Let f:R to R be a continuously differentiable | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given the equation $\int_6^{f(x)} 4t^3 \,dt = (x - 2)g(x)$.We need to find $\lim_{x 	o 2} g(x) = \lim_{x 	o 2} \frac{\int_6^{f(x)} 4t^3 \,dt}{x

Vedclass · Accepted Answer

$18$

Vedclass · Answer

(B) Given the equation $\int_6^{f(x)} 4t^3 \,dt = (x - 2)g(x)$.We need to find $\lim_{x 	o 2} g(x) = \lim_{x 	o 2} \frac{\int_6^{f(x)} 4t^3 \,dt}{x - 2}$.Since $f(2) = 6$,the limit is of the form $\frac{0}{0}$. Applying $L$'$H$ôpital's rule:$\lim_{x 	o 2} g(x) = \lim_{x 	o 2} \frac{\frac{d}{dx} \int_6^{f(x)} 4t^3 \,dt}{\frac{d}{dx} (x - 2)}$.Using the Leibniz integral rule,the derivative of the numerator is $4(f(x))^3 \cdot f'(x)$.Thus,$\lim_{x 	o 2} g(x) = \lim_{x 	o 2} \frac{4(f(x))^3 \cdot f'(x)}{1} = 4(f(2))^3 \cdot f'(2)$.Substituting the given values $f(2) = 6$ and $f'(2) = \frac{1}{48}$:$\lim_{x 	o 2} g(x) = 4 \cdot (6)^3 \cdot \frac{1}{48} = 4 \cdot 216 \cdot \frac{1}{48} = \frac{864}{48} = 18$.

Let $f:R \to R$ be a continuously differentiable function such that $f(2) = 6$ and $f'(2) = \frac{1}{48}$. If $\int_6^{f(x)} 4t^3 \,dt = (x - 2)g(x)$,then $\lim_{x \to 2} g(x)$ is equal to

Similar Questions

If $f(x) = \begin{cases} ax+b, & \text{if } x \leq 1 \\ ax^2+c, & \text{if } 1 < x \leq 2 \\ \frac{dx^2+1}{x}, & \text{if } x > 2 \end{cases}$ is differentiable on $\mathbb{R}$,then $ad-bc = $

At the point $x=1$,the function $f(x) = \begin{cases} x^3-1, & 1 < x < \infty \\ x-1, & -\infty < x \leq 1 \end{cases}$ is

If $f(x) = \begin{cases} \frac{8}{x^3} - 6x, & x \le 1 \\ \sqrt{x} + 1, & x > 1 \end{cases}$,then at $x = 1$,$f$ is:

Let $a_1, a_2, \ldots, a_{100}$ be non-zero real numbers such that $a_1+a_2+\ldots+a_{100}=0$. Then,

Let $f(x) = \begin{cases} x^{3/5} & \text{if } x \le 1 \\ -(x - 2)^3 & \text{if } x > 1 \end{cases}$. Then the number of critical points on the graph of the function is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module