Linear approximation a. Write The equation of the line that represents the linear approximation to the following functions at the given point a. b. Graph the function and the linear approximation near x? = a ? . c. Use the linear approximation to estimate the given function value. d. Compute the percent error in your approximation:? 100 • lapprox-exact| / |exact| f?(?x?)=sin ?x?: ?a? = ?/4; ?f?(0.75)
Solution 8E SOLUTION STEP 1 Given f(x) = sin x;a = /4;f(0.75) f x) = cos x (a). Equation of the line of linear approximation y = L(x) = f(a) + f (a)(x a) Thus f(a) = sin /4 = 1 2 1 f a) = cos /4 = 2 Therefore y = L(x) = f(a) + f (a(x a) y = 1 + 1 (x 4 = 1 (x 4 + 1) 2 2 2 1 Thus we get y = 2(x 4 + 1) STEP 2 (b).Graph the function and the linear approximation near x = a. STEP 3 (c). Use the linear approximation to estimate the given function value. We have f(0.75) L(0.75) 1 2(0.75 4+ 1) 0.682 Therefore f(0.75) 0.682 STEP 4 (d). |approximate exact| Compute the percent error in your approximation: 100 × |exact| We get the exact value ,f(0.75)= sin(0.75) = 0.681 0.6820.681 Then percent error = 100 × | |0.681| | = 0.14%
