Question

Find the tangent line approximation for sqrt7+x near x=2.

Answer 1

Let T(x) be the equation of the tangent line to the function f(x) at x = a.
Slope of the tangent at x = a is f'(a)
Therefore,
\[T(x) = f(a) + f'(a)(x - a) \\
a = 2 \\
T(x) = f(2) + f'(2)(x - 2) \\
f(x) = \sqrt{7+x} \\
f(2) = \sqrt{7+2} = \sqrt{9} = 3\\
f'(x) = \frac {1}{2\sqrt{7+x}} \\
f'(2) = \frac {1}{2\sqrt{7+2}} = \frac 16\\
T(x) = 3 + \frac 16(x-2)
\]

Answer 2

Then multiply through and add 3?