Question

1. Linear Approximation First, read Section 4.1 and the lecture notes of days 16 and 17. The steps for linear approximation of are as follows 1. Choose an objective function f whose value at r we want to estimate and choose a center value a closed to r 2. Compute the linearization L(x) - f(a)f'(a) (z - a) of the objective function 3. Compute L(x) to get the required approximation 4. Compute the second derivative and decide whether the linear approximation over- or under timates the actual value es 5. If the actual value is known, then also calculate the error and percent error 6. Finally, give an error bound for our approximation For each of the following numbers, please do the following: i. Use a suitable objective function and a center value a to estimate the given numbers ii. Without using the actual value, determine (with justification) whether your approxima- tion is over or under and give an upper bound for the error iii. Use the actual value provided and compute the error and percent error of your approx- imation. (a.) In(0.97) (actual value-0.0304592) (b.) sin(1.8°) (actual value 0.0314108) Note: for (b.), convert to radian first 1.8 0.031415 radians.

Answer 1

Linear Approxi ma tion Henas . can be pn (0.91) (a) Taking centrol value os wherre loc f (x): Lineaγ approximation is a. (x)st , differentia t'ng fir) u. t -12. (a ) z sinc, (a) co oveY-esh' therefo e approximaかon is on Upper esor bound in linerr approxi mat'on f (x)-- 4x)[L ta)-a).fta) L (K) E(x) = 叶(xa 它

EvOY Exatt value App roximate value 304542-o03) o.ooo.1 592. exad value -0,030너59 2 (b) sin () 1.3 sin (o 314108) Considesing cen bral value- o tx) sin() (0) cos t (a) Linear L (x) appro 지 me hron is (X-9 (a ) 千 (7-0). I 1.gx_-_-0.0314159 Llig") こ 30 1oo Also, -sinx f" (x) Co)o Sinu , fu (al τ o , fvnch on cannot be detamined to be over or under es himate

Upper bound for emmor こ sin E TroT Actul value Approximate velve = 0.0314108-0.0314159 ヒ.mo. | 乂100 Actial value -O.OO0OOS Taring absolute value. 0,01623 %,