Leonidas Cook

2022-08-18

Determine whether f'(0) exists. $f\left(x\right)={x}^{2}\frac{\mathrm{sin}1}{x}$ if x is not equal to 0, 0 if x=0

betoosolis7i

Beginner2022-08-19Added 12 answers

Recall that:

$f}^{\prime}\left(a\right)=\underset{x\Rightarrow a}{lim}\frac{f\left(x\right)-f\left(a\right)}{x-a$

$f}^{\prime}\left(0\right)=\underset{x\Rightarrow 0}{lim}\frac{{x}^{2}\mathrm{sin}\left(\frac{1}{x}\right)-0}{x-0$

${f}^{\prime}\left(0\right)=\underset{x\Rightarrow 0}{lim}x\mathrm{sin}\left(\frac{1}{x}\right)$

Squeeze theorem

If

$h\left(x\right)\le f\left(x\right)\le g\left(x\right)$

And

$\underset{x\Rightarrow a}{lim}h\left(x\right)=L$

And

$\underset{x\Rightarrow a}{lim}g\left(x\right)=L$

Then

$\underset{x\Rightarrow a}{lim}f\left(x\right)=L$

Since$-1\le \mathrm{sin}\left(\frac{1}{x}\right)\le 1$

Therefore$-x\le x\mathrm{sin}\left(\frac{1}{x}\right)\le x$

Since

$\underset{x\Rightarrow 0}{lim}-x=0$

And

$\underset{x\Rightarrow 0}{lim}x=0$

By Squeeze Theorem, We have

$\underset{x\Rightarrow 0}{lim}x\mathrm{sin}\left(\frac{1}{x}\right)=0$

Result:

f'(0)=0

Squeeze theorem

If

And

And

Then

Since

Therefore

Since

And

By Squeeze Theorem, We have

Result:

f'(0)=0

What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders?

How do I find the y-intercept of a parabola?

What are the vertices of $9{x}^{2}+16{y}^{2}=144$?

How to determine the rate of change of a function?

Why are the tangents for 90 and 270 degrees undefined?

How to find the center and radius of the circle ${x}^{2}+{y}^{2}-6x+8y=0$?

What is multiplicative inverse of a number?

How to find the continuity of a function on a closed interval?

How do I find the tangent line of a function?

How to find vertical asymptotes using limits?

How to find the center and radius of the circle ${x}^{2}-12x+{y}^{2}+4y+15=0$?

Let f be a function so that (below). Which must be true?

I. f is continuous at x=2

II. f is differentiable at x=2

III. The derivative of f is continuous at x=2

(A) I (B) II (C) I and II (D) I and III (E) II and IIIHow to find the center and radius of the circle given ${x}^{2}+{y}^{2}+8x-6y=0$?

How to find the center and radius of the circle ${x}^{2}+{y}^{2}+4x-8y+4=0$?

How to identify the center and radius of the circle ${(x+3)}^{2}+{(y-8)}^{2}=16$?