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.

Key Concepts

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line is given by the derivative of the function at that point. In this case, to find the equation of the tangent line to y = h(x) at x = 2, we need to evaluate h(2) and h'(2).

Derivative

The derivative of a function measures how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. For the function h(x), we need to apply the quotient rule to find h'(x), which will help us determine the slope of the tangent line at x = 2.

Quotient Rule

The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If h(x) = f(x) / g(x), the derivative h'(x) is given by (g(x)f'(x) - f(x)g'(x)) / (g(x))². In this problem, we will apply the quotient rule to differentiate h(x) = f(x) / (x - 3) to find the necessary slope for the tangent line.