Question

Find the linear approximation of the function g(x)=(1+x)^(1/3) at a=0 and use it to approximate the numbers (0.95)^(1/3) and (1.1)^(1/3). Illustrate by graphing g and the tangent line.

Answer 1

this is the same problem as "find the line tangent to the graph of
\[g(x)=\sqrt[3]{x+1}\] at the point (0,1)

Answer 2

you have the point, and to get the slope take the derivative and replace x by 0

Answer 3

the derivative is
\[g'(x)=\frac{1}{3\sqrt[3]{(x+1)^2}}\] and
\[g'(0)=\frac{1}{3}\]

Answer 4

so your line is
\[y-1=\frac{1}{3}x

Answer 5

oops

Answer 6

so your line is \[y-1=\frac{1}{3}x\]
or
\[y=\frac{1}{3}x+1\]

Answer 7

if you want to approximate
\[\sqrt[3]{.95}\]replace x by
\[-.05\] this is some old school stuff now mainly of historical interest

Answer 8

hope it is clear where the
\[-.05\] came from
i solved
\[x+1=0.95\] for x

Answer 9

what about the approximation of (1.1)^(1/3) and the graph?