Help to Integral Calculus Solutions & Examples

Constructing a function that is continuous and has a max on an open interval, but is not necessarily increasing immediately to the left of the max.
Suppose F(x) is continuous on some open interval I and c is a maximum point inside this interval. Is it true that f(x) must be increasing immediately to the left of c and decreasing immediately to the right of c? Proof or counterexample. (Note: A constant function is considered to be both increasing and decreasing.)

Can anyone confirm if my answer is right?
Find the critical points and the intervals of the increase and decrease of the function $f(x)=(x+5{)}^2(x-<!--\; -\; -->2{)}^5$
The critical points are: $-<!--\; -\; -->3,-<!--\; -\; -->5,2$
The four intervals -
from left to right these open intervals are $(-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->},-<!--\; -\; -->5),(-<!--\; -\; -->5,-<!--\; -\; -->3),(-<!--\; -\; -->3,2)\text{and}(2,\mathrm{\infty <!--\; \infty \; -->})$
This part I am not sure of:
On the interval $(-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->},-<!--\; -\; -->5)$, f(x) is decreasing
On the interval $(-<!--\; -\; -->5,-<!--\; -\; -->3)$, f(x) is decreasing
On the interval $(-<!--\; -\; -->3,2)$, f(x) is increasing
On the interval $(2,\mathrm{\infty <!--\; \infty \; -->})$, f(x) is increasing

Answer true or false to the following statement and explain your answer
Any scatterplot that exhibits curvature can be straightened by the method of transformations.

Derivatives and graphs
$f(x)=\sin <!--\; \; -->(x)+\cos <!--\; \; -->(x)$ for $0\le x\le 2\mathrm{\pi <!--\; \pi \; -->}$.
I have to do the following
1) Find the intervals on which f is increasing or decreasing
2) Find the local maximum and minimum values of f
3) Find the intervals of concavity and the inflections points
I got until this point when trying to solve problem 1:
$\tan <!--\; \; -->(x)=1$. The next step in the manual says that $x=\mathrm{\pi <!--\; \pi \; -->}/4$ or $5\mathrm{\pi <!--\; \pi \; -->}/4$. How did they get that $x=\mathrm{\pi <!--\; \pi \; -->}/4$ or $5\mathrm{\pi <!--\; \pi \; -->}/4$?

Answer true or false to each of the following statement and explain your answer.
It is not always possible to fnd a power transformation of the response variable or the predictor variable (or both) that will straighten the scatterplot.

Find asymptotes of $(2x)/(x-<!--\; -\; -->1{)}^2$
What are the asymptotes of
$\frac{2x}{(x-<!--\; -\; -->1{)}^2}$
? I have problems already on domain.

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places.
${\displaystyle x=\sqrt{y}-y,1\le y\le 4}$

Find the distance between the points with polar coordinates ${\displaystyle (2,\frac{\pi }{3})\text{and}(4,2\frac{\pi }{3})}$.

Set up a definite integral that yields the area of the region. (Do not evaluate the integral.)
${\displaystyle f\left(x\right)=25-x^2}$

Evaluate the definite integral from 1 to 2 ${\displaystyle \frac{e^1}{x}/x^{2}dx}$

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with r < 0 and the other with r > 0.
${\displaystyle (3,\frac{\pi }{2})}$

I want to understand the explicit and the implicit Euler method better. Assume I have the initial value problem $y^{″}+y=0$, $y(0)=0$, $y^{\prime }(0)=1$, which is of course solved by $y=\sin <!--\; \; -->(x)$, and I convert this into a system of first-degree linear equations
$\left(\begin{array}{c}{y}^{\prime }\\ z^{\prime }\end{array}\right)=\left(\begin{array}{cc}0& 1\\ -<!--\; -\; -->1& 0\end{array}\right)\left(\begin{array}{c}y\\ z\end{array}\right)\text{.}$
When I apply the explicit and the implicit Euler methods to the differential equation $y^{″}+y=0$, with stepwidth $h>0$, I get approximation points that satisfy:
$\begin{array}{rcll}{y}_k& =& \frac{ih}{2}((1-<!--\; -\; -->ih{)}^k-<!--\; -\; -->(1+ih{)}^k)& \text{for the explicit Euler method}\\ {\stackrel{~<!--\; ~\; -->}{y}}_k& =& \frac{ih}{2(1+h^2{)}^k}((1-<!--\; -\; -->ih{)}^k-<!--\; -\; -->(1+ih{)}^k)& \text{for the implicit Euler method}\end{array}$
I am interested in seeing how fast these sequences grow with k. My guess is that the first sequence is unbounded, whereas the second converges to zero. At least that is what I get for , but I have no idea how to prove this. Any suggestions?

Rudin's PMA: Theorem 3.29 Proof
If $p>1$, $\underset{n=2}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}\frac{1}{n(logn{)}^p}$ converges; if $p\le <!--\; \le \; -->1$, the series diverges.
Proof: The monotonicity of the logarithmic function implies that $\{logn\}$ increases. Hence $\{1/nlogn\}$ decreases, and we can apply Theorem 3.27 to the series above; this leads us to the series $\underset{k=1}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}{2}^k\cdot <!--\; \cdot \; -->\frac{1}{2^k(log{2}^k{)}^p}=\underset{k=1}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}\frac{1}{(klog2{)}^p}=\frac{1}{(log2{)}^p}\underset{k=1}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}\frac{1}{k^p}$ and Theorem 3.29 follows from Theorem 3.28.
I have two questions:
(1) How can we get the decrease of $\{1/nlogn\}$ from the increase of $\{logn\}$? I didn't see any conncection between them.
(2) The author said that we can apply Theorem 3.27 to the series. However, in order to apply Theorem 3.27, I think we need to show $\{1/n(logn{)}^p\}$ is decreasing. But I don't know how to do that.
Theorem 3.27: Suppose $a_1\ge <!--\; \ge \; -->a_2\ge <!--\; \ge \; -->\cdots <!--\; \cdots \; -->\ge <!--\; \ge \; -->0$. Then the series $\underset{n=1}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}{a}_n$ converges if and only if the series $\underset{k=0}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}{2}^k{a}_{2^k}=a_1+2a_2+4a_4+8a_8+\cdots <!--\; \cdots \; -->$ converges.

Oblique asymptotes
When we find oblique asymptotes, we divide the numerator by the denominator and take only the polynomial portion of the expression as the equation of the slant asymptote. Why? What is the proof that this equation is the slant asymptote?

I have the following system:
$\begin{array}{rl}\stackrel{\dot{}<!--\; \dot{}\; -->}{x}& =-<!--\; -\; -->y\\ \stackrel{\dot{}<!--\; \dot{}\; -->}{y}& =x\end{array}$
Given that $(x_i^n,y_i^n)$ are the points obtained for $i=1,2\dots <!--\; \dots \; -->n^2$ using a time-step $h=1/n$ starting at the initial point $(x_0,y_0)=(1,0)$.
I am struggling to find the following limit:
$\underset{n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}(x_n^n,y_n^n)$
I can intutively think of the above limit ending up somewhere on the unit circle but I am unable to obtain the value of the exact limit. Pardon if its too simple, but I am surely missing out something. Please help out, thanks in advance.

Is there a monotone bijection between the rationals of two intervals?
Given two nontrivial intervals I and J (both open or both closed), do there always exist a monotone bijection between $I\cap <!--\; \cap \; -->\mathbb{Q}$ and $J\cap <!--\; \cap \; -->\mathbb{Q}$?
If the endpoints of I and J are rational numbers, then such a bijection is easy to find (just take the linear function that sends the endpoints of I to those of J). But in general, it's not clear what to do.

For the following questions use the functions:
f(x) = 4x + 2
g(x) = 2x - 3
Evaluate f(g(x))
Describe the transformation of f(x) = (4x + 2) - 1
Describe the transformation of f(x) = 4(x + 3) + 2
Describe the transformation of g(x) = 2(2x - 3)
Describe the transformation of f(x) = 4(3x) + 2

A product transformation _.
multiplies coordinates to transform the imagecombines the coordinates of multiple transformationsfinds the result of combining multiple transformationsuses a scale factor to multiply coordinates of the image

(a) identify the transformation, and(b) graphically represent the transformation for an arbitrary vector in ${\displaystyle R^2}$.
${\displaystyle T(x,y)=(x+y,y)}$

Answer true or false to the following statement and explain your answer .
A power transformation of the response variable will not change its conditional distribution.