Consider the function and the value of a.

Question

f(x) = squareroot(x + 8) , a = 1

Use mtan = lim h→0 (f(a + h) − f(a))/h
to find the slope of the tangent line mtan = f '(a).

Then, find the equation of the tangent line to f at
x = a.(Let x be the independent variable and y be the dependent variable.)

Answer 1

f'(x) = lim_x→1 [(f(x) - f(1))/(x-1)]

 

        = lim_x→1[ (√(x+8) - 3)/(x-1)]

 

Multiply the numerator and denominator by √(x+8) + 3 to get:

 

lim_x→1 [(x-1)/{(x-1)(√(x+8) + 3)}]

 

= lim_x→1 [ 1/(√(x+8) +3)] = 1/6

 

Slope of tangent line = 1/6

 

Point on tangent line:  (1, f(1)) = (1, 3)

 

Equation of tangent line:  y - 3 = (1/6)(x - 1)

 

                                          y = (1/6)x + 17/6