Math PhD Students Hunt for Errors in False Proofs. Try it Yourself!

We test Michigan PhD students on "false proofs" to see if they can spot the errors. Try them yourself and then watch them take a crack at each!
*Small hints were allowed.
*Annaliese wanted to note that she is not ~technically~ in the PhD program; she is doing her Masters.

[@BehnamEsmayli]

Q1)For positive real numbers, square root is ALWAYS a positive quantity. So, root of 4 is +2 and never -2. The confusion comes from x^2=4 ==> x=+/-2 and the incorrect/lazy way some instructors do it by writing x^2=4 ==> x=
oot{4} ==> x= +/-2 which should instead be x=+/-
oot{4}.
Q2) OMG! Why isn't everyone shouting out the simple refutation that "what is $x$ many times if $x$ is not an integer?!" Because something is written down on a page does not mean it is valid in to begin with. Since we intend to take d/dx we must consider x being non-integer.