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Calculus and Analysis
Integral Calculus

Describe the vertical asymptotes) and holes) for the graph of $y = \frac{(x + 2) (x + 4)}{(x + 4) (x + 1)}$
A. asymtotes: x=-1 and hole: x=4
B. asymtotes: x=2 and hole: x=-1
C. asymtotes: x= -1 and hole: x=-4
D. asymtotes: x=1 and hole: x=4

nuseldW4r 2022-11-25

Which of the following statements is true for the function
f(x)=x 2 +x−6 2x 2 −2 ?
a)
y=1 2 is a horizontal asymptote and x=2 is a vertical asymptote.
b)
y=1 2 is a horizontal asymptote and x=1 is a vertical asymptote.
c)
y=3 is a horizontal asymptote and x=-1 is a vertical asymptote.
d)
y=3 is a horizontal asymptote and x=2 is a vertical asymptote

vegetatzz8s 2022-11-24

$1. \int_{0}^{π} \int_{0}^{1} \int_{0}^{\sqrt{1 - y^{2}}} y \sin d y d x Find the above triple integral$

intrigahQNn 2022-11-24

How to parameterize the paraboloid $z = 9 - x^{2} - y^{2}$ ? I know that I can parameterize it in cartesian coordinates as $r (x, y) = (x, y, 9 - x^{2} - y^{2})$ but I see in a book this parameterization of it that is $r (t) = (3 c o s (t), s i n (t), 0)$ with $0 \leq t \leq 2 π$ and I don't know how to deduce it.

Goundoubuf 2022-11-24

How to prove that $\int_{- π / 2}^{π / 2} \frac{\cos (2 x)}{e^{x} + 1} = 0$

Justine Pennington 2022-11-22

I am going to program Eulers method in Octave to approximate gravity in 1-dimension. I understand the formula for Eulers method, which is equal to:
$y_{i + 1} = y_{i} + (△ t ⋆ f (t_{i}, y_{i}))$
What I don't understand is what my function f(t,y) is in this case. What do I have to insert into the formula to get my next y-point?

Ty Moore 2022-11-22

Interval Notation for Increasing and Decreasing Intervals of a Function
This was brought up by another student in one of my pre-calculus classes.
The graph was a simple quadratic $x^{2}$ . The teacher stated that the graph was decreasing from $(- \infty, 0)$ , and increasing from $(0, \infty)$ .
Why would zero not be included? i.e: decr. $(- \infty, 0]$ and incr. $[0, \infty)$

E Laurie2022-11-21

PLEASE SHOW ALL WORK. FOLLOW INSTRUCTIONS CAREFULLY

E Laurie2022-11-20

SHOW ALL WORK PLEASE. FOLLOW DIRECTIONS CAREFULLY

Aleah Avery 2022-11-20

I am trying to calculate the global error bound for Euler's method, but I am having trouble. I am given the formula $| y (t_{i}) - u_{i} | \leq \frac{1}{L} (\frac{h M}{2} + \frac{δ}{h}) (e^{L (t_{i} - a)} - 1) + | δ_{0} | e^{L (t_{i} - a)}$ where $u_{i}$ is the Euler approxmation. I am also given $M, L, a, δ, δ_{0}$ , h. If I am not mistaken this will give me the error for each step, but how do I find the upper bound for the total error?

sbrigynt7b 2022-11-20

How to prove that
$\int_{0}^{1} (1 + x^{n})^{- 1 - 1 / n} d x = 2^{- 1 / n}$

Sophie Marks 2022-11-20

Prove that
$\int_{0}^{x} \int_{0}^{y} \int_{0}^{z} f (t) d t d z d y = \frac{1}{2} \int_{0}^{x} (x - t)^{2} f (t) d t$

kunguwaat81 2022-11-19

How do you find vertical and horizontal asymptotes
Whenever i am doing curve sketching in calculus class i find it incredibly difficult to find the vertical and horizontal asymptotes and as such i always get up to that point correct and anything from asymptotes and further wrong. Sometimes even my curve is a bit off

E Laurie2022-11-18

What is the reason behind the multiplication of the function's derivative with the step size (and the subsequent addition) in the numerical Euler method?
$y_{n + 1} = y_{n} + h f (t_{n}, y_{n})$
I can't figure out why exactly this would work for generating a new value. How can scaling the output of the function $f$ and subsequently adding it to the prior value $y_{n}$ approximate a new value? Is there some formal or graphical explanation of this? The formula seems somewhat counterintuitive.

Rihanna Bentley 2022-11-17

This problem is giving me a headache.
$\int \frac{4 / 7 + \sqrt{x \sqrt{x}}}{\sqrt{4 - x (1 + x^{3 / 4})}} d x$

همس الهمس2022-11-16

nyle2k8431 2022-11-16

Find increasing and decreasing intervals of a function
The given function is $f (x) = x^{200} - x^{100}$ , and I'm supposed to find it's decreasing and increasing intervals. Also, I should find them not by using derivatives but by doing function composition, like this:
$f_{1} (x) = x^{100}$
$f_{2} (x) = x (x - 1)$
$f (x) = f_{2} (f_{1} (x))$
I know that $f_{1}$ is decreasing on the interval $(- \infty, 0]$ and increasing on $[0, + \infty)$ , and $f_{2}$ is decreasing on $(- \infty, \frac{1}{2}]$ and increasing on $[\frac{1}{2}, + \infty)$ , but I'm not really sure what to do next.

Jared Lowe 2022-11-16

Negation of monotonicity of a continuous function
So I am given a continuous function mapping a connected domain to the reals, i.e $f : (a, b) \to R$ .
I want to show that if f is not strictly monotone and f is continuous, we have $x, y, z \in (a, b)$ with $x < y < z$ such that:
$f (x) \leq f (y) and f (y) \geq f (z)$
or $f (x) \geq f (y) and f (y) \leq f (z)$
As for what I've tried, using the negation of strictly monotone, we have that f must be increasing and decreasing in two intervals $[x_{1}, x_{2}]$ and $[x_{3}, x_{4}]$ i.e.:
$f (x_{1}) \leq f (x_{2}) and f (x_{3}) \geq f (x_{4})$ or $f (x_{1}) \geq f (x_{2}) and f (x_{3}) \leq f (x_{4})$
The problem comes when I look to combine the two into a single inequality. In an effort to combine these, the only way I see of doing it (directly) would be casewise. If anyone could provide a hint or (less preferred) a complete direct proof I would appreciate it.

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