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What is the equation of the tangent line of #r=cos(theta-pi/4) +sin^2(theta+pi)-theta# at #theta=(-13pi)/4#?

Answer 1

Christopher Allers

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Answer 2

Aurora Alvarado

To find the equation of the tangent line, we first need to find the derivative of the polar curve ( r = \cos(\theta - \frac{\pi}{4}) + \sin^2(\theta + \pi) - \theta ). The derivative of a polar equation ( r = f(\theta) ) is given by ( \frac{{dr}}{{d\theta}} = \frac{{dr}}{{dx}}\frac{{dx}}{{d\theta}} + \frac{{dr}}{{dy}}\frac{{dy}}{{d\theta}} ), where ( x = r \cos(\theta) ) and ( y = r \sin(\theta) ).

After finding ( \frac{{dr}}{{d\theta}} ), we evaluate it at ( \theta = -\frac{{13\pi}}{4} ) to find the slope of the tangent line. Then, we use the point-slope form of a line, ( y - y_1 = m(x - x_1) ), where ( (x_1, y_1) ) is the point of tangency, and ( m ) is the slope of the tangent line. Finally, we substitute the given values to find the equation of the tangent line.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.