Calculus Answers

Let D c R[sup]2[/sup] be the unit disk in the Euclidean plane, and let S c R[sup]2[/sup] be the filled in square of unit side-length. Can you decompose D as the disjoint union of two connected sets, D= D1∪ D2, and S as the disjoint union of two connected sets, S=S1∪S2, such that D1 is similar to S1 and D2 is similar to S2?(here "similar to" means they differ only by scaling;in other words, by th linear transformation L(x)=aXfor some positive number a∈ R.) You can present your proposed solutuion in the form of drwan pictures

Determine the values of c∈R so as to make the following function continuous on R.

f(x)= {c(x^2) + 2x, x < 2
{x^3 - cx x ≥ 2

evaluate (a) lim_(x→0)(√(x^2+x^4 )*sin(π/x));
(b) lim_(x→0+)(√(x) (1+sin^2(π/x)));

sketch the graph of [[x]]

The floor function, ⌊x⌋, is defined as: ⌊x⌋ equals the largest integer less than or
equal to x.
Determine if
(a) limx →n ⌊x⌋ and
(b) limx→n+1/2 ⌊x⌋ exist,
when n ∈ Z. If the limit exists give it.

Using the definition of the derivative, determine f ′ (0) for
f (x) = cos x, x < 0
1 x ≥ 0.
Is f ′ continuous at 0?

Meaning of Calculus

G(u)=(1+u^2)^5(2-3u^2)8

f(x)=5(x^3-x)^4

Find the derivative f(t)=(5t^2-10t+4)^3/2