An equation of the line tangent to the graph of g at x = 3 is y = 5x + 4. Find g(3) and g′(3).

Verified step by step guidance

1. Identify that the equation of the tangent line is given as y = 5x + 4, which is in the form y = mx + b, where m is the slope of the tangent line.
2. Recognize that the slope of the tangent line, m, is equal to the derivative of the function g at the point x = 3, so g'(3) = 5.
3. Understand that the point of tangency (x, y) on the graph of g is also a point on the tangent line. Since the tangent line is y = 5x + 4, substitute x = 3 into this equation to find the y-coordinate.
4. Calculate the y-coordinate by substituting x = 3 into the equation y = 5x + 4, which gives y = 5(3) + 4.
5. Conclude that the point (3, y) is on the graph of g, so g(3) is equal to the y-coordinate found in the previous step.

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 represents the instantaneous rate of change of the function at that point, which is equivalent to the derivative of the function.

Derivative

The derivative of a function at a specific point quantifies how the function's output changes as its input changes. It is denoted as g′(x) and can be interpreted as the slope of the tangent line to the graph of the function at that point. In this case, g′(3) is the slope of the tangent line at x = 3.

Function Value

The function value g(3) represents the output of the function g when the input is 3. It is the y-coordinate of the point on the graph of g where x equals 3. This value can be determined by substituting x = 3 into the function g, which is related to the tangent line equation provided.