Exploring Second Derivatives of Parametric Equations

What is the main topic of this lesson?

• Second derivative of parametric equations

• Integration of parametric equations

• Limits of parametric equations

• First derivative of parametric equations

What is the first step in finding the second derivative of a parametric equation?

• Find the second derivative of y with respect to x

• Find the first derivative of y with respect to x

• Find the first derivative of y with respect to t

• Find the second derivative of x with respect to t

What is a common mistake when finding the second derivative?

• Not using the chain rule

• Incorrectly simplifying the first derivative

• Forgetting to include x prime of t in the denominator

• Forgetting to take the derivative of y with respect to t

In the first example, what is the simplified form of dy/dx?

• t^(3/2)

• 2t^(3/2)

• t^(1/2)

• 2t^(1/2)

What is the second derivative of y with respect to x in the first example?

• 3t

• 3t^(1/2)

• 6t^(1/2)

• 6t

In the second example, what is the simplified form of dy/dx?

• 4/3 cot(t)

• -4/3 tan(t)

• -4/3 cot(t)

• 4/3 tan(t)

What is the second derivative of y with respect to x in the second example?

• -4/9 sec^3(t)

• 4/9 sec^3(t)

• 4/9 cosec^3(t)

• -4/9 cosec^3(t)

Why is it important to simplify expressions when finding the second derivative?

• To avoid using the chain rule

• To make it easier to take the second derivative

• To avoid using the product rule

• To make it easier to integrate

What is the second derivative of y with respect to x when t = 1 in the third example?

• 2/3

• 2/9

• 1/3

• 1/9

What does a positive second derivative indicate about the concavity of the function?

• The function is linear

• The function is concave up

• The function has a point of inflection

• The function is concave down