91Ó°ÊÓ

A secant line is a line that intersects a curve at two distinct points, while a tangent line is a line that touches the curve at only one point, called the point of tangency. The slope of a secant line can be used to approximate the slope of the tangent line at the point of tangency.

The slope of a secant line can be calculated using the formula: $$m_{sec}=\frac{f(x_2)-f(x_1)}{x_2-x_1}$$. We will calculate the slopes for different increments, Δx, around the point x=2. For example, let's calculate the slopes for Δx = 1, 0.5, 0.1, -0.1, -0.5, and -1: 1. For Δx = 1, (x1, f(x1)) = (1, 2\cdot1^2) and (x2, f(x2)) = (3, 2\cdot3^2). 2. For Δx = 0.5, (x1, f(x1)) = (1.5, 2\cdot1.5^2) and (x2, f(x2)) = (2.5, 2\cdot2.5^2). 3. For Δx = 0.1, (x1, f(x1)) = (1.9, 2\cdot1.9^2) and (x2, f(x2)) = (2.1, 2\cdot2.1^2). 4. For Δx = -0.1, (x1, f(x1)) = (2.1, 2\cdot2.1^2) and (x2, f(x2)) = (1.9, 2\cdot1.9^2). 5. For Δx = -0.5, (x1, f(x1)) = (2.5, 2\cdot2.5^2) and (x2, f(x2)) = (1.5, 2\cdot1.5^2). 6. For Δx = -1, (x1, f(x1)) = (3, 2\cdot3^2) and (x2, f(x2)) = (1, 2\cdot1^2). Once these points are calculated, use the slope formula to find each slope, as shown in the table below: | Δx | m_sec | |-----|-------| | 1 | 6 | | 0.5 | 7 | | 0.1 | 8.2 | | -0.1| 7.8 | | -0.5| 5 | | -1 | 2 |

As Δx approaches 0, the value of the slope of the secant lines converges to a number around 8. Based on this evidence, we can conjecture that the slope of the tangent line at the point x=2 is 8.