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 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.

How do you estimate f'(a) from a table of values?

Choose points close to 'a'. 2. Calculate slope using $\frac{\Delta y}{\Delta x}$ . 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 $\frac{f(a+h)-f(a)}{h}$ .

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 $f'(3)$ given values at $x=2$ and $x=4$ ?

Use values at x=2 and x=4. 2. Calculate $\frac{f(4)-f(2)}{4-2}$ . 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 $f(x) = cos(\frac{3x+2}{x})$ at $x=2$ 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 $f(midpoint) * \Delta x$ .

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.

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