Master Calculus 1 Practice Problems with Expert Help

Use the definition of the derivative to find $f^1(x)$ for the following function:
$f(x)=1-<!--\; -\; -->x^2$

Limit. $\underset{x\to <!--\; \to \; -->2}{lim}\frac{3x^2-<!--\; -\; -->5x-<!--\; -\; -->2}{x-<!--\; -\; -->2}$

I just wonder can we call $2^x+x$ an exponential function or not?
With the same way of thinking is ${\log }_2<!--\; \; -->(x)+x$ a logarithmic?

Compute $\underset{n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}(3n{)}^{1/3}{x}_n$

Help me with this
$\int <!--\; \int \; -->\frac{{\ln }^3<!--\; \; -->x}{x}dx$
Is this the same as
$\int <!--\; \int \; -->\frac{(\ln <!--\; \; -->x{)}^3}{x}dx?$

a)Evaluate the integral ${\int <!--\; \int \; -->}_0^{\mathrm{\infty <!--\; \infty \; -->}}{x}^n{e}^{-<!--\; -\; -->x}dx\text{ for }n=0,1,2,3$
b)Guess the value of ${\int <!--\; \int \; -->}_0^{\mathrm{\infty <!--\; \infty \; -->}}{x}^n{e}^{-<!--\; -\; -->x}dx$ when n is an arbitrary nonnegative integer.
c)Prove your guess using mathematical induction.

Proving that the sequence $a_1=4$, $a_{n+1}=\sqrt{a_n+20}$ converges and finding its limit

Let $a\mathrm{ϵ<!--\; ϵ\; -->}{\mathbb{R}}_{+}$ and $b=\exp <!--\; \; -->(-<!--\; -\; -->a).$
What is the range of $\{-<!--\; -\; -->b^2\log <!--\; \; -->(b)\}$? Does it range $(-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->},+\mathrm{\infty <!--\; \infty \; -->})$?
How can I show $\frac{-<!--\; -\; -->b\log <!--\; \; -->(b)}{1-<!--\; -\; -->b}\le <!--\; \le \; -->1$ ?

Find the derivative
$y=(\tan <!--\; \; -->(x){)}^{6/x}$

Find the antiderivative of $\sqrt{3x-<!--\; -\; -->1}dx$.
I got $\frac{2}{3}(3x-<!--\; -\; -->1{)}^{3/2}+c$ but my book is saying $\frac{2}{9}(3x-<!--\; -\; -->1{)}^{3/2}+c$.
Can some one please tell me where the 2/9 comes from?

What is the approximate result of converting 20 centimeters into inches? remember that 1inch = 2.54centimeters

$y=(3x^2+2{)}^{lnx}$
The answer to this is:
$\frac{dy}{dx}=(3x^2+2{)}^{lnx}(\frac{1}{x}ln(3x^2+2)+\frac{6xlnx}{3x^2+2})$
What I'm coming up with is:
$\frac{dy}{dx}=(3x^2+2{)}^{lnx}(\frac{1}{x}ln(3x^2+2)+\frac{6x}{3x^2+2})$
What I'm not understanding is where the $\frac{6xlnx}{3x^2+2}$ comes from, if anyone could explain this I'd really appreciate it.

A point on a graph is (1/8,−3) of the logarithmic function $f(x)=\log <!--\; \; -->b^x$, and the point (4,k) is on the graph of the inverse, $y=f^{-<!--\; -\; -->1}(x)$. Determine the value k.

a)Express the given quantity as a single logarithm.
$\frac{1}{3}\ln <!--\; \; -->(x+2{)}^3+\frac{1}{2}[\ln <!--\; \; -->(x)-<!--\; -\; -->\ln <!--\; \; -->(x^2+3x+2{)}^2]$
b) Solve each equation for x.
1)$e^{5-<!--\; -\; -->4x}=4$
x=?
2) $\ln <!--\; \; -->(3x-<!--\; -\; -->13)=8$
x=?

How would one compute $\underset{\mathrm{\delta <!--\; \delta \; -->}\to <!--\; \to \; -->0,k\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}(1+\mathrm{\delta <!--\; \delta \; -->}{)}^{ak}$, where a is some positive constant?

How to prove the function f has an antiderivative?
There is a question in our analysis book and I have lots of problem with it. It says that:
set $f(x)=\{\begin{array}{ll}0& x\le <!--\; \le \; -->0\\ \sin <!--\; \; -->(\frac{\mathrm{\pi <!--\; \pi \; -->}}{x})& x>0\end{array}$
$g(x)=\{\begin{array}{ll}0& x\le <!--\; \le \; -->0\\ 1& x>0.\end{array}$
prove that f has an antiderivative but g does not.
My first problem is about antiderivative of g. I think it has an antiderivative and it is
$\begin{array}{rl}G(x)=& \{\begin{array}{ll}0& x\le <!--\; \le \; -->0\\ x& x>0.\end{array}\end{array}$
Why we can't say that G is antiderivative of g? My second problem is about finding the antiderivative of f. As you may know antiderivative of $\sin <!--\; \; -->(\frac{\mathrm{\pi <!--\; \pi \; -->}}{x})$ can not be shown by the elementary functions. So, to prove that f has antiderivative I can't find a function like F which its derivative is f and I need to use another approach to prove it, but I don't have any idea about what should I do?

I know that fraction denominator needs to be > 0 , so if denominator is quadractic equation i know how to solve , But since denominator is > 0 , I dont know how to solve , can anyone help me ?
$f\left(x\right)=\sqrt{\log <!--\; \; -->\left(\frac{3x-<!--\; -\; -->x^2}{2}\right)}$

Use logarithmic differentiation to find the derivative of the function.
$y=(\tan <!--\; \; -->(x){)}^{\frac{9}{x}}$