Calculus: Find the Error?

http://www.dougshaw.com/findtheerror/





1) Find the error: Differentiation


2)Find the error: Fundamental Theorem of Calculus


3) Find the error: U-substitution


4)Find the error: Trigonometric Integration


5)Find the error: Integration by Parts (part 1)


6)Find the error: Integration by Parts (part 2)


7)Find the error: L'Hopital's Rule


8)Find the error: Related Rates


9)Find the error: Separation of Variables


10) Find the error: Separation of Variables - Exponential Growth


11)Find the error: Improper Integrals and Taylor Series





If anyone can explain all of these (or most of them), I will give best answer.

Calculus: Find the Error?
for the 1st


In the first part writing x^2=x+x+x+x+....x times is the error.


first of all x has to be an integer.


Again speaking of the eqn.


x^2=x+x+x+x+x+x... x times is not differentiable. and hence the fault.





For the 2nd


After the third line it has been written that F'(x)=f(x)


and hence on integrating it we get





F(x)={{{{{f(x)+c(which is missing)


here {{{{{denotes the integration sign.





The third goes like this





putting u=1/x^5 is not valid because we have to take care in the mind the function(u)and function to be integrated is differentiable throughout the region -1 to 1








For the Fourth


There is no constant of integration.so its wrong to compare both the equations.








For the fifth and the sixth


again uv is directly written and constant of integration has not been taken into account again








For the eighth


we cannot put x=6 directly first and then differentiate.


so the answer would be like


x^2 + y^2 = 100.


differentiating w.r.t. time


2x dx/dt + 2y dy/dt =0


or


dy/dt = (-x dx/dt)/y


now put x=6 in the eqn and we get y=8








i.e. dy/dt = -6/8 is the answer.








For the 10th


{{{{{{dy/y is taken as ln(y)


whereas its gotta be ln|y|


|x| represents the modulus and hence the rectification can be made.
Reply:Gosh! 10 whole points?????


I'll get right on it!!!
Reply:Now this looks like a fun page! I sure hope you're not a student of this guy and trying to cheat on an assignment. Here are the first few:





1) Using their own logic, splitting up x in to "1 + 1 + 1 +..." (x times) and taking the derivative implies d/dx = 0, when it should be 1. Also "1 plus 1 plus 1 and so on, x times" doesn't take into account values of x that aren't integers. It's not the same thing as the function f(x) = x, which is continuous. Also, if you use the formal definition of the derivative, it's hard to say what f(x + Δx) would be in the "1 + 1 + 1..." case.





2) If you're evaluating the integral like this, then you have to take F(x) - F(a) on the right-hand side. Or if you define F(x) as that integral in the first place, the deriviative rule follows, but not necessarily vice-versa. Go here to see a more formal explanation:


http://mathworld.wolfram.com/Fundamental...





3) If u = x^(-4), then


du = -4x^(-5) dx


dx = du / (-4x^5)


dx = (-x^(-5))/4 du


dx = (-x^(-4)x^(-1))/4 du


dx = (-u x^(-1))/4 du


But the problem puts a different value back in the equation.





4) This is what happens when you miss the "+ C" when you integrate without any bounds. If you evaluate each integeral the bounds of a to b and work some more trig identities, you'll find that they're the same.
Reply:1) These are not functions and thus not differentiable. It is true that if x = 10 , then 10^2 = 10*10. If we differentiate this identity on both sides we get 0=0. Otherwise, we would have to say x^2 = 10x which is also true but does not lead to the result 2=1.





2 The fundamental theorem of calculus states that the integral = F(b)-F(a) = sin x + 5 - sina - 5 = sinx -sina which is correct.


The definite integral was being likened to an indefinite integral so that the constant did not drop out.





The u substitution does not work because it makes the limits of integartion equal to each other and thus the value of the integral is 0. The u substitution used here simply does not work.