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3 years ago

Which of the following functions have graphs that contain vertical asymptotes?

1 answer:

6 0

A vertical asymptotic is a point at which y approaches positive or negative infinite, or both, and y is undefined at that x value.

Of the options, the reciprocal has a vertical asymptote. When approaches zero, y approaches 1÷0, or infinite.
The logarithmic function also has a vertical asymptote at x=0. Let's use f(x) =log base 10 (x) as an example. This function describes the exponent you'd have to put on 10 to get x. When x=0, we have a bit of a problem. What power can you raise 10 to to get 0? The answer is negative infinite, which would be
1/ (10^∞) = 0.

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3 years ago

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<img src="https://tex.z-dn.net/?f=%20%5Csf%20%20%5Clarge%20%5C%3A%20find%20%5C%3A%20%20%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%20%20%5C%3A

solong [7]

Given :

$\: \: \:$

$\rm \large \: x = a ( \Theta + Sin \Theta )$

$\: \: \:$

$\rm \large y = a ( 1 - cos \: \Theta )$

$\: \:$

Now , x = a ( θ + sin θ )

 $\: \:$ 

$\: \: \:$

$\rm \large\frac{dx}{dθ } = a \: \frac{d}{dθ} (θ + \sin\theta )$

$\: \:$

$\boxed{ \rm \large\underline{ \frac{dx}{d \theta} = a(1 + \cos \theta ) }}$

$\: \:$

Now y = a ( 1-cosθ)

$\: \:$

Diff w.r.t " θ " we get .

$\: \:$

$\rm \large \frac{dy}{d \theta} = a \frac{d}{d \theta} (1 - cos \theta)$

$\: \: \:$

$\boxed{ \rm \large \underline{ \frac{dy}{d \theta} = a \sin \theta}}$

$\: \: \:$

From eqn ( 1 ) & ( 2 )

$\: \:$

$\rm \large \frac{dy}{dx} = \frac{ \frac{dy}{d \theta} }{\frac{dx}{d \theta}}$

$\: \: \:$

$\: \: \: \rm \large = \frac{ \cancel{a} \: sin \theta}{ \cancel a \: (1 + cos \theta)}$

$\: \:$

$\rm \large \: = \frac{sin \theta}{1 + \cos \theta }$

$\: \: \:$

$\rm \large \: \frac{2 \sin( \frac{\theta }{2} ) cos\frac{\theta }{2} }{2 \: cos ^{2} \frac{\theta }{2}} \: \: ......(sin \: a \: = 2 \: sin \frac{a}{2} \: cos \frac{a}{2} 1 + \: cos \: a \: = 2cos ^{2} \frac{a}{2} )$