Solve Integral Calculus Problems with Ease

College
Calculus and Analysis
Integral Calculus

Find the Intervals of Increase or Decrease, the concavity, and point of inflection for: $f (x) = (1 - x) e^{- x}$ .
$\begin{aligned} f (x) & = (1 - x) e^{- x} \\ = \frac{1 - x}{e^{x}} \end{aligned}$
Quotient Rule:
$\begin{aligned} f^{'} (x) & = e^{x} \cdot [1 - x]^{'} - (1 - x) \cdot [e^{x}]^{'} \\ = e^{x} (- 2 + x) \end{aligned}$
What do I do now? Take the derivative again?

Dillan Valenzuela 2022-08-11

Question regarding the Asymptotes of Hyperbola
Take a curve ,some curve which converges to a point as one variable( say x)tends to $\infty$
$lim_{x \to \infty}$
,the value that it approaches (or) converges to ( y value) ,is the value of the asymptote ,the value that the curve tries to reach(y reluctantly tries to get to )but never reaches (or) only reaches the value at $\infty$ .
Now the equation of Hyperbola is given by
$\frac{y ²}{b ²}$
To find the asymptotes we substitute $\frac{y ²}{b ²}$
Why do we do that ?
What is happening here?
Is there any geometrical reasoning for this?
To find the asymptotes we take the RHS of the equation as 0,why so?

Carsen Patel 2022-08-11

fundamental theorem of calculus and second derivative
Using the fundamental theorem of calculus, find the second derivative of
$\int_{\sqrt{(} x)}^{x} x - e^{t} d t$
I've looked up the theorem on wikipedia but I can't really see what I'm meant to do.

Taliyah Reyes 2022-08-11

Is $f (x) = x \sqrt{4 - x}$ decreasing at $x = 4$ ?
I am solving a single variable calculus problem and it asks me to determine the decreasing and decreasing intervals of the function $f (x) = x \sqrt{4 - x}$ . It's pretty clear that from $] - \infty, \frac{8}{3} [$ the function is increasing but, since the domain of the function goes only until $x = 4$ , should I write that the decreasing interval of the function is $] \frac{8}{3}, 4 [$ or $] \frac{8}{3}, 4]$ ?

orkesruim40 2022-08-11

Why the staggeblack Euler (Euler-Backward) method is not runge-kutta method?

The method is given by
$x_{n + 1} = x_{n} + h g (p_{n + 1})$
$p_{n + 1} = p_{n} + h g (x_{n})$
I am not very familiar with the conditions of the Runge-kutta method, can someone help me with this? thank you.

Silvina2b 2022-08-11

Figuring out when $f (x) = \sin (x^{2})$ is increasing and decreasing
Regarding the function $f (x) = \sin (x^{2})$ , I'm supposed to figure out when it is increasing/decreasing.
So far, I've found the derivative to be $f^{'} (x) = 2 x \cos (x^{2})$ .
So long as I can solve the inequality $\cos (x^{2}) > 0$ . I can figure the rest out, but this is where I'm stuck.
I've narrowed it down to $- \frac{π}{2} < x^{2} < \frac{π}{2}$ meaning $x^{2} < \frac{π}{2} \pm n 2 π, n \in N$ .
Then I get $x < \pm \sqrt{\frac{π}{2} \pm n 2 π}$ .
Am I on the right track here? I can't seem to find a path from here.

lexi13xoxla 2022-08-11

Graphing using derivatives
Sketch the graph of the following equation. Show steps of finding out critical numbers, intervals of increase and decrease, absolute maximum and minimum values and concavity.
$y = x e^{x^{2}}$
I found the first derivative which is $y^{'} = (2 x^{2} + 1) e^{x^{2}}$ and I know that in order to find min and max the zeroes for y′ must be found, but y′ doesn't have any real zeroes, and I'm confused about how to go on with solving the problem.
If someone could help me out, that would be appreciated. Thank you in advance.

bsmart36 2022-08-11

Is a Square Bracket Used in Intervals of Increase/Decrease?
For example, the I.O.I of $y = x^{2}$ is (0,infinite), with the round brackets meaning that the value is excluded. Are there any scenarios where a square bracket would be used when stating the intervals of increase/ decrease for a function? If it narrows it down, the only functions I deal with are: linear, exponential, quadratic, root, reciprocal, sinusoidal, and absolute

targetepd 2022-08-10

Find the rate of change of main dependent variable
We have $f : R \to R, f (x) = x^{2} + x \sin (x)$ , and we need to find intervals of monotonicity. Here is all my steps:
$f^{'} (x) = 2 x + x \cos (x) + \sin (x)$
$f^{'} (x) = 0 \Rightarrow x = 0$ the only solution.
Now I need to find where f′ is positive and negative. I don't want to put value for f′ to find the sign, I want another method. So I tried to differentiate the function again to see if f′ is increasing or decreasing:
$f^{″} (x) = 2 - x \sin (x) + \cos (x)$ , but I don't know if $f^{″} (x) \geq 0$ or $f^{″} (x) \leq 0$ .
How can I find the sign for f′ to determine monotony of f ?

Janiya Rose 2022-08-10

How to find out whether a function has vertical asymptotes
I got the following question wrong in my exame
$\frac{16 x - 64}{| x | - 4}$
My answer was, -16,16 for the horizontal asymptotes and 4,-4 for the vertical asymptotes.
For the horizontal asymptotes the answer is right, But there aren't any vertical asymptotes.
I thought if there wasn't a sum of squares in the denominator There would be vertical asymptotes.
So I proceeded this way, as I need to equal the denominator to 0 in order to find the vertical asymptotes.
$| x | - 4 = 0$
x−4=0
x=4
−x−4=0
−x=4(−1)
x=−4
There are 3 rules to find the HA's, but what are the rules to find the VA's ?

musicbachv7 2022-08-10

Suppose we have a system of ODE's: a $a^{'} = - a - 2 b$ and $b^{'} = 2 a - b$ with initial conditions $a (0) = 1$ and $b (0) = - 1$ .

How can we find the maximum value of the step size such that the norm a solution of the system goes to zero (if we apply the forward Euler formula)?

Edit: the main part is to calculate the eigenvalues of the following matrix, based on the Euler method, this becomes
$(\begin{matrix} - 1 - h & - 2 - 2 h \\ 2 + 2 h & - 1 - h \end{matrix})$
The eigenvalues are $(- 1 + 2 i) (1 + h)$ and $(- 1 - 2 i) (1 + h)$

cofak48 2022-08-09

I was messing around with a program I wrote to trace through slope fields using Euler's method, and I noticed this equation looked particularly funny:
$\frac{d y}{d x} = \frac{y - x}{y \cdot \sin (y)}$
Is there any solution, implicit or explicit?

Max Macias 2022-08-09

How can one determinate the variationt of f(g(x))
Given the two functions:
$f (x) = x^{2} - 2 x$
$g (x) = \sqrt{x + 1}$
The question is determinate the variation of f(g(x))
We have $D_{f (g (x))} = (- 1, + \infty)$
For f it's decreasing for $x < 1$ .
Increasing for $x > 1$
For g its increasing for $x > - 1$
To determinate the variationf of f(g(x))
In interval $(- 1; + \infty)$
We have g is increasing
$X > - 1$ means that $g (x) > g (- 1)$ so $g (x) > 0$ .
So $g ([- 1; + \infty)) = [0, + \infty)$ .
But the problem is that f in that interval is increasing and decreasing Im stuck here.
Can one write a methode to answer any question like this theoricaly.

ljudskija7s 2022-08-08

Convert the polar equation to a rectangular equation.
r = 2

crazygbyo 2022-08-08

Graphing functions with calculus
I am stuck on graphing $y = (x - 3) \sqrt{x}$
I am pretty sure that the domain is all positive numbers including zero. The y intercept is 0, x is 0 and 3.
There are no asymptotes or symmetry.
Finding the interval of increase or decrease I take the derivative which will give me $\sqrt{x} + \frac{x - 3}{2 \sqrt{x}}$ .
Finding zeroes for this I subtract $\sqrt{x}$ and then multiply by the denominator $2 x = x - 3$ .
This gives me 3 as a zero, but this isn't correct according to the book so I am stuck. I am not sure what is wrong.

brasocas6 2022-08-07

Finding asymptotes of a function using limits
I need to find asymptotes of a function. I didn't find any vertical asymptotes and I think there aren't any. But when trying to find a horizontal asymptote I evaluate the limit as x approaches infinity but can't successfully find it.
$f (x) = x \arctan (x)$

Brandon Monroe 2022-08-07

$y^{'} (t) = f (t, y), a \leq t \leq b, y (a) = α$
apply for the extrapolation technique
we let $h_{0} = \frac{h}{2}$ and use Euler method with $w_{0} = α$
$w_{1} = w_{0} + h_{0} f (a, w_{0})$
and then midpoint method
$t_{i - 1} = a, t_{i} = a + h_{0} = a + \frac{h}{2}$ to produce
approximation to $y (a + 2 h_{0})$
$w_{2} = w_{0} + 2 h_{0} f (a + h_{0}, w_{1})$
$y_{1, 1} = \frac{1}{2} [w_{2} + w_{1} + h_{0} f (a + 2 h_{0}, w_{2})]$
then why this form results $O (h_{0}^{2})$ approximation to $y (t_{1})$

sublimnes9 2022-08-07

One method consists in computing the numerical solution for an arbitrary $h$ and then $2 h$ . The Richardson extrapolation gives an estimate of $e = max_{t} | y (t, 2 h) - y (t, h) |$ of the error. When the error is smaller than the tolerance, we keep the result and start from $2 y (t, h) - y (t, h)$ . If the error is larger we restart with $h / 2$ until we reach the tolerance.
( $y (t, 2 h)$ means approximation with $2 h$ )
I don't understand why the Richardson extrapolation is mentioned. For what do I have to use it? Can I not just calculate $y (t, 2 h)$ and $y (t, h)$ and see the error?

makeupwn 2022-08-07

Find the standard form of the equation of the circle having the following properties.
Center at the point (8,-6) Tangent to the x-axis
type the standard form of the equation of this circle

Gauge Roach 2022-08-06

Convert the decimal $0. \bar{6}$ to a fraction by writing it as an infinite series. $0. \bar{6} =$

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