Help to Integral Calculus Solutions & Examples

the percentage of alcohol in a person's bloodstream t hr after drinking 8 fluid oz of whiskey is given by $A(t)=0.23t{e}^{-<!--\; -\; -->0.4t}(0\le <!--\; \le \; -->t\le <!--\; \le \; -->12)$
a) What is the percentage of alcohol in a person's bloodstream after $\frac{1}{2}$ hr? And After 4 hr?
b) How fast is the percentage of alcohol in a person's bloodstream changing after $\frac{1}{2}$ hr? And After 4 hr?

Which statement is symmetric?
Hint: Reflexive Property: a = a
Symmetric Property: If a = b, then b = a
Transitive Property: If a b and b = c, then a = c
Distributive Property: a(b+c) = ab + ac
A)24=24
B)4(x+2)=4x+8
C)13+7=7+13

I am struggling to find advantages and disadvantages of the following: Forward Euler Method, Trapezoidal Method, and Modified Euler Mathod (pblackictor-corrector).
Any help or books I can use to get these?

Vertical/horizontal asymptotes
We know that for rational functions
$f(x)/g(x)$
vertical asymptotes are defined as the lines $x=x_0$ where $g(x_0)=0$, and for horizontal asymptotes it is the limit as x approaches $\mathrm{\infty <!--\; \infty \; -->}$ or x approaches $-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->}$. My question is the following: what is the relationship between vertical and horizontal asymptotes? For instance, if we have
$y=f(x)$
and we express x as function of y, or
$x=f^{-<!--\; -\; -->1}(y),$
is the horizontal asymptote of inverse function the same as the vertical one of the original?
Thanks in advance.

Does cancellation impact vertical asymptotes?
Question: Let $r(x)=\frac{(x^2+x)}{(x+1)(2x-<!--\; -\; -->4)}$. Does the graph has x=1 as one of its asymptotes?
Answer: No.
My reasoning: $\frac{(x^2+x)}{(x+1)(2x-<!--\; -\; -->4)}=\frac{x(x+1)}{(x+1)(2x-<!--\; -\; -->4)}=\frac{x}{(2x-<!--\; -\; -->4)}$ and so, it cannot have x=−1 as one of its asymptotes.
However, what if I don't calcel and then say that −1 is a vertical asymptote? Will I be wrong?

First Derivative vs Second Derivative in relation to minima and maxima
I am struggling with understanding the difference between them and I need to write about them intuitively. The way my teacher explained it, the sign of the first derivative is used to determine if there is a minimum, maximum, or neither. For example, if the derivative is increasing to the left of x and decreasing to the right of x, than a maximum is present. If the derivative is decreasing to the left of x and is increasing to the right of x, there is a minimum. I know that the second derivative is the derivative if the first derivative and if it is positive on a certain interval, it is concave up. Is that not telling me the same thing as the first derivative?

If $1\le <!--\; \le \; -->\mathrm{\alpha <!--\; \alpha \; -->}$ show show that the gamma density has a maximum at $\frac{\mathrm{\alpha <!--\; \alpha \; -->}-<!--\; -\; -->1}{\mathrm{\lambda <!--\; \lambda \; -->}}$.
So using this form of the gamma density function:
$g(t)=\frac{{\mathrm{\lambda <!--\; \lambda \; -->}}^{\mathrm{\alpha <!--\; \alpha \; -->}}{t}^{\mathrm{\alpha <!--\; \alpha \; -->}-<!--\; -\; -->1}{e}^{-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}t}}{{\int <!--\; \int \; -->}_0^{\mathrm{\infty <!--\; \infty \; -->}}{t}^{\mathrm{\alpha <!--\; \alpha \; -->}-<!--\; -\; -->1}{e}^{-<!--\; -\; -->t}dt}$
I would like to maximize this. Now i was thinking of trying to differentiate with respect to t, but that is becoming an issue because if i remember correctly the denominator is not a close form when the bound goes to infiniti

I need to solve the equation below with Euler's method:
$y^{″}+\mathrm{\pi <!--\; \pi \; -->}y{e}^{x/3}(2y^{\prime }\sin <!--\; \; -->(\mathrm{\pi <!--\; \pi \; -->}x)+\mathrm{\pi <!--\; \pi \; -->}y\cos <!--\; \; -->(\mathrm{\pi <!--\; \pi \; -->}x))=\frac{y}{9}$
for the initial conditions $y(0)=1$, $y^{\prime }(0)=-<!--\; -\; -->1/3$
So I know I need to turn the problem into a system of two first order differential equations.
Therefore $u_1=y^{\prime }$ and $u_2=y^{″}$ I can now write the system as:
$$\begin{gathered} u_1=y^{\prime } \\ {u}_2={\displaystyle \frac{y}{9}}-<!--\; -\; -->\mathrm{\pi <!--\; \pi \; -->}y{e}^{x/3}(2u_1\sin <!--\; \; -->(\mathrm{\pi <!--\; \pi \; -->}x)-<!--\; -\; -->\mathrm{\pi <!--\; \pi \; -->}y\cos <!--\; \; -->(\mathrm{\pi <!--\; \pi \; -->}x)) \end{gathered}$$
How do I proceed from here?

Suppose I am given a Cauchy-Euler form second order differential equation
$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+y=f(x).$
The usual textbook method for solving the Cauchy Euler equation is to blackuce it to a linear differential equation with constant coefficients by the transformation $x=e^t$. But I have a fundamental doubt here, we know that $e^t>0$ $\mathrm{\forall <!--\; \forall \; -->}t\in <!--\; \in \; -->\mathbb{R}$. But when we are using the above transformation we are subconsciously assuming $x>0$. How does this make sense?

Substituting $t=\ln <!--\; \; -->(x)$ also makes no difference as ln is defined on ${\mathbb{R}}_{>0}$. So I now doubt the validity of the method followed to solve the Cauchy-Euler equation.

Can someone give me a proper explanation of what exactly going on here and why the process is valid?

I've just discoveblack that I can use Euler's method to estimating the slope of $y=(x+c{)}^2$ with perfect accuracy, regardless of what $\mathrm{\Delta <!--\; \Delta \; -->}x$ is used. What causes this?

Does it happen with other functions, what must they look like for it to happen?

I am using this method to estimate the slope, $m$
$m\approx <!--\; \approx \; -->\frac{(x+\mathrm{\Delta <!--\; \Delta \; -->}x+c{)}^2-<!--\; -\; -->(x-<!--\; -\; -->\mathrm{\Delta <!--\; \Delta \; -->}x+c{)}^2}{2\mathrm{\Delta <!--\; \Delta \; -->}x}$
Doesn't matter what $\mathrm{\Delta <!--\; \Delta \; -->}x$ is, the slope is always correct ...I would expect this only to work for tiny $\mathrm{\Delta <!--\; \Delta \; -->}x$, but works with large ones too.

Sum of monotonic increasing and monotonic decreasing functions
Consider an interval $x\in <!--\; \in \; -->[x_0,x_1]$. Assume there are two functions f(x) and g(x) with $f^{\prime }(x)\ge <!--\; \ge \; -->0$ and $g^{\prime }(x)\le <!--\; \le \; -->0$. We know that $f(x_0)\le <!--\; \le \; -->0$, $f(x_1)\ge <!--\; \ge \; -->0$, but $g(x)\ge <!--\; \ge \; -->0$ for all $x\in <!--\; \in \; -->[x_0,x_1]$. I want to show that $q(x)\equiv <!--\; \equiv \; -->f(x)+g(x)$ will cross zero only once. We know that $q(x_0)\le <!--\; \le \; -->0$ and $q(x_1)\ge <!--\; \ge \; -->0$.

Find integral values k such that sum of expression is minimized
Given n values $X_1,X_2,....,X_n$, where $X_i$ can be positive or negative.
The absolute values of $X_i$ will be less than 100000, also $n<=100000$.
What should be the possible value(s) of k such that $f(k)=|k|+|X_1+k|+|X_1+X_2+k|+\cdots <!--\; \cdots \; -->+|X_1+X_2+X_3+\cdots <!--\; \cdots \; -->+X_n+k|$ is minimized .
$|a|$ absolute value of a .

$Y^{\prime }=AY$
$A=\left[\begin{array}{ccc}1& 2& -<!--\; -\; -->1\\ 1& 0& 1\\ 4& -<!--\; -\; -->4& 5\end{array}\right]$
$Y(0)=\left(\begin{array}{c}-<!--\; -\; -->1\\ 0\\ 0\end{array}\right)$
I have to solve this system using Eulers method, but what's Eulers method, like I know how to approach this using the eigenvalue method, but the problem explicitly states that it has to be solved using that method, yet my textbook doesn't provide an algorithm for such a method.

For ${\displaystyle f\left(t\right)=(\frac{1}{2t-3},t{e}^t)}$ what is the distance between ${\displaystyle f\left(0\right)}$ and ${\displaystyle f\left(1\right)}$ ?

This question was asked in CSIR. please help me to find out correct choice

Let $y(t)$ satisfy the differential equation
$y^{\prime }=\mathrm{\lambda <!--\; \lambda \; -->}y;y(0)=1$
. Then the backward Euler method for $n>1$ and $h>0$
$\frac{y_n-<!--\; -\; -->y_{n-<!--\; -\; -->1}}{h}=\mathrm{\lambda <!--\; \lambda \; -->}{y}_n;y(0)=1$
yields
1. a first order approximation to $e^{\mathrm{\lambda <!--\; \lambda \; -->}nh}$
2. a polynomial approximation to $e^{\mathrm{\lambda <!--\; \lambda \; -->}nh}$
3. a rational function approximation to $e^{\mathrm{\lambda <!--\; \lambda \; -->}nh}$
4. a Chebyshev polynomial approximation to $e^{\mathrm{\lambda <!--\; \lambda \; -->}nh}$

Integrate ${\int <!--\; \int \; -->}_0^{\mathrm{\infty <!--\; \infty \; -->}}\frac{{\sin }^2<!--\; \; -->x}{\cosh <!--\; \; -->x+\cos <!--\; \; -->x}\frac{dx}{x}$

Which rule for numerical integration is more accurate, the Trapezoidal Rule or the Simpson Rule?

For ${\displaystyle f\left(t\right)=(\frac{{\left(\ln t\right)}^2}{t},t^3)}$ what is the distance between f(2) and f(4)?

What is the distance between the following polar coordinates?:
${\displaystyle (1,\frac{5\pi }{4}),(5,\frac{3\pi }{8})}$

Finding the asymptotes
Find the asymptotes for $F(x)=ln(x^2=-<!--\; -\; -->3)$. I realise there is no vertical asymptote since $X^2=-<!--\; -\; -->3$. I want to know if the horizontal asymptote exist