Taylor’s theorem shows that any function can be approximated
Taylor’s theorem shows that any function can be approximated in the vicinity of any convenient point by a series of terms involving the function and its derivatives. Here we look at some applications of the theorem for functions of one and two variables.
a. Any continuous and differentiable function of a single variable, f (x), can be approximated near the point a by the formula
Using only the first three of these terms results in a quadratic Taylor approximation. Use this approximation together with the definition of concavity given in Equation 2.85 to show that any concave function must lie on or below the tangent to the function at point a.
b. The quadratic Taylor approximation for any function of two variables, f (x, y), near the point
(a, b) is given by
f(x, y) = f (a, b) + f1(a, b)(x –a) +f2(a, b)(y –b)
+0.5[f11(a,b)(x –a)2+2f12(a, b)(x –a)(y –b) +f22(y –b)2].
Use this approximation to show that any concave function (as defined by Equation 2.98) mustlie on or below its tangent plane at (a, b).
Â
Answers
Status NEW
Posted 18 May 2017 08:05 AM
My Price 7.00
-----------
Attachments
file 1495098720-Answer.docx preview (215 words )
T-----------ayl-----------orâ-----------€™s----------- th-----------eor-----------em -----------sho-----------ws -----------tha-----------t a-----------ny -----------fun-----------cti-----------on -----------can----------- be----------- ap-----------pro-----------xim-----------ate-----------d i-----------n t-----------he -----------vic-----------ini-----------ty -----------of -----------any----------- co-----------nve-----------nie-----------nt -----------poi-----------nt -----------by -----------a s-----------eri-----------es -----------of -----------ter-----------ms -----------inv-----------olv-----------ing----------- th-----------e f-----------unc-----------tio-----------n a-----------nd -----------its----------- de-----------riv-----------ati-----------ves-----------. H-----------ere----------- we----------- lo-----------ok -----------at -----------som-----------e a-----------ppl-----------ica-----------tio-----------ns -----------of -----------the----------- th-----------eor-----------em -----------for----------- fu-----------nct-----------ion-----------s o-----------f o-----------ne -----------and----------- tw-----------o v-----------ari-----------abl-----------es.----------- a-----------. A-----------ny -----------con-----------tin-----------uou-----------s a-----------nd -----------dif-----------fer-----------ent-----------iab-----------le -----------fun-----------cti-----------on -----------of -----------a s-----------ing-----------le -----------var-----------iab-----------le,----------- f -----------(x)-----------, c-----------an -----------be -----------app-----------rox-----------ima-----------ted----------- ne-----------ar -----------the----------- po-----------int----------- a -----------by -----------the----------- fo-----------rmu-----------la ----------- -----------Usi-----------ng -----------onl-----------y t-----------he -----------fir-----------st -----------thr-----------ee -----------of -----------the-----------se -----------ter-----------ms
Not Rated(0)