Use linear approximation, i.e. the tangent line, to approximate cube root of 27.05 as follows. Let f(x) = cube root of x and find the equation of the tangent line to f(x) at x = 27 in the form y = mx+b.

Question

Note: The values of m and b are rational numbers which can be computed by hand. You need to enter expressions which give m and b exactly. You may not have a decimal point in the answers to either of these parts.

A. m = ?

B. b= ?

C.Using these values, find the approximation. Cube root of 27.05 = ?

Answer 1

this is just algebra I, after finding the slope of the tangent line...

since y = x^(1/3)
y' = 1/3 x^-(2/3)
y(27) = 3
y'(27) = 1/9

So, now you have a point and a slope, so the line is

y-3 = 1/9 (x-27)

I'll let you massage the equation.
Now you can use that line to find y(27.05)

Or, you can note that

dy = 1/9 dx
and dx = 0.05,
so you can add dy to y to approximate y(27.05)

Hint, using the line or using the differentials will give the same answer.