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Find equations for the lines that are tangent and normal to the curve $y=f(x)$ at the given point.$$y= an (2 x),(0,0)$$...

Question

Find equations for the lines that are tangent and normal to the curve $y=f(x)$ at the given point.$$y= an (2 x),(0,0)$$

Find equations for the lines that are tangent and normal to the curve $y=f(x)$ at the given point. $$y=\tan (2 x),(0,0)$$



Answers

Find equations for the lines that are tangent and normal to the curve $y=f(x)$ at the given point. $$y=\tan (2 x),(0,0)$$

It's clear sailing right here. So the curve has two phones functions associated with it, which is plus and minus X to the three house power one has a graph above the X axis, and the other one's below the X axis. We know that why it's bigger than zero at the point. One comma one. So we're gonna take positive X three hands. We have our slope, which is equal to D over D, acts for X 23 house power for access equal to one, and this is equal to three halves and the tangent line equation becomes lie. Minus one is equal to three house X minus one, which is equal to three house acts minus on her. We have our slope of our normal line, which is negative 2/3 since they have to multiply to become negative one. And it's why minus one is equal to 2/3 negative 2/3 X minus one, which is equal to negative 2/3 acts plus 5/3

It's Clarissa when you right here. So we have acts is equal to two x e to the X To find the derivative, we got D over DX. To exceeds the X, we used the product Rule two x d to the axe eat the X plus eat the X d over DX for two X this becomes equal to eat. The axe comes to X plus two and the slope of detention at 00 becomes too. So the equation of attention is why equals two X? We have to note that while we multiply perpendicular lines, there slope is negative one. So the normal normal line has a slope of negative 1/2. So the equation of the normal line is gonna be Y minus zero over X minus zero, which is equal to negative one tough ex. So this becomes a Y is equal to negative 1/2 X for the normal line.

He It's clear. So when you right here, So we have of X is equal to two x over X square plus one. We're gonna difference she using the product quotient rule to get X square plus one D over DX for two x minus two X d over D X for X square plus one all over X square plus one square. This becomes equal to X square plus one times two minus two X tones to X all over X square plus one square, which is equal to two minus two X square over X square plus one square. A slip of detention at one common one become zero, and the equation of change in is why minus one over X minus or unequal. Ciro. So why is equal to one? And we know that the normal line is perpendicular, so it's going to be access equal to one

This question asked us to find equations, the tangent line and normal line to the curb at the given point. What we knows that we're looking at the formula. Why is a x plus be with a being the slope we know that we're looking for Why, prime of zero this is the slope of the tangent line at point X equals zero We know the derivative of X to the fork is for X cubed and the derivative of two e to the X is also to be to the ex plug n zero and we end up with simply to cause this cancels and then eat of the zero is just 11 times two is two. So we now have the formula. Why is two x plus b? Therefore, given the fact that tangent passed through the 0.2 we know the equation of the tension is to expose too. And then we know the slope of the Tangela at the same point is gonna be the negative reciprocal so negative 1/2 x and it passes through the 0.0 juice to give 1/2 exposed to


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