Sums and Approximations 1.5 In(1+a) The curve goes through the origin, so a trivial approximation is In(1+a)0 when c is small. In general f(a)a f(o) when r is small Sometimes this crude approximation is good enough, but not often. A line tangent to the curve at =0 provides a better approximation. The slope of tangent is given by the derivativ evaluated at r =0, which is 1/(1+0)=1. This gives the approximation In(1+r)Na, which is a significant improvement 1.5 0.5 In general ≈ This approximation is accurate enough in many situations. This is great, because linear functions are very simple and easy to manipulate This linear approximation has both the correct value at a =0 and the correct first derivative. We can go one more step and get an approximation with the correct second f(x)≈f(0)+xf(0)+。f"(0) when x is small10 Sums and Approximations 6 1.5 y = ln(1 + x) 1 0.5 0 - 0 1 2 3 The curve goes through the origin, so a trivial approximation is ln(1 + x) ≈ 0 when x is small. In general: f(x) ≈ f(0) when x is small Sometimes this crude approximation is good enough, but not often. A line tangent to the curve at x = 0 provides a better approximation. The slope of the tangent is given by the derivative d 1 ln(1 + x) = dx 1 + x evaluated at x = 0, which is 1/(1 + 0) = 1. This gives the approximation ln(1 + x) ≈ x, which is a significant improvement: 6 1.5 y = x y = ln(1 + x) 1 0.5 0 - 0 1 2 3 In general: f(x) ≈ f(0) + xf � (0) when x is small This approximation is accurate enough in many situations. This is great, because linear functions are very simple and easy to manipulate. This linear approximation has both the correct value at x = 0 and the correct first derivative. We can go one more step and get an approximation with the correct second derivative: 2 x f(x) ≈ f(0) + xf � (0) + f��(0) when x is small 2