zagonek34

2022-01-12

How can we verify this limit via $\u03f5-\delta$ method?

I have to calculate the limit

$\underset{x\to 0}{lim}\frac{1-\mathrm{cos}\left(x\right)}{3{\mathrm{sin}}^{2}\left(x\right)}$

I have to calculate the limit

Lakisha Archer

Beginner2022-01-13Added 39 answers

We have to show that

$\mathrm{\forall}\u03f5>0:\mathrm{\exists}\delta \in (0,\frac{\pi}{2}]:\mathrm{\forall}x\in R\text{}\left\{0\right\}$

$:(\left|x\right|<\delta \Rightarrow \left|f\left(x\right)\right|<\u03f5)$

where$f\left(x\right)=\frac{{(1-\mathrm{cos}x)}^{2}}{6{\mathrm{sin}}^{2}x}=\frac{{(1-\mathrm{cos}x)}^{2}}{6(1-{\mathrm{cos}}^{2}x)}$

Let$t=\mathrm{cos}x$ . Then, because $t\ne 1\text{}\text{whenever}\text{}x\ne 0\text{}\text{and}\text{}\left|x\right|\frac{\pi}{2}$ ,

$f\left(x\right)=\frac{{(1-t)}^{2}}{6(1-{t}^{2})}=\frac{(1-t)}{6(1+t)}=\frac{(2-1-t)}{6(1+t)}$

$=\frac{1}{3(1+t)}-\frac{1}{6}$

f(x) strictly decreases w.r.t.t.

Find t that makes$f\left(x\right)=\u03f5$ . After all, $t=\frac{2}{(6\u03f5+1)}-1$ . Since $t=\mathrm{cos}x$ , choosing

$\delta =\mathrm{arccos}(\frac{2}{6min\{\u03f5,\frac{1}{6}\}+1}-1)$

proves the claim. I think you can show that$\delta$ is an increasing function of $\u03f5$ .

where

Let

f(x) strictly decreases w.r.t.t.

Find t that makes

proves the claim. I think you can show that

Mary Nicholson

Beginner2022-01-14Added 38 answers

You can also multiply and divide by $1+\mathrm{cos}x$ :

$\underset{x\to 0}{lim}\frac{1-\mathrm{cos}x}{3{\mathrm{sin}}^{2}x}=\underset{x\to 0}{lim}\frac{1-{\mathrm{cos}}^{2}x}{3{\mathrm{sin}}^{2}x(1+\mathrm{cos}x)}=$

$\underset{x\to 0}{lim}\frac{{\mathrm{sin}}^{2}x}{3{\mathrm{sin}}^{2}x(1+\mathrm{cos}x)}=\frac{1}{3}\underset{x\to 0}{lim}\frac{1}{1+\mathrm{cos}x}=\frac{1}{6}$

If you need to use the definition, you can easily see that

$|\frac{1-\mathrm{cos}x}{3{\mathrm{sin}}^{2}x}-\frac{1}{6}|=\frac{1}{3}|\frac{1}{1+\mathrm{cos}x}-\frac{1}{2}|\to 0(x\to 0)$

Since there is no indetermination in this limit, you can use Heine's definition in a very straightforward way (instead of Cauchy's). Or, you can go on to obtain

$|\frac{1}{1+\mathrm{cos}x}-\frac{1}{2}|=\frac{1}{2}\left|\frac{1-\mathrm{cos}x}{1+\mathrm{cos}x}\right|\le \frac{1}{2}|1-\mathrm{cos}x|\le \frac{1}{2}\left|x\right|$

If you need to use the definition, you can easily see that

Since there is no indetermination in this limit, you can use Heine's definition in a very straightforward way (instead of Cauchy's). Or, you can go on to obtain

What is 4 over 16 simplified to?

Using impulse formula, derive p_1 i+p_2 i=p_1 f+p_2 f formula where i-is initial state, f-is the final state after collision. Thanks

Averaging Newton's Method and Halley's Method.

Are these two methods identical? / Do they return the same result per iteration?Difference between Newton's method and Gauss-Newton method

Prove that Newton's Method applied to $f(x)=ax+b$ converges in one step? Would it be because the derivative of $f(x)$ is simply $a$?

simlifier tan(arcsin(x))

$dy/dx=\mathrm{sinh}(x)$ A tangent line through the origin has equation $y=mx$. If it meets the graph at $x=a$, then $ma=\mathrm{cosh}(a)$ and $m=\mathrm{sinh}(a)$. Therefore, $a\mathrm{sinh}(a)=\mathrm{cosh}(a)$.

Use Newton's Method to solve for $a$Using Newton's method below:

${x}_{n+1}={x}_{n}-\frac{f({x}_{n})}{{f}^{\prime}({x}_{0})}$

using this chord formula where the chord length $c$ is $1$ cm:

$c=2r\mathrm{sin}\frac{\theta}{2}$

supposing the radius is $1.1$ cm and the angle $\theta $ is unknown, show the iterative Newton's Method equation you would use to find an approximate value for $\theta $ in the context of this problem (using the appropriate function and derivative).Estimate the number of iterations of Newton's method needed to find a root of $f(x)=\mathrm{cos}(x)-x$ to within ${10}^{-100}$.

Why the bisection method is slower than Newton's method from a complexity point of view?

To find approximate $\sqrt{a}$ we can use Newton's method to approximately solve the equation ${x}^{2}-a=0$ for $x$, starting from some rational ${x}_{0}$.

Newton's method in general is only locally convergent, so we have to be careful with initialization.

Show that in this case, the method always converges to something if ${x}_{0}\ne 0$.Consider the function $f(x,y)=5{x}^{2}+5{y}^{2}-xy-11x+11y+11$. Consider applying Newton's Method for minimizing f. How many iterations are needed to reach the global minimum point?

1. If you weigh 140 lbs on Earth, what is your mass in kilograms

2.Using the answer from problem I (hopefully it's correct), determine your weight in Newton's if you were on the moon. Yes, you have to look up something to complete this problem.

3. You measured the mass of a rock to be 355g. What is its weight?

4.You are now holding the rock in your hand from problem 3. What force is the rock on your hand? How much force do you need to exert on the rock to hold in stationary.A solution containing 5.24 mg/100 mL of A (335 g/mol) has

**a transmittance of 55.2% in a 1.50-cm cell at 425 nm**.find molar absorbtivityFind the cube root of 9, using the Newton's method.