This study guide covers local linearity and linearization to approximate function values using tangent lines. It explains how to use the point-slope form and linearization formula to construct tangent line equations. The guide also discusses how to determine overestimations and underestimations based on concavity. Finally, it provides practice questions and solutions, including AP-style examples, covering how to find tangent lines, approximate function values, and determine over/underestimation.

#Approximating Function Values with Local Linearity and Linearization 🚀

Imagine zooming in super close on a curve until it looks almost perfectly straight. That's the idea behind local linearity! We use tangent lines to approximate function values near a known point. It's like using a straight path to guess where a winding trail goes. Let's dive in!

#↗️ Linearization and Tangent Line Approximation

Remember tangent lines from Unit 2? The slope of the line tangent to a graph at a point is the function's derivative at that point. We use this slope and the point's coordinates to build the tangent line equation. This equation is our linearization.

#Point-Slope Form

The most reliable way to build the tangent line equation is using the point-slope formula:

$y - y_1 = m(x - x_1)$

Where:

$(x_1, y_1)$ is the point of tangency
$m$ is the slope of the tangent line (the derivative at $x_1$ )

#Linearization Formula

While you might see the linearization formula, it's just a fancy version of point-slope. You can use it if you like, but point-slope is your friend!

$L(x) = f(a) + f'(a)(x - a)$

$L(x)$ is the linearization of $f(x)$ at $x=a$
$f(a)$ is the function value at $x=a$
$f'(a)$ is the derivative of the function at $x=a$

Key Concept

Essentially, both formulas do the same thing: they build a line that closely approximates the function near a specific point. Point-slope is often easier to remember!

#How to Approximate

Find the point of tangency: $(x_1, y_1)$
Calculate the derivative $f'(x)$ and evaluate it at $x_1$ to find the slope, $m = f'(x_1)$ .
Build the tangent line equation...

#Approximating Function Values with Local Linearity and Linearization 🚀

#↗️ Linearization and Tangent Line Approximation

#Point-Slope Form

The most reliable way to build the tangent line equation is using the point-slope formula:

$y - y_1 = m(x - x_1)$

Where:

$(x_1, y_1)$ is the point of tangency
$m$ is the slope of the tangent line (the derivative at $x_1$ )

#Linearization Formula

While you might see the linearization formula, it's just a fancy version of point-slope. You can use it if you like, but point-slope is your friend!

$L(x) = f(a) + f'(a)(x - a)$

$L(x)$ is the linearization of $f(x)$ at $x=a$
$f(a)$ is the function value at $x=a$
$f'(a)$ is the derivative of the function at $x=a$

Key Concept

Essentially, both formulas do the same thing: they build a line that closely approximates the function near a specific point. Point-slope is often easier to remember!

#How to Approximate

Find the point of tangency: $(x_1, y_1)$
Calculate the derivative $f'(x)$ and evaluate it at $x_1$ to find the slope, $m = f'(x_1)$ .
Build the tangent line equation...

If $f(x) = x^2$ , what is the slope of the tangent line at $x = 3$ ? 🤔

Approximating Values of a Function Using Local Linearity and Linearization

Hannah Hill

#Approximating Function Values with Local Linearity and Linearization 🚀

#↗️ Linearization and Tangent Line Approximation

#Point-Slope Form

#Linearization Formula

#How to Approximate

How are we doing?

Approximating Values of a Function Using Local Linearity and Linearization

Hannah Hill

#Approximating Function Values with Local Linearity and Linearization 🚀

#↗️ Linearization and Tangent Line Approximation

#Point-Slope Form

#Linearization Formula

#How to Approximate

How are we doing?