Problem 30 - Calculus Early Transcendentals

Question

Find all points on the interval (1,3) at which the slope of the tangent line equals the average rate of change of \(f\) on \([1,3] .\) Reconcile your results with the Mean Value Theorem.

Step-by-Step Solution

Answer

Question: Find the points on the interval (1, 3) at which the slope of the tangent line equals the average rate of change of the function \(f\) on \([1,3]\). Answer: To find these points, first calculate the average rate of change of the function using the formula \(\dfrac{f(3)-f(1)}{3-1}\). Next, solve the equation \(f'(x) = \dfrac{f(3)-f(1)}{2}\) for \(x\) in the interval (1, 3). The Mean Value Theorem guarantees that there is at least one point on the interval (1, 3) where the slope of the tangent line equals the average rate of change of the function. To find the exact points, compare the results of solving the equation from Step 2 with the result from the Mean Value Theorem.

Step 1: Find the average rate of change of the function on the interval \([1,3]\)

To find the average rate of change, we will use the formula: $$\text{Average rate of change} = \dfrac{f(3)-f(1)}{3-1}$$

Step 2: Determine the conditions where the slope of the tangent line equals the average rate of change

We know that the slope of the tangent line at a point is equal to the derivative of the function at that point. In other words, to find all points on the interval \((1,3)\) where the slope of the tangent line equals the average rate of change, we need to solve the equation: $$f'(x) = \dfrac{f(3)-f(1)}{2}, \text{ for } x \in (1,3)$$

Step 3: Apply the Mean Value Theorem

According to the Mean Value Theorem, if the function \(f\) is continuous on the closed interval \([1,3]\) and differentiable on the open interval \((1,3)\), there exists at least one point \(c\) in the interval \((1,3)\) such that $$f'(c) = \dfrac{f(3)-f(1)}{2}$$

Step 4: Reconcile the results with the Mean Value Theorem

The Mean Value Theorem ensures us that there is at least one point on the interval \((1,3)\) where the slope of the tangent line equals the average rate of change of the function on \([1, 3]\). Solving the equation from Step 2 will give us the exact points with this property. Once we have those points, we can compare them with the result obtained from the Mean Value Theorem to check for consistency.