Tangent lines Suppose f(2)=2 and f′(2) =3. Let g(x) = x²f(x) and h(x) = f(x) / x−3.
b. Find an equation of the line tangent to y = h(x) at x=2.

Verified step by step guidance

1. Step 1: Identify the function h(x) = \frac{f(x)}{x - 3} and the point of tangency x = 2.
2. Step 2: Calculate h(2) using the given f(2) = 2. Substitute x = 2 into h(x) to find h(2).
3. Step 3: Use the quotient rule to find h'(x). The quotient rule states that if h(x) = \frac{u(x)}{v(x)}, then h'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}. Here, u(x) = f(x) and v(x) = x - 3.
4. Step 4: Calculate h'(2) using the values f(2) = 2, f'(2) = 3, and v(2) = 2 - 3. Substitute these into the derivative found in Step 3.
5. Step 5: Use the point-slope form of a line, y - y_1 = m(x - x_1), where m = h'(2) and the point (x_1, y_1) is (2, h(2)), to write the equation of the tangent line.