All Flashcards
How do you estimate f'(a) from a table of values?
- Choose points close to 'a'. 2. Calculate slope using . 3. Interpret with units.
Steps to estimate a derivative graphically?
- Draw tangent line at the point. 2. Find two points on the tangent line. 3. Calculate the slope.
How to estimate derivative using limit definition?
- Identify f(x) and 'a'. 2. Choose a small 'h'. 3. Calculate .
Steps to estimate derivative with TI-Nspire?
- Menu > Calculus > Numerical Derivative. 2. Input function and point. 3. Read the result.
How to interpret an estimated derivative?
- State the point of interest. 2. Give the rate of change. 3. Include units.
How to estimate given values at and ?
- Use values at x=2 and x=4. 2. Calculate . 3. Simplify.
How to find the slope of a tangent line from a graph?
- Identify two clear points on the tangent line. 2. Calculate the rise and run. 3. Divide rise by run.
How to estimate the derivative of at using Desmos?
- Input the function. 2. Type f'(2). 3. Read the result.
How do you estimate the total distance traveled using a midpoint Riemann sum?
- Divide the interval into subintervals. 2. Find the midpoint of each subinterval. 3. Calculate the sum of .
How do you check if acceleration is negative from a table?
- Look for decreasing velocity values in the table. 2. If velocity decreases, acceleration is negative.
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.