Find an equation of the tangent line to the graph of the function f through the point (x0, y0) not on the graph. To find the point of tangency (x, y) on the graph of f, solve the following equation

Question

Find an equation of the tangent line to the graph of the function f through the point (x₀, y₀) not on the graph. To find the point of tangency (x, y) on the graph of f, solve the following equation of f '(x).

f '(x) =	y0 − y
x0 − x

f(x) = √x

(x₀, y₀) = (-16, 0)

y = 

 

Answer 1

As instructed determine the first derivative of f(x),

Use f'(-16) as the slope in two-point form (Algebra 1)

Answer 2

f(x) = x^1/2; f'(x) = 1/2x^-1/2. Let (a, b) is tangent point. Then b = a^1/2; f'(a) = 1/2 a^-1/2_.

Now equation of tangent line: y = f(a) + f'(a)(x - a); y = a^1/2 + 1/2a^-1/2(x - a). Point (- 16, 0) on this line.

So, 0 = a^1/2 + 1/2a^-1/2(-16 - a); 0 = 2a - 16 - a; a = 16;

So, equation of tangent line: y = 4 + 1/8(x - 16); y = 1/8x + 2