}{1 + e^{-2x}}. $$ Define the composite function $f \circ g^{-1}$ by $(f \circ g^{-1})(x) = f(g^{-1}(x))$, where $g^{-1}$ is the inverse of the function $g$. Then the value of the derivative of the composite function $f \circ g^{-1}$ at $x=2$ is _____.

Let $ \mathbb{R} $ denote the set of all real numbers. Let $ f: \mathbb{R} \to \mathbb{R} $ be defined by $$ f(x) = \begin{cases} \dfrac{6x + \sin x}{2x + \sin x}, & \text{if } x \ne 0 \\ \dfrac{7}{3}, & \text{if } x = 0 \end{cases} $$ Then which of the following statements is (are) TRUE?

Let $ \alpha $ and $ \beta $ be the real numbers such that $$ \lim_{x \to 0} \frac{1}{x^3} \left( \frac{\alpha}{2} \int_{0}^{x} \frac{1}{1 - t^2} \, dt + \beta x \cos x \right) = 2. $$ Then the value of $ \alpha + \beta $ is __________.

Find the value of $ \frac{d}{dx} (3x^2 + 4x + 5) $.

If log$_{10}$(x + 1) = 2, what is the value of x?

Find the derivative of $ y = \sin(x^2) $ with respect to $ x $.

Evaluate: $$ \lim_{x \to 0} \frac{\sqrt{1 + x} - \sqrt{1 - x}}{x} $$

If $\log_2 (x-1) + \log_2 (x-3) = 3$, find the value(s) of $ x $.

Let $\mathbb{R}$ denote the set of all real numbers. Let $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to (0,4)$ be functions defined by $$ f(x) = \log_e (x^2 + 2x + 4), \quad \text{and} \quad g(x) = \frac{4}{1 + e^{-2x}}. $$ Define the composite function $f \circ g^{-1}$ by $(f \circ g^{-1})(x) = f(g^{-1}(x))$, where $g^{-1}$ is the inverse of the function $g$. Then the value of the derivative of the composite function $f \circ g^{-1}$ at $x=2$ is _____.

Let $ \alpha $ and $ \beta $ be the real numbers such that $$ \lim_{x \to 0} \frac{1}{x^3} \left( \frac{\alpha}{2} \int_{0}^{x} \frac{1}{1 - t^2} \, dt + \beta x \cos x \right) = 2. $$ Then the value of $ \alpha + \beta $ is __________.

Let $ f : \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \rightarrow \mathbb{R} $ be a differentiable function such that $ f(0) = \frac{1}{2} $. If the $ \lim_{x \to 0} \frac{\int_{0}^{x} f(t) \, dt}{e^{x^2} - 1} = \alpha $, then $ 8\alpha^2 $ is equal to:

If \[ x = \frac{9t^2}{1 + t^4}, \quad y = \frac{16t^2}{1 - t^4} \] then \[ \frac{dy}{dx} = \]

If the interval in which the real-valued function \[ f(x) = \log\left(\frac{1+x}{1-x}\right) - 2x - \frac{x^{3}}{1-x^{2}} \] is decreasing in $ (a,b) $, where $ |b-a| $ is maximum, then {a}⁄{b} =

For a given function $ y = f(x) $, $ \delta y $ denotes the actual error in $ y $ corresponding to actual error $ \delta x $ in $ x $ and $ dy $ denotes the approximate value of $ \delta y $. If $ y = f(x) = 2x^2 - 3x + 4 $ and $ \delta x = 0.02 $, then the value of $ \delta y - dy $ when $ x = 5 $ is: \vspace{0.5cm}

The general solution of the differential equation $(3x^2 - 2xy)dy + (y^2 - 2xy)dx = 0$ is

If the equation of the curve which passes through the point $ (1,1) $ satisfies the differential equation: \[ \frac{dy}{dx} = \frac{2x - 5y + 3}{5x + 2y - 3}, \] then the equation of the curve is: \vspace{0.5cm}

Find $ \frac{dy}{dx} $ for the given function:

\[ y = \tan^{-1} \left( \frac{\sin^3(2x) - 3x^2 \sin(2x)}{3x \sin(2x) - x^3} \right). \]

If $ f(x) $ is given as: \[ f(x) = \begin{cases} \frac{8}{x^3} - 6x, & \text{if } 0 < x \leq 1 \\ \frac{x - 1}{\sqrt{x}-1}, & \text{if } x > 1 \end{cases} \] is a real-valued function, then at $ x = 1 $, $ f $ is: