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What is the limit definition of the derivative at a point a?

$f'(a) = \lim_{{h \to 0}} \frac{{f(a + h) - f(a)}}{h}$

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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.

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.

All Flashcards

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.

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.