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.