Tangent Lines and Polar Curves

The video tutorial explains how to find the slope of the tangent line to a polar curve R = 2 - 4 cos(Theta) at Theta = pi/3. It begins by determining the point of tangency, which is at the pole, and then proceeds to find the slope of the tangent line using derivatives. The process involves calculating the derivative of the function, simplifying the expression, and evaluating it at the specified angle. The tutorial concludes with a verification of the positive slope found.

What is the form of the points on a polar curve?

• m, n

• x, y

• r, theta

• a, b

What is the value of r when theta equals pi/3 for the curve R = 2 - 4 cos(theta)?

• 0

• 4

• 1

• 2

What is the derivative of the function R = 2 - 4 cos(theta) with respect to theta?

• -4 cos(theta)

• 4 cos(theta)

• -4 sin(theta)

• 4 sin(theta)

In the expression for dy/dx, what does the numerator consist of?

• F of theta times sin(theta) minus frime of theta times cos(theta)

• frime of theta times cos(theta) minus F of theta times sin(theta)

• F of theta times cos(theta) plus frime of theta times sin(theta)

• frime of theta times sin(theta) plus F of theta times cos(theta)

What is the simplified form of the denominator in the dy/dx expression before evaluation?

• 4 sin(theta) cos(theta) - 2 sin(theta)

• 8 sin(theta) cos(theta) + 2 sin(theta)

• 4 sin(theta) cos(theta) + 2 sin(theta)

• 8 sin(theta) cos(theta) - 2 sin(theta)

What is the value of sin(pi/3)?

• 0

• 1/2

• sqrt(3)/2

• 1

What is the value of cos(pi/3)?

• 1/2

• 0

• sqrt(3)/2

• 1

What is the approximate decimal value of the slope of the tangent line at theta = pi/3?

• 3.73

• 2.73

• 1.73

• 0.73

What is the exact slope of the tangent line at theta = pi/3?

• 3/sqrt(3)

• 4/sqrt(3)

• 2/sqrt(3)

• 1/sqrt(3)

Why is it important to verify the slope of the tangent line?

• To determine the angle of the tangent line

• To calculate the area under the curve

• To ensure the calculation is correct

• To find the length of the tangent line