Understanding Tangent Lines to Parametric Curves

The video tutorial explains how to find the exact slope of a tangent line to a parametric curve defined by x = 6 cos(t) and y = 7 sin(t) at t = π/3 radians. It covers evaluating dy/dx using derivatives of parametric equations, finding the point of tangency, and using trigonometric identities to determine coordinates. The tutorial also discusses calculating the exact and approximate slope of the tangent line, including rationalizing the denominator for precision.

What is the purpose of finding the slope of the tangent line to a parametric curve?

• To calculate the curve's area

• To determine the curve's orientation

• To find the curve's length

• To understand the curve's steepness at a point

How do you find the x-coordinate of the point of tangency?

• By finding the derivative of x with respect to t

• By evaluating y at t equals pi over three

• By evaluating x at t equals pi over three

• By finding the derivative of y with respect to t

What is the cosine of pi over three?

• One

• Zero

• One half

• Square root of three

Which method can be used to find trigonometric values besides reference triangles?

• Numerical approximation

• Algebraic equations

• Unit circle

• Graphing calculator

What is the derivative of seven sine t with respect to t?

• Seven cosine t

• Negative seven cosine t

• Negative seven sine t

• Seven sine t

What is the derivative of six cosine t with respect to t?

• Negative six sine t

• Six cosine t

• Six sine t

• Negative six cosine t

What is the formula for dy/dx in terms of parametric equations?

• dy/dt divided by dx/dt

• dx/dt divided by dy/dt

• dy/dt times dx/dt

• dx/dt times dy/dt

What is the exact slope of the tangent line after rationalizing the denominator?

• Negative seven square root three divided by six

• Negative seven divided by eighteen

• Negative seven square root three divided by eighteen

• Negative seven divided by six

What is the approximate decimal value of the slope of the tangent line?

• Negative 0.674

• Negative 1.5

• Negative 0.25

• Negative 0.5

What is the significance of the orientation of the curve?

• It shows the direction in which the curve is traced as t increases

• It determines the length of the curve

• It indicates the steepness of the curve

• It helps in finding the area under the curve