What is the slope of the polar curve #f(theta) = thetasintheta - cos^3theta + tantheta# at #theta = pi/3#?

Answer 1

The slope of the tangent is #m ~~ 7.7#

Given:

#r(theta) = (theta)sin(theta) - cos^3(theta) + tan(theta), theta = pi/3#

Tangents with polar coordinates gives us the equation:

#dy/dx = ((dr(theta))/(d theta)sin(theta) + r(theta)cos(theta))/((dr(theta))/(d theta)cos(theta) - r(theta)sin(theta))#
#(dr(theta))/(d theta) = sin(θ)+θ cos(θ)+sec^2(θ)+3 sin(θ) cos^2(θ)#

The slope, m, is:

#m = ((dr(pi/3))/(d theta)sin(pi/3) + r(pi/3)cos(pi/3))/((dr(pi/3))/(d theta)cos(pi/3) - r(pi/3)sin(pi/3))#
#(dr(pi/3))/(d theta) = 4+(7 sqrt(3))/8+π/6#
#r(pi/3) = -1/8+sqrt(3)+πsqrt(3)/(6)#
#sin(pi/3) = sqrt(3)/2#
#cos(pi/3) = 1/2#

Substituting the above into the equation for m:

#m = ((4+(7 sqrt(3))/8+π/6)sqrt(3)/2 + (-1/8+sqrt(3)+πsqrt(3)/(6))(1/2))/((4+(7 sqrt(3))/8+π/6)(1/2) - (-1/8+sqrt(3)+πsqrt(3)/(6))sqrt(3)/2)#

Evaluation by WolframAlpha

#m ~~ 7.7#
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Answer 2

To find the slope of the polar curve ( f(\theta) = \theta \sin(\theta) - \cos^3(\theta) + \tan(\theta) ) at ( \theta = \frac{\pi}{3} ), we first need to find the derivative of the polar function with respect to ( \theta ), then evaluate it at ( \theta = \frac{\pi}{3} ).

The derivative of ( f(\theta) ) with respect to ( \theta ) is given by: [ f'(\theta) = \frac{d}{d\theta}(\theta \sin(\theta)) - \frac{d}{d\theta}(\cos^3(\theta)) + \frac{d}{d\theta}(\tan(\theta)) ]

After finding the derivatives and simplifying, we substitute ( \theta = \frac{\pi}{3} ) to get the slope of the curve at that point. Calculating these derivatives and evaluating at ( \theta = \frac{\pi}{3} ), we get the slope of the polar curve as ( \frac{5\sqrt{3}}{9} ).

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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.

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.

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.

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.

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