 Find the equation of the tangent line at that point. function f(x)=1/ 3+x^2. slope =-1/24 x=3

To find the equation of the tangent line to the function \( f(x) = \frac{1}{3 + x^2} \) at the point where \( x = 3 \) and the slope of the tangent line is given as \(-\frac{1}{24}\), we can proceed with the following steps:

### Step 1: Find the value of \( y \) at \( x = 3 \)
We need to evaluate the function \( f(x) \) at \( x = 3 \):

\[
f(3) = \frac{1}{3 + 3^2} = \frac{1}{3 + 9} = \frac{1}{12}
\]

So, the point of tangency is \( (3, \frac{1}{12}) \).

### Step 2: Find the equation of the tangent line
The general equation of a line is given by:

\[
y - y_1 = m(x - x_1)
\]

where \((x_1, y_1)\) is the point of tangency, and \( m \) is the slope of the tangent line.

Here, \( (x_1, y_1) = (3, \frac{1}{12}) \) and \( m = -\frac{1}{24} \).

Substituting the values:

\[
y - \frac{1}{12} = -\frac{1}{24}(x - 3)
\]

### Step 3: Simplify the equation
To simplify the equation, we distribute the slope on the right side:

\[
y - \frac{1}{12} = -\frac{1}{24}x + \frac{3}{24}
\]

\[
y - \frac{1}{12} = -\frac{1}{24}x + \frac{1}{8}
\]

Adding \(\frac{1}{12}\) to both sides:

\[
y = -\frac{1}{24}x + \frac{1}{8} + \frac{1}{12}
\]

Finding a common denominator for the fractions:

\[
\frac{1}{8} = \frac{3}{24}, \quad \frac{1}{12} = \frac{2}{24}
\]

So:

\[
y = -\frac{1}{24}x + \frac{3}{24} + \frac{2}{24}
\]

\[
y = -\frac{1}{24}x + \frac{5}{24}
\]

### Final Equation of the Tangent Line:
\[
\boxed{y = -\frac{1}{24}x + \frac{5}{24}}
\]

Would you like further details on any step or have any questions? Here are some related topics:

1. How to find the derivative of a function.
2. Finding the slope of a tangent line at a specific point.
3. Understanding tangent line equations in calculus.
4. Graphing a function and its tangent line.
5. Applications of tangent lines in optimization problems.

**Tip:** The tangent line represents the instantaneous rate of change of the function at a specific point and is useful for approximations and linearization.

Find the equation of the tangent line at that point. function f(x)=1/ 3+x^2. slope =-1/24 x=3

Learn how to find the equation of the tangent line to the function f(x) = 1 / (3 + x^2) at x = 3 with a slope of -1/24. This problem covers calculus concepts including tangent lines, derivatives, and slope calculations. Suitable for high school calculus students (Grades 11-12).

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