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

Consider the function , which has derivative .

The linear approximation of for some value within a neighborhood of is given by



Let . Then can be estimated to be




Since for , it follows that must be strictly increasing over that part of its domain, which means the linear approximation lies strictly above the function . This means the estimated value is an overestimation.

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