Find $$ f $$ if $$ f^{\prime}(x)=4 / \sqrt{1-x^{2}} $$ and $$ f\left(\frac{1}{2}\right)=10 $$. $$ f(x)=\square $$

Question

UpStudy AI · Accepted Answer

To find the function $$ f(x) $$ given that $$ f^{\prime}(x) = \frac{4}{\sqrt{1 - x^2}} $$ and $$ f\left(\frac{1}{2}\right) = 10 $$, we follow these steps:

1. **Integrate $$ f^{\prime}(x) $$ to find $$ f(x) $$:**

$$
   f(x) = \int \frac{4}{\sqrt{1 - x^2}} \, dx = 4 \sin^{-1}(x) + C
   $$
   
   Here, $$ C $$ is the constant of integration.

2. **Determine the constant $$ C $$ using the initial condition $$ f\left(\frac{1}{2}\right) = 10 $$:**

$$
   10 = 4 \sin^{-1}\left(\frac{1}{2}\right) + C
   $$
   
   We know that $$ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} $$ radians. Substituting this value in:

$$
   10 = 4 \cdot \frac{\pi}{6} + C \quad \Rightarrow \quad 10 = \frac{2\pi}{3} + C
   $$
   
   Solving for $$ C $$:

$$
   C = 10 - \frac{2\pi}{3}
   $$

3. **Write the final expression for $$ f(x) $$:**

$$
   f(x) = 4 \sin^{-1}(x) + 10 - \frac{2\pi}{3}
   $$

**Final Answer:**

$$
f(x) = 4\,\sin^{-1}(\,x\,) \;+\; 10\;-\;\tfrac{2}{3}\pi
$$
$$ f(x) = 4\,\sin^{-1}(x) + 10 - \frac{2\pi}{3} $$

Find if and .

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Calculus Questions