All Flashcards
How to find the direction of motion at a given time?
- 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?
- 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]?
- Calculate (x(t2) - x(t1)) / (t2 - t1) for x(t). 2. Calculate (y(t2) - y(t1)) / (t2 - t1) for y(t).
Explain how x(t) relates to horizontal motion.
If x(t) is increasing, the object moves right. If x(t) is decreasing, the object moves left.
Explain how y(t) relates to vertical motion.
If y(t) is increasing, the object moves upward. If y(t) is decreasing, the object moves downward.
Explain why different parametric equations can describe the same curve.
The parameter 't' can be scaled or shifted, changing the velocity and acceleration, but not the path itself.
Explain the relationship between position, velocity, and acceleration vectors.
Velocity is the rate of change of position, and acceleration is the rate of change of velocity. They describe the motion of an object.
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.