Define velocity vector.
A vector showing the direction and speed of an object at a specific moment.
Define acceleration vector.
A vector indicating how the velocity of an object is changing.
Define position vector.
A vector that specifies the location of a point with respect to a reference origin.
What is a parametric function?
A function where the x and y coordinates are defined in terms of a third variable, usually 't'.
Define average rate of change.
The average change in a function's value over a specific interval.
How to find the direction of motion at a given time?
1. Find x'(t) and y'(t). 2. Evaluate x'(t) and y'(t) at the given time. 3. Determine if x'(t) and y'(t) are positive or negative to determine the direction.
How to find the slope of a parametric curve at a point?
1. Find dy/dt and dx/dt. 2. Calculate (dy/dt) / (dx/dt). 3. Evaluate the expression at the given point (t-value).
How to calculate average rate of change for x(t) and y(t) over [t1, t2]?
1. Calculate (x(t2) - x(t1)) / (t2 - t1) for x(t). 2. Calculate (y(t2) - y(t1)) / (t2 - t1) for y(t).
What is the formula for average rate of change?
$$\frac{f(t_2) - f(t_1)}{t_2 - t_1}$$
What is the formula for the slope of a parametric curve?
$$\frac{dy/dt}{dx/dt}$$
Formula for slope using average rates of change?
$$\frac{\Delta y}{\Delta x} = \frac{y(t_2) - y(t_1)}{x(t_2) - x(t_1)}$$