Comparing Quadratic Approximations to Calculator Computations In a previous worked example, we explored linear approximations to the sine function at the point x = 0. In this example, we use the quadratic approximation for ex to calculate values of the exponential function near x = 0 and again compare the results to decimal approximations on a scientiﬁc calculator. Find the quadratic approximation to ex at the point x = 0 and use your answer to approximate the values of e.01 , e.1 and e. Check your answer on a calculator. Solution: In the lecture, you learned that quadratic approximation to a function f (x) at a point x = a was given by a particular quadratic polynomial. This polynomial Q(x) should be chosen so that Q(a) = f (a), Q" (a) = f " (a) and Q"" (a) = f "" (a). In the case where f (x) = ex , we saw from lecture that the quadratic approximation at x = 0 was given by: Q(x) = f (0) + f " (0)(x − 0) + f "" (0) e0 (x − 0)2 = e0 + e0 (x − 0) + (x − 0)2 2 2 1 2 = 1+x+ x . 2 In short, we write ex ≈ 1 + x + 12 x2 when x ≈ 0. (It’s very illuminating to again draw a picture of the exponential curve and its quadratic approximation at x = 0 to illustrate this.) So we would approximate the values of sine above as follows: e.01 ≈ 1 + .01 + (.01)2 /2 = 1.01005 e.1 ≈ 1 + .1 + (.1)2 /2 = 1.105 e1 ≈ 1 + 1 + 12 /2 = 2.5 As in the previous worked example, we expect the approximations at values closest to x = 0 (where the quadratic approximation agrees with the function) will be the most accurate. The calculator conﬁrms this. e.01 = 1.0100501670... e.1 = 1.1051709180... e = 2.7182818284... where we’ve only recorded the ﬁrst ten digits of the decimal expansion. Notice that e.01 only diﬀers from our estimate 1.01005 by less than .00000017, an extremely accurate approximation! Yet e diﬀers from 2.5 by more than .21. Again, the approximation will be poor for large values of x (i.e. far from x = 0) since a quadratic function grows much more slowly than an exponential function. In general, these approximations should only be relied upon for values near the point x = a at which we perform the approximation. Much later in the course, we’ll have a quantitative estimate for the error often referred to as “Taylor’s theorem with remainder.”

Comparing Quadratic Approximations to Calculator Computations

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