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 = g(x) at x = 3.

Verified step by step guidance

1. Step 1: Identify the function g(x) and its components. We have g(x) = x^2 + f(x).
2. Step 2: To find the equation of the tangent line, we need the slope at x = 3. This requires finding g'(x), the derivative of g(x).
3. Step 3: Differentiate g(x) = x^2 + f(x) to get g'(x) = 2x + f'(x).
4. Step 4: Substitute x = 3 into g'(x) to find the slope of the tangent line: g'(3) = 2(3) + f'(3).
5. Step 5: Use the point-slope form of a line, y - y_1 = m(x - x_1), where m is the slope from Step 4, and (x_1, y_1) is the point (3, g(3)). Calculate g(3) using g(3) = 3^2 + f(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. For a function g(x), the equation of the tangent line at x = a can be expressed as y = g(a) + g'(a)(x - a).

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. In this context, f'(3) = 4 indicates that at x = 3, the function f is increasing at a rate of 4 units of output for every 1 unit of input.

Function Composition

Function composition involves combining two functions to create a new function. In this case, g(x) = x² + f(x) combines the polynomial function x² with the function f(x). Understanding how to differentiate composite functions is essential for finding the derivative g'(x), which is necessary for determining the slope of the tangent line.