What is the slope of the polar curve #f(theta) = theta^2 - sectheta # at #theta = (3pi)/4#?

Answer 1

The slope:

#m = (9pi^2+32sqrt(2)-24pi)/(24pi + 9pi^2)#

#m ~~ 0.357#

From the reference Tangents with Polar Coordinates we obtain the equation

#dy/dx = ((dr)/(d theta)sin(theta)+rcos(theta))/((dr)/(d theta)cos(theta)-rsin(theta))" [1]"#
We are given #r = f(theta) = theta^2 - sec(theta)#; substitute this into equation [1]:
#dy/dx = ((dr)/(d theta)sin(theta)+(theta^2 - sec(theta))cos(theta))/((dr)/(d theta)cos(theta)-(theta^2 - sec(theta))sin(theta))" [2]"#
Compute #(dr)/(d theta)#:
#(dr)/(d theta) = 2theta-tan(theta)sec(theta)#

Substitute this into equation [2]:

#dy/dx = ((2theta-tan(theta)sec(theta))sin(theta)+(theta^2 - sec(theta))cos(theta))/((2theta-tan(theta)sec(theta))cos(theta)-(theta^2 - sec(theta))sin(theta))" [3]"#
The slope, m, of the tangent line at #theta = (3pi)/4# is the above equation evaluated at #theta = (3pi)/4#
#m = ((2(3pi)/4-tan((3pi)/4)sec((3pi)/4))sin((3pi)/4)+(((3pi)/4)^2 - sec((3pi)/4))cos((3pi)/4))/((2(3pi)/4-tan((3pi)/4)sec((3pi)/4))cos((3pi)/4)-(((3pi)/4)^2 - sec((3pi)/4))sin((3pi)/4))" [4]"#
#m = (9pi^2+32sqrt(2)-24pi)/(24pi + 9pi^2)#
The slope #m ~~ 0.357#
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Answer 2

To find the slope of the polar curve ( f(\theta) = \theta^2 - \sec(\theta) ) at ( \theta = \frac{3\pi}{4} ), you need to find the derivative of the function with respect to ( \theta ) and then evaluate it at ( \theta = \frac{3\pi}{4} ).

The derivative of the given function is:

[ \frac{df}{d\theta} = 2\theta - \sec(\theta)\tan(\theta) ]

Evaluating this derivative at ( \theta = \frac{3\pi}{4} ), we get:

[ \frac{df}{d\theta}\Bigg|_{\theta=\frac{3\pi}{4}} = 2\left(\frac{3\pi}{4}\right) - \sec\left(\frac{3\pi}{4}\right)\tan\left(\frac{3\pi}{4}\right) ]

Simplify this expression to find the slope of the polar curve at ( \theta = \frac{3\pi}{4} ).

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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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