What is the slope of the tangent line of #r=theta-cos(-4theta+(2pi)/3)# at #theta=(7pi)/4#?

Answer 1

#m = 0.0564#

The reference Tangents with Polar Coordinates gives us this formula for #dy/dx#:

#dy/dx = ((dr)/(d theta)sin(theta) + rcos(theta))/((dr)/(d theta)cos(theta) - rsin(theta))" [1]"#

Substitute #theta - cos(-4theta+ (2pi)/3)# for every instance of r in equation [1]:

#dy/dx = ((dr)/(d theta)sin(theta) + (theta - cos(-4theta+ (2pi)/3))cos(theta))/((dr)/(d theta)cos(theta) - (theta - cos(-4theta+ (2pi)/3))sin(theta))" [2]"#

Substitute #1 - 4cos(pi/6 - 4theta)# for every instance of (dr)/(d theta) in equation [2]:

#dy/dx = ((1 - 4cos(pi/6 - 4theta))sin(theta) + (theta - cos(-4theta+ (2pi)/3))cos(theta))/((1 - 4cos(pi/6 - 4theta))cos(theta) - (theta - cos(-4theta+ (2pi)/3))sin(theta))" [3]"#

The slope, m, is equation [3] evaluated at #theta = (7pi)/4#:

#m = ((1 - 4cos(pi/6 - 7pi))sin((7pi)/4) + ((7pi)/4 - cos(-7pi+ (2pi)/3))cos((7pi)/4))/((1 - 4cos(pi/6 - 7pi))cos((7pi)/4) - ((7pi)/4 - cos(-7pi+ (2pi)/3))sin((7pi)/4))#

#m = 0.0564#

Here is a graph of r #0 <= theta < 2pi#, the specified point, and the tangent line with the above slope:

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the slope of the tangent line of ( r = \theta - \cos(-4\theta + \frac{2\pi}{3}) ) at ( \theta = \frac{7\pi}{4} ), you first need to find the derivative of ( r ) with respect to ( \theta ) (denoted as ( \frac{dr}{d\theta} )). Then, evaluate ( \frac{dr}{d\theta} ) at ( \theta = \frac{7\pi}{4} ) to get the slope of the tangent line at that point.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7