f(x)=x^2sin(1/x) for x is not equal to 0, f(0)=0. use the chain rule and product
ID: 3081005 • Letter: F
Question
f(x)=x^2sin(1/x) for x is not equal to 0, f(0)=0. use the chain rule and product rule to show that f is differentiabl at each c not equal to 0 and find f'(c).Explanation / Answer
First, for x not =0, we can use the usual differentiation formulas; We first need the product rule df/dx = x^2*d[sin(1/x)]/dx + sin(1/x)*d[x^2]/dx= x^2*cos(1/x) *( -1/x^2) + 2xsin(1/x) [we used the chain rule to differentiate sin(1/x)]. This simplifies to: df/dx = -cos(1/x) + 2xsin(1/x) So f'(c) = -cos(1/c) + 2csin(1/c) for c not zero. To get the derivative at x=0, we need to use the limit definition of the derivative f'(c) since the usual formulas are not valid at zero: f'(0) = lim x->0 of [f(x) - f(0)] / (x-0) = x^2sin(1/x) / x = xsin(1/x). As x goes to zero, sine stays bounded (between -1 and +1) but x is going to zero, so the entire limit goes to zero. Thus, f'(0) = 0. The derivative function itself is not continuous at x=0 because for x not 0, f'(x) = -cos(1/x) + 2xsin(1/x) . Taking the limit of this as x goes to zero, we see that the limit does not exist:. Why: as x goes to zero, 1/x goes to infinity so cos(1/x) and sin(1/x) just keep oscillating, so the limit doesn't exist. Thus, the function f'(x) can't be continuous at x=0 (by the definition of continuity you need to have lim x->a of the function equal to the value of the function at x=a). This example illustrates (among other things) that although a function must be continuous to be differentiable, the derivative function need not be continuous. Also note that since the derivative function is not continuous at x=0, there can't be a second derivative at x=0 (continuity is necessary- but not sufficient- for differentiability). The second derivative does exist at all other points, however. You should repeat this exercise for a similar case: f(x) = xsin(1/x) for x not 0, and f(0) = 0. (The result is not quite the same.)
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.