Question

For fun question

aproximate the value of ∛63.97 using differentials
leave in fraction form (if applicable)
show all work
also, show if this is an overestimate or an underestimate

Answer 1

Consider the function
`f(x)=x^(1/3)`, which has derivative
`f'(x)=\frac13x^(-2/3)`.

The linear approximation of
`f(x)` for some value
`x` within a neighborhood of
`x=c` is given by


`f(x)\approx f'(c)(x-c)+f(c)`

Let
`c=64`. Then
`(63.97)^(1/3)` can be estimated to be


`f(63.97)\approxf'(64)(63.97-64)+f(64)`

`\sqrt[3]{63.97}\approx4-(0.03)/(48)=3.999375`

Since
`f'(x)>0` for
`x>0`, it follows that
`f(x)` must be strictly increasing over that part of its domain, which means the linear approximation lies strictly above the function
`f(x)`. This means the estimated value is an overestimation.

Indeed, the actual value is closer to the number 3.999374902...