All Flashcards
What is a limit?
The value that a function approaches as the input approaches some value.
What is instantaneous rate of change?
The rate of change of a function at a specific point, found using limits.
What is a secant line?
A line that intersects a curve at two or more points.
What is a tangent line?
A line that touches a curve at only one point, representing the instantaneous rate of change.
What is the average rate of change?
The slope of the secant line between two points on a curve.
Explain the concept of limits.
Limits describe the behavior of a function as the input approaches a specific value. They are fundamental to calculus.
Explain instantaneous rate of change.
It's the rate of change at a single point, found by taking the limit of the average rate of change as the interval approaches zero.
Explain why limits matter.
Limits allow us to analyze function behavior at points where the standard rate of change formula would be undefined.
Explain the relationship between secant and tangent lines.
As the distance between two points on a curve approaches zero, the secant line approaches the tangent line.
Explain the connection between limits and derivatives.
The derivative of a function at a point is defined as the limit of the difference quotient as the change in x approaches zero.
What does the graph of a function tell us about its limit as x approaches a certain value?
The graph shows the y-value that the function approaches as x gets closer and closer to the specified value. Discontinuities can affect the existence of the limit.