Linear and quadratic approximation

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Dominick York
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1 Linear an quaratic approximation November 11, 2013 Definition: Suppose f is a function that is ifferentiable on an interval I containing the point a. The linear approximation to f at a is the linear function L(x) = f(a) + f (a)(x a), for x in I. Now consier the graph of the function an pick a point P not he graph an look at the tangent line at that point. As you zoom in on the tangent line, notice that in a small neighbourhoo of the point, the graph is more an more like the tangent line. In other wors, the values of the function f in a neighbourhoo of the point are being approximate by the tangent line. Suppose f is ifferentiable on an interval I containing the point a. The change in the value of f between two points a an a+ x is approximately y = f (a). x, where a+ x is in I. Uses of Linear approximation To approximate f near x = a, use f(x) L(x) = f(a) + f (a)(x a). Alternately, f(x) f(a) f (a)(x a). To approximate the change y in the epenent variable given a change x in the inepenent variable, use y f (a) x. Differentials Let f be ifferentiable on an interval containing x. A small change in x is enote by the ifferential x. The corresponing change in y = f(x) is approximate by the ifferential y = f (x)x; that is y = f(x + x) f(x) y = f (x)x. 1

2 Quaratic approximation Recall that if a function f is ifferentiable at a point a, then it can be approximate near a by its tangent line. We say that the tangent line provies a linear approximation to f at the point a. Recall also that the tangent line at the point (a, f(a)) is given by y f(a) = f (a)(x a) a or y = f(a) + f (a)(x a). The linear approximation function p 1 (x) is the polynomial p 1 (x) = f(a) + f (a)(x a) of egree one; this is just the equation to the tangent line at that point. This polynomial has the property that it matches f in value an slope at the point a. In other wors, p 1 (a) = f(a) an p 1(a) = f (a). We woul like to o better, namely we woul also like to get a better approximation to the function in that we match concavity of f as well at this point. Recall that the concavity information is coe in the secon erivative. We thus create a quaratic approximating polynomial p 2 (x) of egree two efine by: p 2 (x) = f(a) + f (a)(x a) + c 2 (x a) 2 ; p 2 (x) = p 1 (x) + c 2 (x a) 2, where c 2 = 1/2f (a). So we have Note that p 2 (x) = f(a) + f (a)(x a) f (a)(x a) 2 = p 1 (x) f (a)(x a) 2. p 2 (a) = f(a), p 2(a) = f (a), p 2(a) = f (a), where we assume that f an its first an secon erivatives exist at a. The polynomial p 2 (x) is the quaratic approximating polynomial for f at the point a. The quaratic approximation gives a better approximation to the function near a than the linear approximation. In solving linear approximation problems, you shoul first look for the function f(x) as well as the point a, so that you can approximate f at a point close to a. The avantage of linear approximation is the following; the function f that one is consiering might be very complicate, but *near* a point a in its omain, we are able to estimate the value of the function using a polynomial of egree one (i.e. a linear function) which is far simpler than the original function. For example, let us consier the following PROBLEM: Approximate the number

3 We use the fact that a curve lies very close to its tangent (linear approximation) near the point of tangency. So let f(x) = 4 x. First we look for a point a close to 1.1, such that f(a) is easy to compute, but f(x) is ifficult to compute for x close to a. So we use the tangent line at (a, f(a)) an use it as an approximation to the curve y = f(x). Clearly, 1 is close to 1.1 an 4 1 is easy to compute! So we take a = 1. Then the tangent line is y = f(a) + f (a)(x a). As f(a) = 1 an f (a) = 1 4a 3/4. The equation to the tangent line is an Putting a = 1 an x = 1.1, we have Unerestimates an overestimates f(a) + f (a)(x a), f(x) f(a) + f (a)(x a) /4 (1.1 1) = (0.1) = If f(x) is concave up in a neighbourhoo of (a, f(a)), then the tangent line lies below the graph of f, an the approximate value is an unerestimate. In this case, the value obtaine by linear approximation is less than the actual value of f(x) in a neighbourhoo of a. If the tangent line lies above the graph of f, then the approximate value is an overestimate. We remake that linear approximation gives goo estimates when x is close to a but the accuracy of the approximation gets worse when the points are farther away from 1. Also, a calculator woul give an approximation for 4 1.1, but linear approximation gives an approximation over a small interval aroun 1.1. Percentage Error Suppose you have use linear or quaratic approximation centre aroun a to approximate f(c), for a point c close to a.the percentage error is calculate using the formula (T rue value) (approximate value) x100. (T rue V alue) Thus if you have use linear approximation using the linear polynomial, the approximate value at a point c close to a is given by p 1 (c) an the percentage error is given by f(c) p 1 (c) f(c) 100. If you have use quaratic approximation using the formula p 2 (x), we woul use p 2 (c) instea of p 1 in the above formula. 3

4 Taylor series Recall that p 1 (x) = f(a) + f (a)(x a) is the linear approximating polynomial for f(x) centre aroun a an p 2 (x) = f(a) + f (a)(x a) + f (a)/2(x a) 2 is the quaratic polynomial. The polynomial p 1 (x) is the equation of the tangent line at the centre point (a, f(a), an these are use to estimate values of f(x) for points x that are close to a. For example, you shoul check that for f(x) = Sin x, with centre a = 0, we have p 1 (x) = x. The quaratic approximation gives a better estimate for f(x) for x near a when compare with the linear polynomial f 1 (x). Suppose a function f has erivatives f (k) (a) of all orers at the point a. The Taylor polynomial of orer (or egree) n is efine by p n (x) = c 0 + c 1 (x a) + c 2 (x a) c n (x a) n where c k = f (k) (a) k!. Of course, p 1 (x) an p 2 (x) are as above. If we let n, we obtain a power series, calle the Taylor series for f centre at a. The special case of a Taylor series centre at 0 is calle a Maclaurin series. The higher orer Taylor polynomials give better an better approximations for f(x) in a neighbourhoo of the centre a. Errors Definition: The error in the linear approximation of a function f(x) is M where R(x) = f(x) p 1 (x) M. To fin M, we shoul fin an upper boun for the ifference between the actual value an the approximate value p 1 (x). Notice that the ifference between p 1 an f increases as we mover farther away from the centre a. Notice also that the ifference increases if the function bens away from the tangent. In other wors, the larger the absolute value of the secon erivative at a, f (a), the greater the eviation of f(x) from the tangent line. The linear approximation at a is more accurate for f, when the rate of change of f, which is nothing else but f is smaller. Remainer In general, write f(x) = p n (x) + R n (x), where n = 1 or 2 so that p 1 (x) is the linear approximating polynomial an p 2 (x) is the quaratic approximation polynomial. To estimate the remainer term R n (x), we fin a number M such that f (n+1 )(c) M, for all c between a an x inclusive, Then the remainer satisfies R n (x) = f(x) p n (x) M x a n+1, (n + 1)! 4

5 where p 1 (x) (respectively p 2 (x)) is the linear (respectively quaratic) approximating polynomial centre aroun a. A similar formula hols for the remainer term for the n-th orer Taylor polynomial. Derivatives of inverse trigonometric functions x (sin 1 x) = 1 1 x 2 for 1 < x < 1. x (cos 1 x) = 1 1 x 2 for 1 < x < 1. x (tan 1 x) = 1 1+x 2 for < x <. x (cot 1 x) = 1 1+x 2 for < x <. x (sec 1 1 x) = x for x > 1. x 2 +1 x (cosec 1 1 x) = x for x > 1. x 2 1 5

Math 115 Section 018 Course Note

Math 115 Section 018 Course Note Course Note 1 General Functions Definition 1.1. A function is a rule that takes certain numbers as inputs an assigns to each a efinite output number. The set of all input numbers is calle the omain of