Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x² + f(x) and h(x) = 3f(x).
Find an equation of the line tangent to y = h(x) at x = 3.

Verified step by step guidance

1. Step 1: Identify the function h(x) and its derivative. We have h(x) = 3f(x). To find the derivative h'(x), use the constant multiple rule: h'(x) = 3f'(x).
2. Step 2: Evaluate h(x) and h'(x) at x = 3. We know f(3) = 1 and f'(3) = 4. Therefore, h(3) = 3f(3) = 3 * 1 = 3 and h'(3) = 3f'(3) = 3 * 4 = 12.
3. Step 3: Use the point-slope form of the equation of a line to find the tangent line. The point-slope form is y - y_1 = m(x - x_1), where m is the slope and (x_1, y_1) is a point on the line.
4. Step 4: Substitute the values into the point-slope form. Here, the slope m is h'(3) = 12, and the point (x_1, y_1) is (3, h(3)) = (3, 3).
5. Step 5: Write the equation of the tangent line using the point-slope form: y - 3 = 12(x - 3).

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 tangent line to h(x) at x = 3, we need to calculate h'(3) to determine the slope.

Derivative

The derivative of a function measures how the function's output changes as its input changes. It is a fundamental concept in calculus that provides the slope of the tangent line at any point on the curve. For the function h(x) = 3f(x), we will use the chain rule to find h'(x) and evaluate it at x = 3.

Function Composition

Function composition involves combining two functions where the output of one function becomes the input of another. In this problem, g(x) and h(x) are defined in terms of f(x). Understanding how to differentiate composite functions is essential for finding the derivative of h(x) and subsequently the equation of the tangent line.