Question

Consider the function f(x) = x ln(3x+1)

(a) Find the derivative

(b) Write the linearization of f at x = 2

(c) Use your linearization to estimate f(2.5)

(d) Draw a sketch of the function in the space below, using a solid line for f(x). On the same coordinate plane, draw a sketch of the linearization using a dotted line. Please use values 0<x<5(or equal to) on the x-axis

(e) Is your estimate from part c an overestimate or underestimate?

Answer 1

Hello Student! sot flow) = x ln (Burs) (a) Derivative = & fix) = of {xeln Butll) (chor form) an = In Bn+1). W + a. 1 .s 3nt = In (31+1) + 31 327] (6) f(a) = 2 ln (3 (2) + 1) = 2 ln 7 f (2) = ln {3 (2) + 1] + 3(2) 3(2)+1 = ln t + Linearization : Lla) = f(2) + f (2) (21-2). Lla) = 2 ln 7 & 1 lnt + 6 (c) 162,5) = 29 + ln ? $)(2.5-2) - 2 bn 7 & le 7 - 2 le 7 & 0.5lond & ma

= 2.5 los 7 + 3 / = 5.2934 (d) At no : fla) = f(o) = o ln (W = 0 At nEO : L(a) = LO) = 2ln 7 & flat to ) (-2) = 1.2143

fln) will have asymptote where food isn't defined or an anal) isn't defined. dn (30+1) = ln (o) asrlz (Not to be de considered as groph goes from 0 to as per question) Select few points and check value of fon) and L(1) f(n) [(x) dan 2 3 .89 3.892 6.91 6.695 10,26 9.498 bannt

single degree Graph of linearization will be linear as of an is involved At n=2 ; [(2) = 2 ln 7 = 3,892 Don't slightly get confused by the curved and Lles) firring, will be fisi) will be a straight line.

fly into 6.91 6 6957 3.892 -1,7143 5) = 2.5 ln ? (Bx2.5) + 17 = 2.5 ln (8.5) = 5.3502 And I (2.5) = 5, 2934 ? 2(2-5) < f (2.8): It is an Kindly put a Thumbs up) underestimate