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Linearization & Differentials

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Linearization & Differentials - Complete Interactive Lesson

Part 1: The Tangent Line Approximation

Linearization & Differentials

Part 1 of 7 — The Tangent Line Approximation

Topic Overview

Part	Topic
1	Tangent line approximation
2	Approximating values
3	Differentials
4	Error analysis
5	Applications & related rates
6	AP-style workshop
7	Comprehensive assessment

Local Linearization Formula

$\boxed{L(x) = f(a) + f'(a)(x - a)}$

Component	Meaning
$a$	Base point (choose a "nice" value)
$f(a)$	Known function value at $a$
$f'(a)$

Worked Example

Approximate $\sqrt{4.1}$ using linearization.

$f(x) = \sqrt{x}$ , $a = 4$ (nearest perfect square).

$f(4) = 2$ , $f'(x) = \frac{1}{2\sqrt{x}}$ ,

$L(x) = 2 + \frac{1}{4}(x-4)$

Actual: $\sqrt{4.1} \approx 2.02485$ . Error $\approx 0.00015$ !

Key Fact: Choose $a$ to be a nearby value where $f(a)$ and $f'(a)$ are easy to compute. The closer $x$ is to , the better the approximation.

Practice — Linearization 🎯

Build the linearization. 🔍

Approximate. ✍️

Key Takeaways — Part 1

Linearization: $L(x) = f(a) + f'(a)(x-a)$
Choose near where and are easy

Part 2: Differentials

Linearization & Differentials

Part 2 of 7 — Approximating Values

Common Linearizations at $a = 0$

Function	Linear Approximation near $0$
$\sin x$	$\approx x$

Part 3: Over/Underestimates

Linearization & Differentials

Part 3 of 7 — Differentials

Differentials vs. Derivatives

$\boxed{dy = f'(x)\,dx}$

Part 4: Percentage Error

Linearization & Differentials

Part 4 of 7 — Error Analysis

Error Terminology

Term	Symbol	Formula
Approximate change	$dy$	$f'(x)\,dx$
Exact change

Part 5: Linearization with Tables

Linearization & Differentials

Part 5 of 7 — Applications

Propagation of Error

If a measurement $x$ has uncertainty $dx$ , then the uncertainty in $f(x)$ is approximately:

$\boxed{df = |f'(x)|\,dx}$

Part 6: Problem-Solving Workshop

Linearization & Differentials

Part 6 of 7 — AP-Style Workshop

AP FRQ Patterns

Pattern	What They Ask
Table + tangent line	"Use the tangent line at $x = a$ to approximate $f(b)$ "
Over/under	"Is your estimate an over- or underestimate? Justify."
Differential	"What is $dy$ when and ?"

Part 7: Final Assessment

Linearization & Differentials

Part 7 of 7 — Comprehensive Assessment

Formula Reference

Formula	Expression
Linearization	$L(x) = f(a) + f'(a)(x-a)$

f'(4) = \frac{1}{4}

L (4.1) = 2 + \frac{1}{4} (0.1) = 2.025

The approximation improves as

x \to a

This is the tangent line at

x = a

used as an approximation

Key Fact: These linearizations appear frequently on the AP exam, especially $e^x \approx 1+x$ and $\sin x \approx x$ for small $x$ .

Over/Under Estimates from Concavity

$\boxed{\text{Concave up} \Rightarrow \text{Tangent line underestimates}}$ $\boxed{\text{Concave down} \Rightarrow \text{Tangent line overestimates}}$

Concavity at $a$	Tangent line is a...
$f''(a) > 0$ (concave up)	Underestimate
$f''(a) < 0$ (concave down)	Overestimate

Approximate $e^{0.1}$ using linearization. Is it an over- or underestimate?

$f(x) = e^x$ , $a = 0$ . $L(x) = 1 + x$ . $L(0.1) = 1.1$ .

$f''(x) = e^x > 0$ everywhere — concave up — so tangent line is an underestimate.

Actual: $e^{0.1} \approx 1.10517$ . Indeed $1.1 < 1.10517$ .

Practice — Approximations 🎯

Classify each approximation. 🔍

Calculate. ✍️

Key Takeaways — Part 2

Memorize common linearizations: $\sin x \approx x$ , $e^x \approx 1+x$ , $\ln(1+x) \approx x$
Concave up ( $f'' > 0$ ) $\Rightarrow$ underestimate
Concave down ( $f'' < 0$ ) $\Rightarrow$ overestimate
AP frequently asks "is this an over- or underestimate?"

Concept	Notation	Meaning
Derivative	$\frac{dy}{dx} = f'(x)$	Instantaneous rate of change
Differential of $y$	$dy = f'(x)\,dx$	Approximate change in $y$
Actual change	$\Delta y = f(x+\Delta x) - f(x)$	Exact change in $y$

$\Delta y \approx dy \quad \text{when } dx = \Delta x \text{ is small}$

The differential $dy$ is the change along the tangent line. The actual change $\Delta y$ is the change along the curve.

$y = x^3$ . Find $dy$ when $x = 2$ and $dx = 0.01$ .

$dy = f'(x)\,dx = 3x^2\,dx = 3(4)(0.01) = 0.12$

Actual change: $\Delta y = (2.01)^3 - 8 = 8.120601 - 8 = 0.120601$

$dy = 0.12$ vs $\Delta y \approx 0.1206$ — very close!

Key Fact: $dy$ is a linear approximation to $\Delta y$ . The smaller $dx$ , the better the approximation.

Practice — Differentials 🎯

Compare $dy$ and $\Delta y$ . 🔍

Compute the differential. ✍️

Key Takeaways — Part 3

$dy = f'(x)\,dx$ is the differential (change along tangent)
$\Delta y = f(x+dx) - f(x)$ is the actual change
$dy \approx \Delta y$ for small $dx$
The derivative $dy/dx$ is the ratio of differentials

Over/Under with Error Bounds

$\boxed{\text{Error} \approx \frac{1}{2}f''(a)(x-a)^2}$

This is the next term in the Taylor expansion. For small $(x-a)$ , the error is approximately quadratic in the displacement.

Approximate $\sqrt{26}$ using $a = 25$ . Estimate the error.

$L(26) = 5 + \frac{1}{10}(1) = 5.1$

Error $\approx \frac{1}{2}f''(25)(1)^2 = \frac{1}{2}\left(-\frac{1}{4 \cdot 125}\right) = -\frac{1}{1000} = -0.001$

So $\sqrt{26} \approx 5.1 - 0.001 = 5.099$ . Actual: $5.0990\ldots$ — extremely close!

AP Tip: The sign of $f''(a)$ tells you whether $L$ overestimates ( $f'' < 0$ ) or underestimates ( $f'' > 0$ ). The magnitude tells you how large the error is.

Practice — Error Analysis 🎯

Error classification. 🔍

Calculate the error. ✍️

Key Takeaways — Part 4

Error in linearization $\propto (x-a)^2$ (quadratic)
Concave up $\Rightarrow$ underestimate; concave down $\Rightarrow$ overestimate
Absolute error $= |\Delta y - dy|$ ; relative error $= |\Delta y - dy|/|\Delta y|$
Stay close to $a$ for smaller errors

df = ∣ f^{'} (x) ∣ d x

Worked Example: Sphere Volume

A sphere has radius $r = 5$ cm with measurement error $dr = 0.02$ cm. Estimate the error in the volume.

$V = \frac{4}{3}\pi r^3$ , $V'(r) = 4\pi r^2$

$dV = 4\pi(25)(0.02) = 2\pi \approx 6.28 \text{ cm}^3$

Relative error: $\frac{dV}{V} = \frac{4\pi r^2\,dr}{\frac{4}{3}\pi r^3} = \frac{3\,dr}{r} = \frac{3(0.02)}{5} = 0.012 = 1.2\%$

Key Fact: For $V = \frac{4}{3}\pi r^3$ , the relative error in volume is 3 times the relative error in radius: $\frac{dV}{V} = 3\frac{dr}{r}$ .

Linearization from a Table

AP problems often give a table of $f$ and $f'$ values and ask you to approximate $f$ at a nearby point:

$x$	$f(x)$	$f'(x)$
$2$	$5$	$3$
$4$	$11$	$-1$

Approximate $f(2.1)$ : $L(2.1) = f(2) + f'(2)(0.1) = 5 + 3(0.1) = 5.3$

Approximate $f(3.9)$ : $L(3.9) = f(4) + f'(4)(-0.1) = 11 + (-1)(-0.1) = 11.1$

Practice — Applications 🎯

AP table problem. 🔍

Propagation of error. ✍️

Key Takeaways — Part 5

Error propagation: $df \approx |f'(x)|\,dx$
Relative error: $df/f$ . For $V = cr^n$ : $dV/V = n\,dr/r$
Table-based linearization: $L(x) = f(a) + f'(a)(x-a)$ using given values
AP loves "approximate $f(2.1)$ given $f(2)$ and $f'(2)$ " questions

Full Worked AP Problem

The table below gives values of a twice-differentiable function $f$ .

$x$ $0$ $1$ $2$ $3$
$f(x)$ $4$ $7$ $3$ $1$
$f^{'}$

(a) Write the linearization of $f$ at $x = 2$ .

(b) Use your answer to approximate $f(1.8)$ .

(c) Given $f''(2) = -1$ , is your estimate an over- or underestimate? Justify.

Solution (a): $L(x) = f(2) + f'(2)(x-2) = 3 + (-3)(x-2) = 3 - 3(x-2)$

Solution (b): $L(1.8) = 3 - 3(1.8-2) = 3 - 3(-0.2) = 3 + 0.6 = 3.6$

Solution (c): Since $f''(2) = -1 < 0$ , $f$ is concave down near $x = 2$ . The tangent line lies above the curve, so $3.6$ is an overestimate.

AP Tip: For "justify," you must state the concavity ( $f'' < 0$ ), explain what it means (concave down), and conclude (tangent above curve = overestimate).

AP-Style Practice 🎯

Justify your reasoning. 🔍

AP Problem. ✍️

Key Takeaways — Part 6

AP FRQs often combine table data with linearization
Always justify over/underestimate with concavity
Three-part justification: sign of $f''$ , concavity, conclusion
The tangent line approximation is exact when $f$ is linear

Common Linearizations at $a = 0$

$f(x)$	$L(x)$
$\sin x$	$x$
$\cos x$	$1$
$e^x$	$1 + x$
$\ln(1+x)$	$x$
$(1+x)^n$	$1 + nx$

Mistake	Correct Approach
Choosing $a$ far from $x$	Pick $a$ as close as possible
Forgetting $f'(a)$ in $L(x)$	$L(x) = f(a) + f'(a)(x-a)$ , not
Incomplete justification	State $f''$ sign, concavity, AND over/under
Confusing $dy$ and $\Delta y$	$dy$ = tangent line change; $\Delta y$ = actual change

Quiz Set 1 — Core Skills 🎯

Quiz Set 2 — Applications 🎯

Final review. 🔍

Final Challenge. ✍️

🎉 Topic Complete!

You've mastered Linearization & Differentials:

Part	Topic	Status
1	Tangent line approximation	✅
2	Approximating values	✅
3	Differentials	✅
4	Error analysis	✅
5	Applications	✅
6	AP-style workshop	✅
7	Comprehensive assessment	✅

Key Fact: Linearization $L(x) = f(a) + f'(a)(x-a)$ is the foundation for approximation in calculus. On the AP exam, always pair your estimate with "over/underestimate" justified by concavity.

Linearization & Differentials

Over/Under Estimates from Concavity

Worked Example

Key Takeaways — Part 2

Relationship

Worked Example

Key Takeaways — Part 3

Over/Under with Error Bounds

Worked Example

Key Takeaways — Part 4

Worked Example: Sphere Volume

Linearization from a Table

Key Takeaways — Part 5

Full Worked AP Problem

Key Takeaways — Part 6

Common Linearizations at $a = 0$

Common AP Mistakes

🎉 Topic Complete!

Linearization & Differentials

Linearization & Differentials - Complete Interactive Lesson

Part 1: The Tangent Line Approximation

Linearization & Differentials

Topic Overview

Local Linearization Formula

Worked Example

Key Takeaways — Part 1

Part 2: Differentials

Linearization & Differentials

Common Linearizations at a=0a = 0a=0

Part 3: Over/Underestimates

Linearization & Differentials

Differentials vs. Derivatives

Part 4: Percentage Error

Linearization & Differentials

Error Terminology

Part 5: Linearization with Tables

Linearization & Differentials

Propagation of Error

Part 6: Problem-Solving Workshop

Linearization & Differentials

AP FRQ Patterns

Part 7: Final Assessment

Linearization & Differentials

Formula Reference

Over/Under Estimates from Concavity

Worked Example

Key Takeaways — Part 2

Relationship

Worked Example

Key Takeaways — Part 3

Over/Under with Error Bounds

Worked Example

Key Takeaways — Part 4

Worked Example: Sphere Volume

Linearization from a Table

Key Takeaways — Part 5

Full Worked AP Problem

Key Takeaways — Part 6

Common Linearizations at a=0a = 0a=0

Common AP Mistakes

🎉 Topic Complete!

Common Linearizations at $a = 0$

Common Linearizations at $a = 0$