Find the slope m and an equation of the tangent line to the graph of the function f at the specified point.
F(X)=x^2/x+6;(3,1)
m = ??
y = ??
To find the slope (m) of the tangent line to the graph of the function f at the point (3, 1), we need to find the derivative of the function at that point.
Step 1: Differentiate the function f(x) with respect to x.
f(x) = x^2 / (x + 6)
We can find the derivative using the quotient rule:
f'(x) = ([(x + 6) * (2x)] - [x^2 * 1]) / (x + 6)^2
= (2x^2 + 12x - x^2) / (x + 6)^2
= (x^2 + 12x) / (x + 6)^2
Step 2: Substitute the x-coordinate of the given point (3, 1) into the derivative function to find the slope (m).
m = f'(3) = (3^2 + 12 * 3) / (3 + 6)^2
= (9 + 36) / (9)^2
= 45 / 81
= 5 / 9
So, the slope of the tangent line at the point (3, 1) is m = 5/9.
Step 3: Find the equation of the tangent line using the point-slope form.
Using the point-slope form (y - y1) = m(x - x1) with the given point (3, 1):
y - 1 = (5/9)(x - 3)
Simplifying, we get:
y - 1 = (5/9)x - 5/3
Rearranging the equation, we have:
y = (5/9)x - 5/3 + 1
y = (5/9)x - 5/3 + 3/3
y = (5/9)x - 2/3
So, the equation of the tangent line to the function f at the point (3, 1) is y = (5/9)x - 2/3.