Question

Use the linear approximation (1+x)^kapprox 1+kx to find an approximation for the function f(x) for values of x near zero.

a.) f(x)=(1-x)^(8) approx

b.) f(x)=-8/(1-x) = approx

c.) f(x)= 1/(sqrt(1+x)) = approx

d.) f(x)=sqrt(4+x^2) approx =

e.) f(x)=(6+3x)^(1/3) approx =

How do I go about attacking a problem like this?

Answer 1

Use the linear approximation (1+x)^k approx = 1+kx to find an approximation for the function f(x) for values of x near zero.

Answer 2

for the first one it should be straightfoward.... just use (-x) in palce of x for the approximation given and your k value is 8

Answer 3

Ohh hahah I think I'm making this more difficult than it is

Answer 4

okay. for B note that 1/y=y^(-1)

Answer 5

What would b =? I know all the derivative rules and everything I just don't really get this for some reason hahah.

Answer 6

b=-8*(1-x)^-(1)

Answer 7

So, -8*(1+x)! Awesome I think I get it. Thank ya

Answer 8

For D, I know that (4+x^2)^(1/2)

so (4+(1/2)x^2). But that's wrong. Can you catch my error? I'm sorry I know this shouldn't be this difficult!

Answer 9

take 4 out

Answer 10

\(\sqrt{4+x^2}=2\sqrt{1+(x/2)^2}\)