f(x)=x−9/(7x+2).

Question

Find an equation for the tangent line to the graph of f at x=11. Tangent line: y =

Answer 1

The function is y = x - [9/(7x + 2)], and x =  11.

If x = 11 then y = 11 - [9/(7*11 + 2)]

= 11 - [9/(77 +  2)]

= 11 - [9/79]

= (869 - 9)/79

= 860/79

= 10.88.

The point is (11, 10.88).

The function y = x - [9/(7x + 2)].

Differentiate with respect to x.

y ' = 1 - [63/(7x + 2)^2]

At x = 11,

y ' = 1 - [63/(7*11 + 2)^2]

= 1 - [63/(77 + 2)^2]

= 1 - [63/(79)^2]

= 1 - [63/6241]

= (6241 - 63)/6241

= 6178/6241

= 0.9899.

⇒ y ' = 0.9899.

This is the slope (m ) of the tangent line to the circle at (11, 10.88).

Slope - intercept form line equation is y = mx + b, where m is slope and b is y - intercept.

Slope(m) = 0.9899.

Now the tangent line equation is y = 0.9899x  + b.

Find the y - intercept by substituting the the point in the tangent line equation say (x, y) = (11, 10.88).

10.88 = (0.9899)(11) + b

b = 10.88 - 10.88

⇒ b = 0.

The tangent line equation is y = 0.9899x.