T Tutorial • 81 Applications of Differentiation 3.8 Tangent Line Approximation and Differentials The line tangent to a curve at a point is the line that best approximates the curve near that point. So, we use the tangent line at the point (c,f(c)) as an approximation to the curve y = f{x) when x is near c. An equation for the tangent line at the point (c,f(c)) is given by y-m = f(cXx-c) or y = m+fXcXx-c) and the approximation f(x)*f{c)+f'{c)(x-c) is called the tangent line approximation of / at c. y=X*). y=Ac)+f(c)(x-c) The linear function K*) = /(c)+/'(c)(*-c) is called the linearization of / at c. Definition of Differentials Let y = f(x) be a differentiable function. The differential dx is an independent variable. The differential dy is dy = f(x)dx. Estimating with Differentials Differentials can be used to approximate function values. f(x+Ax) « f(x)+dy * f(x)+f'{x)dx Example 1 o (a) Find the tangent line approximation of f(x) = -Jx-\ at c = 5 and approximate the number >/3.95. (b) Find the tangent line approximation of f{x) = tan* at c = tt/4 and approximate the number tan 47". 82 • Tutorial Chapter 3 Solution d (a)/(x) = >/x^T r> /(5) = >/5-f = 2 /'(*) = 2,/^T /'(5) = ^ = 2 +41(^-5) 1 3 y=—x+4 4 / — T«-x+— ! 3 VJf-1 4 4_ >^95=V4.95-1 1 2>/5^T 4 Tangent line approximation Simplify. f(x)*f(c)+f(cXx-c) x = 4.95 *-(4.95)+-=1.9875 4 4 (b) f(x) = tan x => /(f) = tan(f) = 1 4 4 ,fl\ „2,*v /'(*) = sec2* => /'(7) = sec2(^-) =2 4 4 ^ = l + 2(x-^-) 4 Tangent line approximation '"te-f+1 Simplify. tan jc « 2* +1 2 f ( x ) ~ fi c ) + f ( c X x - c ) tan 47"* 2(47°-— )180 2 x = 47° and 47° = 47 radian 180 * 1.0698
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