Glossary
Acceleration ($a$)
The vector quantity representing the rate of change of an object's velocity with respect to time, indicating how quickly velocity is changing.
Example:
When a rocket launches, its Acceleration is immense, causing its velocity to increase rapidly.
Amount (Function describing an amount)
A function that directly represents a quantity or total value, rather than how that quantity is changing.
Example:
If V(t) gives the amount of water in a tank at time t, then V(t) would be measured in liters.
Chain Rule
A fundamental calculus rule used to find the derivative of a composite function, essential for linking different rates of change in related rates problems.
Example:
If you know how fast your position changes with respect to time, and how fast your happiness changes with respect to your position, the Chain Rule helps you find how fast your happiness changes with respect to time.
Constant vs. Changing Variables
The distinction between quantities that maintain a fixed value throughout a problem and those whose values are dynamic and change over time.
Example:
In a problem involving a conical tank being filled, the tank's overall height and radius are constant variables, but the water's height and radius are changing variables.
Dependent Variable
The output variable in a function whose value depends on the input variable.
Example:
In the function y = 2x + 5, y is the dependent variable as its value depends on x.
Derivative
A fundamental concept in calculus that measures how a function changes as its input changes, representing the instantaneous rate of change.
Example:
When analyzing the motion of a rocket, the derivative of its position function gives its velocity.
Diagrams
Visual representations used to illustrate the physical setup of a problem, helping to identify relationships between variables and rates.
Example:
When solving a problem about a shadow cast by a moving object, drawing a diagram helps visualize similar triangles and set up the correct equations.
Displacement ($s$)
The vector quantity representing an object's change in position from a fixed reference point, including both magnitude and direction.
Example:
If you walk 5 meters east and then 5 meters west, your final Displacement from your starting point is zero.
Displacement-time graph
A graph that plots an object's displacement on the y-axis against time on the x-axis, where the slope of the curve represents the object's velocity.
Example:
Analyzing the slope of a Displacement-time graph can tell you if a runner is speeding up, slowing down, or moving at a constant pace.
Distance ($d$)
The scalar quantity representing the total path length traveled by an object, always positive or zero.
Example:
Even if you return to your starting point, the total Distance you walked will be the sum of all segments of your journey.
Implicit Differentiation
A technique used to differentiate equations where one variable is not explicitly expressed as a function of another, often applied when variables are implicitly related.
Example:
To find how the radius of a circle changes with respect to its area, even though the equation A = πr² isn't solved for r, you'd use Implicit Differentiation with respect to time.
Independent Variable
The input variable in a function whose value can be chosen freely, determining the value of the dependent variable.
Example:
In the function y = 2x + 5, x is the independent variable as you choose its value to find y.
Initial (or Initially)
A term used to refer to the state or position of a particle at the very beginning of its motion, specifically when time ($t$) is equal to zero.
Example:
To find the starting point of a projectile, you would look at its Initial displacement at t=0.
Instantaneous Rate of Change
The rate at which a function is changing at a specific, single point in time or at a particular input value.
Example:
The speedometer in a car shows the instantaneous rate of change of distance with respect to time, which is the car's speed at that exact moment.
Integral
The inverse operation of differentiation, used to find the total accumulation of a quantity given its rate of change, or the area under a curve.
Example:
If you know the rate of flow of water into a pool, taking the integral over a time interval will tell you the total volume of water added.
Particle
An idealized object in physics, assumed to be a single point in space, meaning its physical dimensions are disregarded.
Example:
When analyzing the trajectory of a thrown ball, we often treat it as a Particle to simplify calculations.
Product Rule
A rule in calculus used to find the derivative of a function that is the product of two or more differentiable functions.
Example:
If you're tracking the area of a rectangle where both length and width are changing over time, you'd use the Product Rule to find the rate of change of the area.
Pythagoras' Theorem
A geometric theorem stating that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (a² + b² = c²).
Example:
When a ladder slides down a wall, the relationship between its height on the wall, its distance from the wall, and its length is governed by Pythagoras' Theorem.
Quotient Rule
A rule in calculus used to find the derivative of a function that is the ratio of two differentiable functions.
Example:
If you have a function describing the concentration of a chemical over time, which is a ratio of two changing quantities, the Quotient Rule would help you find its rate of change.
Rate (Function describing a rate)
A function that represents how quickly a quantity is changing over time or with respect to another variable.
Example:
If R(t) gives the rate at which water flows into a tank, then R(t) would be measured in liters per minute.
Rate of Change
Describes how one quantity (dependent variable) changes in relation to another quantity (independent variable).
Example:
The rate of change of a population might be measured in individuals per year.
Related Rates Problems
Problems that involve finding the rate at which one quantity changes with respect to time, given the rate at which another quantity changes with respect to time.
Example:
Imagine a spherical balloon being inflated; a related rates problem would be finding how fast its radius is growing given the rate at which its volume increases.
Second Derivative
The derivative of the first derivative of a function, representing the rate of change of the rate of change (e.g., acceleration if the first derivative is velocity).
Example:
If s(t) is position, s'(t) is velocity, and s''(t) is acceleration, then s''(t) is the second derivative of position.
Slope of the Tangent
The slope of the straight line that touches a curve at exactly one point, representing the instantaneous rate of change (derivative) of the curve at that point.
Example:
To find the velocity of a ball thrown upwards at a specific time, you'd calculate the slope of the tangent to its height-time graph at that time.
Speed
The scalar quantity representing the magnitude of an object's velocity, indicating how fast an object is moving without regard to direction.
Example:
Whether a car is going 100 km/h forward or 100 km/h backward, its Speed is the same.
Straight Line Motion
A model describing how objects move along a single, linear path, which can have both positive and negative directions.
Example:
Imagine a train moving back and forth on a single track; this is an example of Straight Line Motion.
Time ($t$)
A fundamental quantity representing the progression of events, typically measured in seconds, against which motion variables like displacement, velocity, and acceleration are functions.
Example:
To calculate how far a car travels, you need to know the Time it was in motion.
Velocity ($v$)
The vector quantity representing the rate of change of an object's displacement with respect to time, indicating both speed and direction.
Example:
A car moving at 60 km/h north has a different Velocity than one moving at 60 km/h south.
Velocity-time graph
A graph that plots an object's velocity on the y-axis against time on the x-axis, where the slope of the curve represents the object's acceleration.
Example:
A flat line on a Velocity-time graph indicates constant velocity, meaning zero acceleration.