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Y_Kistochka [10]
3 years ago
15

PLEASE HELP which is equivalent??

Mathematics
1 answer:
Zolol [24]3 years ago
6 0

Answer:

the 3 one

Step-by-step explanation:

the rest would be like 0.9721111111 repeating but the 3 one won't be so its 3 one hive me brainlest

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The answer is attached

Step-by-step explanation:

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Q.04: (11 points) Given the polar curve r = e θ , where 0 ≤ θ ≤ 2π. Find points on the curve in the form (r, θ) where there is a
Citrus2011 [14]

I suppose the curve is r(\theta)=e^\theta.

Tangent lines to the curve have slope \frac{dy}{dx}; use the chain rule to get this in polar coordinates.

\dfrac{dy}{dx}=\dfrac{dy}{d\theta}\dfrac{d\theta}{dx}=\dfrac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}

We have

y(\theta)=r(\theta)\sin\theta\implies\dfrac{dy}{d\theta}=\dfrac{dr}{d\theta}\sin\theta+r(\theta)\cos\theta

x(\theta)=r(\theta)\cos\theta\implies\dfrac{dx}{d\theta}=\dfrac{dr}{d\theta}\cos\theta-r(\theta)\sin\theta

r(\theta)=e^\theta\implies\dfrac{dr}{d\theta}=e^\theta

\implies\dfrac{dy}{dx}=\dfrac{e^\theta\sin\theta+e^\theta\cos\theta}{e^\theta\cos\theta-e^\theta\sin\theta}=\dfrac{\sin\theta+\cos\theta}{\cos\theta-\sin\theta}

The tangent line is horizontal when the slope is 0, which happens wherever the numerator vanishes:

\sin\theta+\cos\theta=0\implies\sin\theta=-\cos\theta\implies\tan\theta=-1

\implies\theta=\boxed{-\dfrac\pi4+n\pi}

(where n is any integer)

The tangent line is vertical when the slope is infinite or undefined, which happens when the denominator is 0:

\cos\theta-\sin\theta=0\implies\sin\theta=\cos\theta\implies\tan\theta=1

\implies\theta=\boxed{\dfrac\pi4+n\pi}

6 0
4 years ago
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