Question

Show all the steps and explain. Don't skip steps and please clear hand written

f(x)=x^m sin(1/x^n) if x is not equal 0 and f(x)=0 if x =0

(a) prove that when m>1+n, then the derivative of f is continuous at 0

limit x to 0 x^n sin(1/x^n) does not exist? but why??? please explain it should be 0*sin(1/x^n)

Answer #1

If m>1 then x^m sin(1/x^n) will be differentiable at 0
However, why?
Q1. if we take limit x to 0 sin(1/x^n), we get sin(1/0) it
doesn't make sense even if m>1
Q2. why when M=1, it will not differentiable at 0?
Q3 Please tell me that limit x to infinity sin(1/x) DNE becasue
limit will be -1 and 1???? is that the reason?

Solve the 2nd Order Differential Equation using METHOD OF
REDUCTION
Please don't skip steps!
(x-1)y"-xy'+y=0 x>1 y1(x)=x

Solve the 2nd Order
Differential Equation using METHOD OF REDUCTION
Please don't skip
steps!
(x-1)y"-xy'+y=0 x>1
y1(x)=x
My professor is
getting y2(x)=e^x and I don't understand how!

(b) Define f : R → R by f(x) := x 2 sin 1 x for x 6= 0, and f(x)
= 0 for x = 0. Does f 0 (0) exist? Prove your claim.

Show all the step and don't be lazy. You will get thumbs down.
Otherwise...Thumb down
Please clear writing
and explain step by step this should be a simple
question
Consider the function
f : [0,1]→R deﬁned by (f(x) =0 if x = 0) and (f(x)=1 if 0
< x≤1)
(i)Compute L(f)
andU(f).
(ii) Is f Riemann
integrable on [0,1]?

Please show all steps and explain every line of proof.
show that if f:[a,b] -> R is differentiable on a closed
interval [a,b] and if f' is continuous on [a,b], then f is lipshitz
on [a.b]

If you use handwritten, please provide clear hand
written
Compact and analysis conception
1. Are all closed interval compact? For example
[0,1]. are they closed and bounded?
2. If i can find the Maximum and Minimum, does that mean the set
is closed and bounded?

If you use handwritten, please provide clear hand
written
Compact and analysis conception
1. Are all closed interval compact?
for example [0,1]. are they closed and bounded?
2. If i can find the Maximum and Minimum, does that mean the set
is closed and bounded?

Prove that a tree with n node has exactly n-1 edges. Please show
all steps. If it connects to Induction that works too.

f(x,y) = (x^4-y^2) / (x^4 + y^2). show the following limit does
not exist, and explain why lim (x,y)->(0,0) f(x,y)

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