Math:Kimberly took her 6 nieces and nephews to a hockey game. She wants to buy them snacks. How much can she spend on snacks for each child if Kimberly wants to spend less than $33 in total

Evaluate the line integral, where C is the given curve
C xy ds
C: $x=t^2,y=2t,0\le t\le 5$

$(x^2+2xy-4y^2)dx-(x^2-8xy-4y^2)dy=0$

Find an equation of the following curve in polar coordinates and describe the curve.         ${\displaystyle x=(1+\cos t)\cos t,y=(1+\cos t)\sin t.0\le t\le 2\pi }$

Find all asymptotes of a function
Find all asymptotes of a function:
$f(x)=\log (x^2-4)$
Domain: $x\in (-\mathrm{\infty },-2)\cup (2,\mathrm{\infty })$
Vertical asymptotes are x=−2 (left) and x=2(right):
$\underset{x\to -2^{-}}{lim}\log (x^2-4)=-\mathrm{\infty }$
$\underset{x\to 2^{+}}{lim}\log (x^2-4)=-\mathrm{\infty }$
I calculate the limits in +/- infinity:
$\underset{x\to +\mathrm{\infty }}{lim}\log (x^2-4)=+\mathrm{\infty }$
$\underset{x\to -\mathrm{\infty }}{lim}\log (x^2-4)=+\mathrm{\infty }$
So I'm looking for the oblique asymptotes of a form y=Ax+B:
$A_{+}=\underset{x\to +\mathrm{\infty }}{lim}\frac{\log (x^2-4)}{x}=\underset{x\to +\mathrm{\infty }}{lim}\frac{\frac{2x}{x^2-4}}{1}=0$
$B_{+}=\underset{x\to +\mathrm{\infty }}{lim}\log (x^2-4)-A_{+}x=+\mathrm{\infty }$
The same for $-\mathrm{\infty }$. How should I interpret this? There are no oblique asymptotes?

Evaluate the improper integral using limits.
${\displaystyle {\int }_1^2\frac{dx}{\sqrt{3x-2}}}$

Concavity and critical numbers
I am attempting to study for a test, but I forgot how to do all the stuff from earlier in the chapter.
I am attempting to find the intervals where f is increasing and decreasing, min and max values and intervals of concavity and inflection points.
I have $f(x)=e^{2x}+e^{-x}$. I know that I have to find the derivative, set to zero to find the critical numbers then test points then find the second derivative and do the same to find the concavity but I don't know how to get the derivative of this. Wolfram Alpha gives me a completely different number than what I get.

Question on Local Maxima and Local Minima
Find the set of all the possible values of a for which the function $f(x)=5+(a-2)x+(a-1)x^2-x^3$ has a local minimum value at some $x<1$ and local maximum value at some $x>1$.
The first derivative of f(x) is:
$f^{\prime }(x)=(a-2)+2x(a-1)-3x^2$
I do know the first derivative test for local maxima and local minima, but I can't figure out how I could use monotonicity to find intervals of increase and decrease of f′(x)
The expression for f′(x) might suggest the double derivative test is the key, considering $f^{″}(x)=2(a-1)-6x$ for which the intervals where it is greater than zero and less than zero can be easily found, but then again I can't think of a way how I could find a c such that $f^{\prime }(c)=0$.