Mathematics

Mathematics

Problem

Let f(x)=ln⁡xf(x)=\ln x. Two tangent lines to ff are considered: Line L1L_1: the tangent at x=1x=1. Line L2L_2: the tangent at x=1.2x=1.2. The goal is to estimate ln⁡(1.1)\ln(1.1). (a) Write an explicit equation for L1L_1 and use it to approximate ln⁡(1.1)\ln(1.1). (b) Write an explicit equation for L2L_2 and use it to approximate ln⁡(1.1)\ln(1.1). (c) The exact value of ln⁡(1.1)\ln(1.1) (to six decimal places) is 0.0953100.095310. Determine which of the two tangent–line estimates obtained in parts (a) and (b) is closer to this exact value. Support your answer with numerical evidence.
Intuitive explanation

The key insight is that a tangent line gives a linear approximation to a curve near the point of tangency. Since 1.11.1 lies between 11 and 1.21.2, you can use both tangent lines to bracket the true value of ln⁡(1.1)\ln(1.1)—one will underestimate and one will overestimate. By evaluating both L1L_1 and L2L_2 at x=1.1x=1.1, you'll get bounds on ln⁡(1.1)\ln(1.1) and can average them for a better estimate.

Intuitive explanation

The key insight is that a tangent line gives a linear approximation to a curve near the point of tangency. Since 1.11.1 lies between 11 and 1.21.2, you can use both tangent lines to bracket the true value of ln⁡(1.1)\ln(1.1)—one will underestimate and one will overestimate. By evaluating both L1L_1 and L2L_2 at x=1.1x=1.1, you'll get bounds on ln⁡(1.1)\ln(1.1) and can average them for a better estimate.

Browse the problem library