Define parametric equations.

Functions where independent functions x and y are connected via a parameter t.

Flip to see [answer/question]

Revise later

SpaceTo flip

If confident

All Flashcards

Define parametric equations.

Functions where independent functions x and y are connected via a parameter t.

What is a second derivative?

The derivative of the first derivative of a function.

What is $\frac{d^2y}{dx^2}$ ?

Notation for the second derivative of y with respect to x.

Define concavity.

The direction of the curve of a function (upward or downward).

What is a cycloid?

A curve traced by a point on a circle as it rolls along a straight line.

Define $\frac{dy}{dx}$ for parametric equations.

The first derivative of y with respect to x, found by $\frac{dy/dt}{dx/dt}$ .

What does the second derivative indicate?

The rate of change of the slope of a function; indicates concavity.

What is the quotient rule?

A rule for finding the derivative of a function that is the ratio of two other functions.

What is a parameter?

An independent variable (often denoted by 't') that relates two dependent variables.

What does it mean for a curve to be concave down?

The curve bends downwards, and its second derivative is negative.

If the graph of a parametric curve is concave up, what does this indicate about the second derivative?

The second derivative is positive.

How can you visually identify concavity on a graph of a parametric curve?

Concave up: the curve opens upwards. Concave down: the curve opens downwards.

What does a point of inflection on a parametric curve's graph indicate about the second derivative?

The second derivative changes sign at that point.

How does the graph of a cycloid relate to its second derivative?

The graph is always concave down, corresponding to a negative second derivative.

What does a horizontal tangent on a parametric curve indicate about $\frac{dy}{dt}$ ?

$\frac{dy}{dt} = 0$ at that point (assuming $\frac{dx}{dt}$ is not also zero).

What does a vertical tangent on a parametric curve indicate about $\frac{dx}{dt}$ ?

$\frac{dx}{dt} = 0$ at that point (assuming $\frac{dy}{dt}$ is not also zero).

How can you use a graph to estimate the sign of the second derivative at a given point?

Observe the concavity: upward curve suggests positive, downward suggests negative.

If the graph of $\frac{dy}{dx}$ is increasing, what does this imply about $\frac{d^2y}{dx^2}$ ?

$\frac{d^2y}{dx^2}$ is positive.

If the graph of $\frac{dy}{dx}$ is decreasing, what does this imply about $\frac{d^2y}{dx^2}$ ?

$\frac{d^2y}{dx^2}$ is negative.

How does the graph of a parametric equation help visualize the relationship between x, y, and t?

It shows the path traced by the point (x(t), y(t)) as t varies.

Formula for $\frac{d^2y}{dx^2}$ in parametric form?

$\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}$

How to find $\frac{dy}{dx}$ given x(t) and y(t)?

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

What is the trigonometric identity for $sin^2(t) + cos^2(t)$ ?

$sin^2(t) + cos^2(t) = 1$

Formula for the derivative of a quotient $\frac{u}{v}$ ?

$\frac{d}{dx}(\frac{u}{v}) = \frac{v(\frac{du}{dx}) - u(\frac{dv}{dx})}{v^2}$

Given $x(t)$ and $y(t)$ , how to find the slope of the tangent line?

Calculate $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ and evaluate at the given t.

What is the formula for $dx/dt$ if $x(t) = 2(t - sin(t))$ ?

$\frac{dx}{dt} = 2 - 2cos(t)$

What is the formula for $dy/dt$ if $y(t) = 2(1 - cos(t))$ ?

$\frac{dy}{dt} = 2sin(t)$

What is the formula for $\frac{d}{dt}(\frac{2}{3(t-1)})$ ?

$\frac{d}{dt}(\frac{2}{3(t-1)}) = -\frac{2}{3(t-1)^2}$

If $\frac{dy}{dx} = \frac{4}{3}t$ , what is $\frac{d}{dt}(\frac{dy}{dx})$ ?

$\frac{d}{dt}(\frac{dy}{dx}) = \frac{4}{3}$

What is the formula for the second derivative of $x = t^3$ and $y = t^4$ ?

$\frac{d^2y}{dx^2} = \frac{4}{9t^2}$