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

The derivative of the first derivative of a function.

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

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

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

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

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

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

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

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

1. Find $\frac{dx}{dt}$ and $\frac{dy}{dt}$. 2. Find $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$. 3. Find $\frac{d}{dt}(\frac{dy}{dx})$. 4. Divide by $\frac{dx}{dt}$.

1. Find $\frac{d^2y}{dx^2}$. 2. Set $\frac{d^2y}{dx^2} > 0$. 3. Solve for t.

1. $\frac{dx}{dt} = 3t^2$, $\frac{dy}{dt} = 4t^3$. 2. $\frac{dy}{dx} = \frac{4}{3}t$. 3. $\frac{d}{dt}(\frac{dy}{dx}) = \frac{4}{3}$. 4. $\frac{d^2y}{dx^2} = \frac{4}{9t^2}$.

1. Find $\frac{dx}{dt} = 3t^2 - 3$ and $\frac{dy}{dt} = 2t + 2$. 2. Find $\frac{dy}{dx} = \frac{2}{3(t-1)}$. 3. Find $\frac{d}{dt}(\frac{dy}{dx}) = -\frac{2}{3(t-1)^2}$. 4. $\frac{d^2y}{dx^2} = \frac{-2}{9(t-1)(t^2-1)}$.

1. Find $\frac{dx}{dt} = 2 - 2cos(t)$ and $\frac{dy}{dt} = 2sin(t)$. 2. Find $\frac{dy}{dx} = \frac{sin(t)}{1 - cos(t)}$. 3. Find $\frac{d^2y}{dx^2} = \frac{-1}{2(1 - cos(t))^2}$. 4. Since the second derivative is always negative, the cycloid is always concave down.

1. Expand: $\frac{cos(t) - cos^2(t) - sin^2(t)}{(1 - cos(t))^2}$. 2. Use $sin^2(t) + cos^2(t) = 1$: $\frac{cos(t) - 1}{(1 - cos(t))^2}$. 3. Simplify: $\frac{-1}{1 - cos(t)}$.

Find the first derivatives: $\frac{dx}{dt} = -sin(t)$ and $\frac{dy}{dt} = cos(t)$.

1. Find $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$. 2. Evaluate $\frac{dy}{dx}$ at the given value of t.

1. Find $\frac{dy}{dt}$. 2. Set $\frac{dy}{dt} = 0$ and solve for t. 3. Ensure $\frac{dx}{dt}$ is not also zero at those t-values.

1. Find $\frac{dx}{dt}$. 2. Set $\frac{dx}{dt} = 0$ and solve for t. 3. Ensure $\frac{dy}{dt}$ is not also zero at those t-values.

$\frac{dy}{dx}$: Find $\frac{dy/dt}{dx/dt}$. $\frac{d^2y}{dx^2}$: Find the derivative of $\frac{dy}{dx}$ with respect to t, then divide by $\frac{dx}{dt}$.

Concave up: second derivative > 0. Concave down: second derivative < 0.

First derivative: slope of the tangent line. Second derivative: concavity.

Parametric: use $\frac{dy/dt}{dx/dt}$. Standard: direct differentiation with respect to x.

Horizontal: set $\frac{dy}{dt} = 0$. Vertical: set $\frac{dx}{dt} = 0$.

Quotient rule: derivative of a ratio. Chain rule: derivative of a composite function.

$\frac{dy}{dx}$ is the rate of change of y with respect to x, while $\frac{dx}{dy}$ is the rate of change of x with respect to y. They are reciprocals of each other.

First derivative test: finds local max/min using sign changes of f'(x). Second derivative test: uses the sign of f''(x) to determine concavity and local extrema.

Explicit: y is directly defined as a function of x (y = f(x)). Implicit: y is defined implicitly in terms of x (e.g., x^2 + y^2 = 1).

Local: max/min within a specific interval. Global: absolute max/min over the entire domain.