Notes 4A (Approximation)

Notes 4A (Approximation) - MIT OpenCourseWare...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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A. APPROXIMATIONS In science and engineering, you often approaimate complicated functions by simpler ones which are easier to calculate with, and which show the relations between the variables more clearly. Of course, the approximation must be close enough to give you reasonable accuracy. For this reason, approximation is a skill, one your other teachers will expect you to have. This is a good place to start acquiring it. Throughout, we will use the symbol a to mean "approximately equal to"; this is a bit vague, but making approximations in engineering is more art than science. 1. The linear approximation; linearizations. The simplest way to approximate a function f(z) for values of z near a jr is to use a linear funcion. The linear function we shall use is the one whose graph is the tangent line to f(z) at x = a. This makes sense because the tangent line at (a, f(a)) gives a good approximation to the graph of f(z), _. i l. T . . a s at a to cose s a (1) height of the graph of f(z) s height of the tangent at (a, f(a))- To turn (1) into calculus, we need the equation for the tangent line. Since the line goes through (a, f(a)) and has slope f'(a), its equation is S f(a) + f'(a)(x - ), and therefore (1) can be expressed as (2) f(z) f() + f'(a)( - a), for Tua. This says that for x near a, the function f(z) can be approximated by the linear function on the right of (2). This function - the one whose graph is the tangent line - is called the linerization of f(z) at z = a. The appraoimation (2) is often written in an equivalent form that you should become familiar with; it makes use of a dependent variable. Writing (3) v = f(z), A = z- a, Ay = f() - f(a), the approximation (2) takes the form (2') AL f'(a) Az, for Az . f(a)Ax In this form, the quantity on the right represents the change in height of the taent line, while the left measures the change in height of the eraph. Here are some examples of linear approximations. In all of them, we are taking a = 0, this being the most important case All can be found by using (2)above and calculating the derivative. You should verify each of them, and memorize the approximation.
A. APPROXIMATIONS Basic Linear Approximations 1 (4) r-a 1 + z, for ax 0 ; (5) (1 + x)' s 1 + rz, for z s 0 ; r is any real number (6) .. . sinz z, for zx 0. Note that (4) becomes a special case of (5) if we take r = -1 and replace x by -z; nonetheless, learn (4) separately since it is very common. As an example of verification, let us check (5): f(x) = (1 + x)' f'() = r(1 + x) - , for any real r ; S f'(O) =r. Therefore, (2) becomes

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Notes 4A (Approximation) - MIT OpenCourseWare...

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