Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

4. Applications of Derivatives

Linearization

Calculus

4. Applications of Derivatives

Linearization

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1:43 minutes

Problem 3.9.14a

Textbook Question

Use the linear approximation (1 + x)ᵏ ≈ 1 + kx to find an approximation for the function f(x) for values of x near zero.

a. f(x) = (1 − x)⁶

Verified step by step guidance

Identify the function f(x) = (1 - x)⁶ and recognize that it is in the form of (1 + x)ᵏ with k = 6 and x replaced by -x.

Apply the linear approximation formula (1 + x)ᵏ ≈ 1 + kx to the function. Here, substitute x with -x and k with 6.

The linear approximation becomes: (1 - x)⁶ ≈ 1 + 6(-x).

Simplify the expression: 1 + 6(-x) becomes 1 - 6x.

Thus, the linear approximation for f(x) = (1 - x)⁶ near x = 0 is f(x) ≈ 1 - 6x.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Linear Approximation

Linear approximation is a method used to estimate the value of a function near a given point using the tangent line at that point. For a function f(x), the linear approximation at x = a is given by f(a) + f'(a)(x - a). This technique is particularly useful for simplifying complex functions near a specific point, often x = 0.

Binomial Expansion

The binomial expansion is a way of expressing powers of binomials, such as (1 + x)ᵏ, as a series. For small values of x, the expansion can be approximated by the first few terms, often just 1 + kx for linear approximation. This simplification is useful for estimating the behavior of functions like (1 - x)⁶ near x = 0.

Function Behavior Near Zero

Understanding the behavior of functions near zero involves analyzing how the function changes as x approaches zero. This often involves using approximations or expansions to simplify the function, making it easier to evaluate or estimate. For f(x) = (1 - x)⁶, using linear approximation helps predict its value when x is close to zero.