What is the slope of the tangent line of #r=theta-sin((10theta)/3-(pi)/8)# at #theta=(pi)/4#?

Answer 1

Slope of the tangent line of
#r=theta-sin((10theta)/3-pi/8)#
at #theta=pi/4# is
#(dy)/(dx)=0.995#

Given: #r=theta-sin((10theta)/3-pi/8)# #(dr)/(d(theta))=1-(10/3)cos((10theta)/3-pi/8)#
wkt #x=rcostheta# #(dx)/(d(theta))=r(-sintheta)+costheta((dr)/(d(theta)))# at #theta = pi/4# #r=pi/4-sin((10pi/4)/3-pi/8)# #=-0.008# #(dr)/(d(theta))=1-(10/3)cos((10pi/4)/3-pi/8)# #=3.029# #(dx)/(d(theta))=-0.008(-sinpi/4)+cos(pi/4)(3.029)# #=0.008(0.707)+0.707(3.029)# #(dx)/(d(theta))=2.147#
#y=rsintheta# #(dy)/(d(theta))=r(costheta)+sintheta((dr)/(d(theta)))# At #theta=pi/4# #(dy)/(d(theta))=-0.008(cospi/4)+sin(pi/4)(3.029)# #(dy)/(d(theta))=2.136# Slope of the tangent line of #r=theta-sin((10theta)/3-pi/8)# at #theta=pi/4# is #(dy)/(dx)=((dy)/(d(theta)))/((dx)/(d(theta)))=2.136/2.147# Thus, #(dy)/(dx)=0.995#
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Answer 2

To find the slope of the tangent line at a given point on a polar curve, you first need to express the curve in Cartesian coordinates and then differentiate the equation with respect to ( \theta ). The slope of the tangent line is given by ( \frac{dy}{dx} ) in Cartesian coordinates.

For the given polar curve ( r = \theta - \sin\left(\frac{10\theta}{3} - \frac{\pi}{8}\right) ), when evaluated at ( \theta = \frac{\pi}{4} ), the Cartesian coordinates are found to be ( x = \frac{\pi}{4} - \sin\left(\frac{5\pi}{6} - \frac{\pi}{8}\right) ) and ( y = \frac{\pi}{4} - \sin\left(\frac{5\pi}{6} - \frac{\pi}{8}\right) ). Then, differentiate both ( x ) and ( y ) with respect to ( \theta ), and find ( \frac{dy}{dx} ) when ( \theta = \frac{\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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