Find the slope m of the tangent line to the graph of the function at the given point and determine an equation of the tangent line.
f(x) = 15/(4 x) at (1,(15/4)
m =
y =
To find the slope (m) of the tangent line to the graph of the function, we need to take the derivative of the function and evaluate it at the given point.
The function is: f(x) = 15/(4x).
First, let's take the derivative of f(x) with respect to x:
f'(x) = d/dx (15/(4x))
= -15/(4x^2)
Now, let's substitute the x-coordinate of the given point (1) into the derivative:
f'(1) = -15/(4(1)^2)
= -15/4
So, the slope (m) of the tangent line is -15/4.
To determine an equation of the tangent line, we can use the point-slope form of a line, which is:
y - y1 = m(x - x1)
Substituting the given point (1, 15/4) and the slope (-15/4) into the equation, we have:
y - 15/4 = -15/4(x - 1)
Next, we can simplify and rewrite the equation in slope-intercept form (y = mx + b):
y - 15/4 = -15/4x + 15/4
y = -15/4x + 15/4 + 15/4
y = -15/4x + 30/4
y = -15/4x + 15/2
So, the equation of the tangent line to the graph of the function at the given point is y = -15/4x + 15/2.