5–7. For each function ƒ and interval [a, b], a graph of ƒ is given along with the secant line that passes though the graph of ƒ at x = a and x = b.




a. Use the graph to make a conjecture about the value(s) of c satisfying the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) .


b. Verify your answer to part (a) by solving the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) for c.




ƒ(x) = x² / 4 + 1 ; [ -2, 4] <IMAGE>

Verified step by step guidance

1. Step 1: Understand the Mean Value Theorem (MVT), which states that for a function ƒ that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one number c in (a, b) such that (ƒ(b) - ƒ(a)) / (b-a) = ƒ'(c).
2. Step 2: Identify the function ƒ(x) = x² / 4 + 1 and the interval [a, b] = [-2, 4]. Calculate ƒ(a) and ƒ(b) by substituting x = -2 and x = 4 into the function.
3. Step 3: Calculate the slope of the secant line using the formula (ƒ(b) - ƒ(a)) / (b-a). This represents the average rate of change of the function over the interval [a, b].
4. Step 4: Find the derivative of the function, ƒ'(x). For ƒ(x) = x² / 4 + 1, use the power rule to differentiate, resulting in ƒ'(x) = x / 2.
5. Step 5: Set the derivative equal to the slope of the secant line: x / 2 = (ƒ(b) - ƒ(a)) / (b-a). Solve this equation for x to find the value(s) of c that satisfy the Mean Value Theorem.

Key Concepts

Secant Line

A secant line is a straight line that intersects a curve at two or more points. In calculus, it is often used to approximate the slope of the curve between those points. The slope of the secant line between points (a, f(a)) and (b, f(b)) is given by the formula (f(b) - f(a)) / (b - a), which represents the average rate of change of the function over the interval [a, b].

Mean Value Theorem

The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) such that the derivative at that point equals the average rate of change over the interval. This theorem is fundamental in connecting the concepts of secant lines and instantaneous rates of change, as it provides a formal justification for finding a point where the slope of the tangent line (f'(c)) matches the slope of the secant line.

Derivative

The derivative of a function at a point measures the instantaneous rate of change of the function with respect to its variable at that point. It is defined as the limit of the average rate of change as the interval approaches zero. In the context of the question, f'(c) represents the slope of the tangent line to the curve at the point c, which is crucial for verifying the relationship established by the Mean Value Theorem.