What is the equation of the tangent line of #f(x) =1-x^2/(x-5)# at # x = 4#?

Question

Ezekiel Alexander · Accepted Answer

#24x-y-73=0#. See the tangent-inclusive Socratic graph.

graph{[0, 4.4, 5.808, 14.8, 15.2]} = -x-6-25/(x-5)-y)(24x-y-81)=0 By precise division, At x = 4, y = -x-6-25/(x-5) = 15. Thus, P(4, 15) is the tangent's point of contact. At x = 4, the tangent's slope, m, is fi. f#'=-1+25/(x-5)^2=24 =m. Consequently, the tangent equation at P is #y-15=24(x-4)#, giving #24x-y-81=0#

Ezra Adams · Answer

undefined To find the equation of the tangent line at x = 4, we need to find the slope of the tangent line and a point on the line.

First, let's find the slope. We can use the derivative of the function f(x) to find the slope at any given point.

The derivative of f(x) = 1 - x^2/(x - 5) can be found using the quotient rule. After simplifying, we get:

f'(x) = (2x^2 - 10x - 5)/(x - 5)^2

Now, let's find the slope at x = 4 by substituting x = 4 into the derivative:

f'(4) = (2(4)^2 - 10(4) - 5)/(4 - 5)^2
       = (32 - 40 - 5)/(-1)^2
       = -13

So, the slope of the tangent line at x = 4 is -13.

Next, let's find a point on the line. We can substitute x = 4 into the original function f(x):

f(4) = 1 - (4^2)/(4 - 5)
     = 1 - 16/(-1)
     = 1 + 16
     = 17

Therefore, a point on the tangent line is (4, 17).

Finally, we can use the point-slope form of a line to find the equation of the tangent line:

y - y1 = m(x - x1)

Substituting the values we found, we get:

y - 17 = -13(x - 4)

Simplifying, we have:

y - 17 = -13x + 52

Rearranging the equation, we get the equation of the tangent line:

y = -13x + 69