Question

(a) If F(x) = 13x/(4 + x2), find F '(3) and use it to find an equation of the tangent line to the curve y = 13x/(4 + x2) at the point (3, 3).

Answer 1

`\[ F'(x) = (-12x^2 + 52)/((4 + x^2)^2) \]`

To find F'(3), substitute x = 3 into the derivative:

`\[ F'(3) = (40)/(49) \]`

The equation of the tangent line at the point (3, 3) is:

`\[ y = (40)/(49)(x - 3) + 3 \]`

Step-by-step explanation:

To find the derivative F'(x), apply the quotient rule. The numerator is obtained by differentiating the product of 13x, which results in 13, and the denominator is the square of
`\( 4 + x^2 \)`. After finding F'(x), plug in x = 3 to get F'(3), which is
`\((40)/(49)\)`.

Now, the equation of the tangent line can be determined using the point-slope form
`\( y - y_1 = m(x - x_1) \)`, where
`\( (x_1, y_1) \)` is the given point. Substituting
`\( m = (40)/(49) \)`,
`\( x_1 = 3 \)`, and
`\( y_1 = 3 \)`, we obtain the equation of the tangent line.