Find linearization of the functions. A) f(x) = x^4 - 3x^2 + 1 at x = 1 B) y = e^(4x) ln(x) at x = 1

Question:

Find linearization of the functions.

A) {eq}f(x) = x^4 - 3x^2 + 1 {/eq} at {eq}x = 1 {/eq}

B) {eq}y = e^{4x} \ln(x) {/eq} at {eq}x = 1 {/eq}

Answer and Explanation:

{eq}a)f(x)=x^4-3x^2+1\\ \text{Linearization is:}\\ L(x)=f(a)+f'(a)(x-a)\\ a=1\\ f(1)=-1\\ f'(x)=4x^3-3x\\ f'(1)=1\\ L(x)=-1+1(x-1)\\ L(x)=-1+x-1\\ L(x)=x-2\\ {/eq}

{eq}b)f(x)=e^{4x}\ln x\\ f(1)=0\\ \text{Differentiating by the product rule:}\\ f'(x)=4e^{4x}\ln x+\frac{e^{4x}}{x}\\ f'(1)=e^4\\ \text{Putting in the formula:}\\ L(x)=e^4(x-1)\\ L(x)=e^4x-e^4\\ \text{is the linearization of the function.} {/eq}


Learn more about this topic:

Derivatives: The Formal Definition

from Math 104: Calculus

Chapter 8 / Lesson 5
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