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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.
What is the limit definition of the derivative at a point *a*?
$f'(a) = \lim_{{h \to 0}} \frac{{f(a + h) - f(a)}}{h}$
How to estimate f'(a) using nearby points?
$f'(a) \approx \frac{f(a + h) - f(a)}{h}$ for a small h
What is the formula to estimate the derivative using two points?
$f'(x) \approx \frac{f(x_2) - f(x_1)}{x_2 - x_1}$
How do you estimate a derivative using symmetric points around 'a'?
$f'(a) \approx \frac{f(a + h) - f(a - h)}{2h}$
What's the formula for approximating f'(2.25) using f(2) and f(2.5)?
$f'(2.25) \approx \frac{f(2.5) - f(2)}{2.5 - 2}$
What is the numerical derivative command on TI-Nspire for $f'(x)$ at $x=a$?
$\frac{d}{dx}(f(x))|_{x=a}$
What is the formula to estimate the derivative using the values from a table?
Choose two points close to the desired point. Use the slope formula: $\frac{\Delta y}{\Delta x}$
How to estimate the derivative graphically?
Draw a tangent line at the point of interest and calculate its slope: $\frac{rise}{run}$
Formula for the slope of a secant line?
$\frac{f(x_2) - f(x_1)}{x_2 - x_1}$
What is the formula for midpoint Riemann sum?
$\Delta x [f(x_1) + f(x_2) + ... + f(x_n)]$, where $x_i$ are the midpoints of the subintervals.
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: $\lim_{{h \to 0}} \frac{{f(a + h) - f(a)}}{h}$
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.