Detailed Explanations for Calculus 2 Equations

Why is Taylor series expansion for $1 / (1 - x)$ valid only for $x \in (- 1, 1)$ ?

alistianyStu 2022-12-02

Prove that the equation $4^{x} = 8 x + 1$ has only one solution

Julius Ho 2022-12-02

Consider the differential equation $\frac{d y}{d t} = a y - b$ .
Find the equilibrium solution $y_{e}$

e3r2a1cakCh7 2022-12-01

Let f be a differentiable function such that f(3) = 2 and f'(3) = 5. If the tangent line to the graph of f at x = 3 is used to find an approximation to a zero of f, that approximation is?

hsupaing moe2022-11-29

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. Find the dimensions of a Norman window of maximum area when the total perimeter is 16 feet.

arponatsdWT 2022-11-29

What is the product of $3$ numbers?

funnyantyLEy 2022-11-29

Evaluate the expression $C 2 5$ ?

Brenda Leach 2022-11-29

Solve the given initial-value problem. The DE is homogeneous.
$x y^{2} \frac{d y}{d x} = \frac{y^{3}}{x^{3}}$

Naomi Rowland 2022-11-27

Let $w = \frac{y z}{x}$ where $x = t^{2}, y = r + t$ and $z = r - t$ . Find $\frac{\partial w}{\partial t}$ and $\frac{\partial w}{\partial r}$ by using Chain Rule.

Mollie Wise 2022-11-27

What is its worth $15^{\circ}$ ?

Harmony Oneal 2022-11-25

Suppose $\sum_{n = 1}^{\infty} x_{n} < \infty$ . Then prove that $\sum_{n = 1}^{\infty} x_{n} y_{n}$ converges.

Jamir Summers 2022-11-25

How to prove that $lim_{n \to \infty} \sqrt[n]{n} = 1$ ?

valahanyHcm 2022-11-25

I'm struggling somewhat to understand how to use implicit differentiation to solve the following equation:
$\cos \cos (x^{3} y^{2}) - x \cot y = - 2 y$
I figublack that the calculation requires the chain rule to differentiate the composite function, but I'm not sure how to 'remove' the y with respect to x from inside the composite function. My calculations are:
$\frac{d y}{d x} [\cos \cos (x^{3} y^{2}) - x \cot y] = \frac{d y}{d x} [- 2 y]$
$\frac{d y}{d x} [\cos \cos (x^{3} y^{2})] = \sin \cos (x^{3} y^{2} \cdot y^{'} (x)) \cdot \sin (x^{3} y^{2} \cdot y^{'} (x)) \cdot 6 x^{2} y \cdot y^{'} (x)$
This seems a bit long and convoluted. I'm also not sure how this will allow me to solve for $y^{'} (x)$ . Carrying on...
$\frac{d y}{d x} [x \cot y] = - \csc^{2} y \cdot y^{'} (x)$
$\frac{d y}{d x} [- 2 y] = - 2$
Is my calculation correct so far? This seems to be a very complex derivative. Any comments or feedback would be appreciated.

hEorpaigh3tR 2022-11-25

What is the value of $\frac{100}{0}$ ?
1) 0
2) 1
3) 6
4) Not defined

Cherish Rivera 2022-11-24

What is the integral of log(z) over the unit circle?
I tried three times, but in the end I am concluding that it equals infinity, after parametrizing, making a substitution and integrating directly (since the Residue Theorem is not applicable, because the unit circle encloses an infinite amount of non-isolated singularities.)
Any ideas are welcome.
Thanks,

Lorena Becker 2022-11-24

My final for my introductory analysis course is tomorrow and my teacher gave us a list of possible theorems to prove. If anyone could please show me a proof for The Intermediate Value Theorem that is short and easy to follow, so even if I still cannot understand it I can at least memorize it. Also, I have looked through numerous texts and the internet, but they all seem to confuse me. I know that itis an insult to all you math experts to memorize proofs, but I am desperate at this point. Thank you

Barrett Osborn 2022-11-23

Use the intermediate value theorem to prove that if $f : [1, 2] \to R$ is a continuous function, that there is at least one number $c$ in the interval $(1, 2)$ such that $f (c) = 1 / (1 - c) + 1 / (2 - c)$
This is a question for my intro calc class that I am having a hard time understanding.

Talia Frederick 2022-11-23

I need to find $\frac{d y}{d x}$ :
$\frac{d}{d x} e^{y} \cos x = 1 + \sin (x y)$
using implicit differentiation.
So far I've gotten to this point which I don't think I'm doing right because then I get stuck:
$e^{y} (- \sin x) + (\cos x) (e^{y}) (\frac{d y}{d x}) = \cos (x y) (x) (\frac{d y}{d x}) + y (1)$
Am I right? Am I wrong? If I'm right I definitely have no clue where to go...and if I'm wrong...then again I have no idea what to do.

Nicholas Hunter 2022-11-23

Suppose $f : R \to R$ is continuous and periodic with period 2a for some $a > 0$ ; that is, $f (x) = f (x + 2 a)$ for all $x \in R$ . Show there is some $c \in [0, a]$ such that $f (c) = f (c + a)$ .
The only way I see to do this question is to apply the intermediate value theorem, but I just don't know how to apply it to this question. I know that $f (x) = f (x + 2 a)$ . Therefore, if i can show that $f (c + a) = f (c + 2 a)$ or $f (c + a) - f (c + 2 a) = 0$ , I'd be done. But I just don't know where to go from there. I tried substituting $x = a$ , getting $f (2 a) - f (3 a)$ , but I can't show that's less than, equal to, or greater than 0. Any ideas?

drogaid1d8 2022-11-23

Show Taylor series of $f (x^{2})$

When you are dealing with any Calculus 2 homework, it is vital to have a look at the various questions and answers that will help you see whether you are correct in your approach to finding solutions. Even if you are dealing with analytical aspects of Calculus 2, it will be helpful as you are looking at provided equations and learn how the answers relate to original questions and problems specified.

Do not be afraid to take a look at the basic integration and related application if Calculus 2 does not sound clear or start with the Calculus 1 first.

Calculus 2

Calculus 2 Equations: Expert Solutions and More