Glossary
Acceleration
The rate of change of velocity with respect to time, indicating how quickly an object's velocity is changing (speeding up, slowing down, or changing direction).
Example:
When a rocket launches, its increasing speed means it has positive acceleration; when it lands, its decreasing speed implies negative acceleration.
Derivative
The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function's graph at that point.
Example:
The derivative of a car's position function with respect to time gives its instantaneous velocity.
Estimating Derivatives By Hand
The process of approximating the derivative of a function at a point using nearby data points, typically from a table, by calculating the slope between two close points.
Example:
Given a table of a city's population over several years, you can estimate derivatives by hand to approximate the rate of population growth in a specific year.
Estimating Derivatives Graphically
The method of approximating a derivative by drawing a tangent line to the function's graph at the point of interest and then calculating the slope of that drawn line.
Example:
To find the steepest point on a mountain trail from its elevation profile graph, you would be estimating derivatives graphically by identifying where the tangent line has the largest slope.
Estimating Derivatives with Technology
The use of graphing calculators or software (like Desmos) to numerically compute or approximate the derivative of a function at a given point.
Example:
When you need to quickly verify your manual calculations for a complex function, estimating derivatives with technology can provide a rapid numerical check.
Instantaneous Rate of Change
The rate at which a function's value is changing at a single, specific moment or point, as opposed to an average rate over an interval.
Example:
Your car's speedometer displays your instantaneous rate of change of position, which is your speed at that exact moment.
Interpretation (of derivative)
Explaining the real-world meaning of a derivative's value, including its sign and units, within the context of the given problem scenario.
Example:
An interpretation of for a population function might be: 'At 10 years, the population is increasing at a rate of 50 individuals per year.'
Limit Definition of a Derivative
A fundamental formula that defines the derivative of a function f(x) at a point 'a' as the limit of the difference quotient: f'(a) = lim (h->0) [f(a+h) - f(a)] / h.
Example:
Using the Limit Definition of a Derivative, you can formally prove that the derivative of is .
Midpoint Riemann Sum
A numerical method for approximating the definite integral of a function by dividing the area under the curve into rectangles whose heights are determined by the function's value at the midpoint of each subinterval.
Example:
To estimate the total distance traveled by a car given its velocity at discrete time points, you could use a Midpoint Riemann Sum for a more accurate approximation than left or right sums.
Radian Mode
A calculator setting that interprets angle measurements in radians rather than degrees, which is crucial for accurate calculations involving trigonometric functions in calculus.
Example:
Always ensure your calculator is in Radian Mode when evaluating derivatives or integrals of trigonometric functions like or to avoid incorrect results.
Secant Line
A straight line that intersects a curve at two or more distinct points.
Example:
The average rate of change of a function between two points is represented by the slope of the secant line connecting those two points on the graph.
Tangent Line
A straight line that touches a curve at exactly one point and has the same slope as the curve at that specific point.
Example:
When a ball is thrown, its instantaneous direction of motion at any point in its trajectory is along the tangent line to its path.
Units (in interpretation)
The specific measurements (e.g., meters/second, degrees Celsius/minute) that must be included with numerical answers to provide meaningful context in real-world calculus problems.
Example:
If a derivative represents the rate of change of water volume in a tank with respect to time, the units for the derivative would be liters per minute (L/min).