Calculating Derivatives and Tangent Lines

The video tutorial explains how to find the slope of a tangent line to a parametric curve defined by x = 4 cos(T) and y = 2 sin(T) at T = π/6 radians. It covers evaluating the point of tangency using parametric equations, calculating trigonometric values using reference triangles and the unit circle, and determining the slope of the tangent line by evaluating dydx. The tutorial concludes with a graphical representation of the tangent line and its slope.

What is the first step in finding the slope of the tangent line to a parametric curve?

• Find the second derivative of the curve.

• Evaluate the parametric equations at a given point.

• Calculate the derivative of y with respect to x.

• Determine the point of intersection with the x-axis.

How do you find the x-coordinate of the point of tangency for the given parametric equations?

• By evaluating x at T = Pi/6.

• By using the midpoint formula.

• By finding the derivative of x with respect to y.

• By setting y equal to zero.

Which method can be used to find trigonometric values for angles like Pi/6?

• Using a calculator.

• Using a reference triangle or unit circle.

• Using the Pythagorean theorem.

• Using the quadratic formula.

What is the derivative of 2 sin T with respect to T?

• -2 cos T

• 2 cos T

• -2 sin T

• 2 sin T

What is the derivative of 4 cos T with respect to T?

• 4 sin T

• -4 sin T

• 4 cos T

• -4 cos T

How do you simplify the expression for dydx in terms of trigonometric functions?

• By dividing dydt by dxdt.

• By multiplying dydt by dxdt.

• By adding dydt and dxdt.

• By subtracting dxdt from dydt.

What is the slope of the tangent line at T = Pi/6?

• 2

• 1

• 0.5

• Square root of 3 divided by 2

What is the approximate decimal value of the slope of the tangent line at T = Pi/6?

• 0.5

• 2.0

• 1.732

• 0.866

What is the significance of the orientation of the curve as T increases?

• It determines the direction of the tangent line.

• It affects the length of the tangent line.

• It changes the slope of the tangent line.

• It has no effect on the tangent line.

What tool can be used to graph the tangent line more accurately?

• Graph paper

• A protractor

• A ruler

• Graphing software