An equation​​ of the line tangent to the graph of f at the point (2,7) is y = 4x−1. Find f(2) and f′(2).

Verified step by step guidance

1. Step 1: Understand that the equation of the tangent line at a point (a, f(a)) on the graph of a function f is given by y = f'(a)(x - a) + f(a).
2. Step 2: Recognize that the given tangent line equation is y = 4x - 1, which can be compared to the general form y = mx + b, where m is the slope of the tangent line.
3. Step 3: Identify that the slope of the tangent line, m, is 4. This means that f'(2) = 4, as the slope of the tangent line at x = 2 is the derivative of the function at that point.
4. Step 4: Use the point (2, 7) given in the problem, which lies on both the function f and the tangent line, to find f(2). Since the point is on the tangent line, substitute x = 2 into the tangent line equation: y = 4(2) - 1.
5. Step 5: Conclude that f(2) = 7, as the y-coordinate of the point (2, 7) is the value of the function at x = 2.

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.

Function Value

The function value f(2) refers to the output of the function f when the input is 2. In the context of the tangent line equation provided, we can confirm that f(2) equals 7, as the point (2,7) lies on the graph of the function.

Derivative

The derivative of a function, denoted as f′(x), represents the slope of the tangent line to the graph of the function at any point x. In this case, since the equation of the tangent line is given as y = 4x - 1, the derivative at x = 2 is simply the slope of this line, which is 4.