Lesson 6: 4-6 Approximating Values of a Function Using Local Linearity and Linearization
Duration of Days: 1
Lesson Objective
Understand the concept of local linearity: that a differentiable function can be approximated by a tangent line near the point of tangency.
Write the equation of a tangent line at a specific point (a, f(a)) and use it as a linear approximation function L(x).
Use L(x) to approximate values of a function that are difficult to calculate.
Use the second derivative (concavity) to determine if a linear approximation is an overestimate or an underestimate.
Why is a tangent line a better approximation for f(x) the closer you are to the point of tangency?
How does the "bend" (concavity) of a graph determine where the tangent line sits in relation to the curve?
If we don't know the actual value of a function, how can we be certain our estimate is "too high" or "too low"?
Linearization L(x)
Local Linearity
Tangent Line
Approximation
Point of Tangency
Concavity (Up/Down)
Overestimate / Underestimate
Differentiability
Topic 4.6: Approximating Values of a Function Using Local Linearity and Linearization.
Standard CHA-3.F: Explain the relationship between the differentiability of a function and the applicability of local linearity.
Mathematical Practice 2.E: Describe the behavior of a function based on the signs of its derivatives.
Description:
Students learn to construct the "Linearization" formula: L(x) = f(a) + f'(a)(x - a). This is simply the point-slope form of a tangent line rearranged. The lesson emphasizes that for values of x very close to a, f(x) approximate L(x). Students then use the sign of f''(x) to justify the accuracy of their estimate.
Purpose: In the "No Calculator" section of the AP Exam, students are often asked to approximate a value. This lesson provides the algebraic tool to do so. It also forces students to connect the first derivative (slope) with the second derivative (concavity) for a conceptual justification.
DOK 1 (Recall): Writing the equation of a tangent line given f(a) and f'(a).
DOK 2 (Skill/Concept): Using the tangent line to estimate a specific value.
DOK 3 (Strategic Thinking): Justifying whether an estimate is an over or under-approximation by analyzing f''(x).
Scaffolding (Support):The "Zoom" Visual: Show a digital graph of y = sin(x) and zoom in repeatedly at the origin until the curve is indistinguishable from the line y = x.
Step-by-Step Recipe:
Find the point: (a, f(a)).
Find the slope: f'(a)
Write the line: $y - y_1 = m(x - x_1)
Plug in the "ugly" number for x.
Extension (Challenge):Error Analysis: Introduce the idea that the "error" in the approximation grows as we move further from a.
Physics Application: Discuss why physicists often use the "small-angle approximation" in pendulum equations—it's just linearization!
The "Over/Under" Trick
If f''(x) > 0 (Concave Up), the curve stays above the tangent line. Therefore, the linear approximation is an underestimate.
If $f''(x) < 0$ (Concave Down), the curve stays below the tangent line. Therefore, the linear approximation is an overestimate.
AP College board Assessments