All Flashcards
What is a derivative?
The instantaneous rate of change of a function at a specific point.
What does f'(x) represent?
The derivative of the function f(x).
What is the limit definition of a derivative?
The derivative at a point a is defined as:
What is a tangent line?
A line that touches a curve at a single point, representing the derivative at that point.
What is a secant line?
A line that crosses a curve at two points.
Define instantaneous rate of change.
The rate of change of a function at a specific instant in time.
What does 'estimating derivatives' mean?
Approximating the value of a derivative when an exact formula is not available or easily calculable.
What is the role of 'h' in the limit definition?
'h' represents a small change in x, approaching zero, used to find the instantaneous rate of change.
What are units in derivative interpretation?
Units are essential and reflect the rate of change of the function's output with respect to its input.
What is radian mode?
Radian mode is a setting on calculators used for trigonometric functions, measuring angles in radians instead of degrees.
What are the differences between secant and tangent lines?
Secant: intersects curve at two points | Tangent: touches curve at one point, represents derivative.
What is the difference between average and instantaneous rate of change?
Average: slope of secant line | Instantaneous: slope of tangent line, the derivative.
What is the difference between estimating derivatives by hand versus using technology?
By hand: uses limit definition or nearby points, approximate | Technology: uses algorithms, more accurate.
Compare estimating derivatives using symmetric vs non-symmetric points.
Symmetric: more accurate approximation | Non-symmetric: less accurate, but still useful.
What is the difference between estimating derivatives from a table and from a graph?
Table: Uses discrete data points to calculate slope | Graph: Requires drawing a tangent line and estimating its slope visually.
Compare the accuracy of estimating derivatives with larger vs. smaller 'h' values.
Larger 'h': Less accurate estimation | Smaller 'h': More accurate estimation, approaching the true derivative.
Compare estimating derivatives for linear vs. non-linear functions.
Linear: Derivative is constant, easy to estimate | Non-linear: Derivative varies, requires more careful estimation.
What is the difference between the limit definition and the power rule for finding derivatives?
Limit definition: fundamental, works for all functions | Power rule: shortcut, only applies to power functions.
Compare the use of calculators versus software (like Desmos) for estimating derivatives.
Calculators: Portable, good for quick calculations | Desmos: Visual, good for graphing and exploring functions.
Compare the use of secant lines and tangent lines for estimating derivatives.
Secant lines: approximate average rate of change | Tangent lines: approximate instantaneous rate of change.
What does a steep tangent line on a graph indicate?
A large magnitude derivative, indicating a rapid rate of change.
What does a horizontal tangent line indicate?
A derivative of zero, indicating a stationary point (local max/min).
What does the sign of the derivative tell you about a function's graph?
Positive: increasing, Negative: decreasing, Zero: horizontal tangent.
How can you identify points where the derivative is undefined on a graph?
Look for sharp corners, cusps, or vertical tangent lines.
What does a secant line approaching a tangent line visually represent?
It represents the process of finding the instantaneous rate of change as the interval shrinks to zero.
How to estimate the derivative at a point from a graph?
Draw a tangent line at the point and determine its slope.
What does a tangent line with a positive slope indicate?
The function is increasing at that point.
What does a tangent line with a negative slope indicate?
The function is decreasing at that point.
What does the x-intercept of the derivative's graph represent?
A point where the original function has a horizontal tangent (critical point).
How can you visually estimate where a function has its maximum rate of change?
Look for the steepest part of the graph where the tangent line would have the greatest slope magnitude.