Skip to main content
Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.9.13
Chapter 3, Problem 3.9.13

Show that the linearization of f(x) = (1 + x)ᵏ at x = 0 is L(x) = 1 + kx.

Verified step by step guidance
1
To find the linearization of a function at a point, we use the formula L(x) = f(a) + f'(a)(x - a), where a is the point of linearization. Here, a = 0.
First, evaluate f(x) at x = 0. For f(x) = (1 + x)ᵏ, substitute x = 0 to get f(0) = (1 + 0)ᵏ = 1.
Next, find the derivative of f(x) = (1 + x)ᵏ with respect to x. Using the power rule, f'(x) = k(1 + x)^(k-1).
Evaluate the derivative at x = 0. Substitute x = 0 into f'(x) to get f'(0) = k(1 + 0)^(k-1) = k.
Substitute f(0) and f'(0) into the linearization formula: L(x) = 1 + k(x - 0) = 1 + kx. This shows that the linearization of f(x) at x = 0 is L(x) = 1 + kx.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

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

Linearization

Linearization is the process of approximating a function near a specific point using its tangent line. For a function f(x), the linearization at a point a is given by L(x) = f(a) + f'(a)(x - a). This method is particularly useful for simplifying complex functions into linear forms that are easier to analyze and compute.
Recommended video:
07:17
Linearization

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In the context of linearization, the derivative at a point provides the slope of the tangent line, which is essential for constructing the linear approximation.
Recommended video:
05:44
Derivatives

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)ⁿ, where n is a non-negative integer. It states that (a + b)ⁿ = Σ (n choose k) a^(n-k) b^k, where the sum is taken over k from 0 to n. In the given function f(x) = (1 + x)ᵏ, this theorem helps in understanding the behavior of the function around x = 0, particularly in determining its value and derivative at that point.
Recommended video:
06:11
Fundamental Theorem of Calculus Part 1
Related Practice
Textbook Question

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.


g(x) = { x²/³, x ≥ 0

x¹/³, x < 0

Textbook Question

Find the derivatives of the functions in Exercises 1–42.


𝔂 = x³ - 3 (x² + π²)

1
views
Textbook Question

Differentiating Implicitly


Use implicit differentiation to find dy/dx in Exercises 1–14.


x cos(2x + 3y) = y sin x

Textbook Question

Find ds/dt when θ = 3π/2 if s = cosθ and dθ/dt = 5.

Textbook Question

Find the derivatives of the functions in Exercises 1–42.

______

𝓻 = √2θ sinθ

Textbook Question

Find the derivatives of all orders of the functions in Exercises 29–32.


y = x⁵ / 120