Question

this is section 3.1 problem 22: for y = f(x)=x - x³, x = 1, and δx = 0.02: δy =, and f(x)δx. round to three decimal places unless the exact answer has less decimal places. hint: follow example 2. resources ebook submit answer 8. -/1 points this is section 3.1 problem 30: use linear approximation to find a decimal approximation of √4.5. answer: √4.5≈

Explanation:

Step1: Recall linear - approximation formula

The linear - approximation formula is $L(x)\approx f(a)+f^{\prime}(a)(x - a)$. Let $y = f(x)=\sqrt{x}$, and we want to approximate $\sqrt{4.5}$. We choose $a = 4$ since it's a perfect - square close to $4.5$. First, find the derivative of $y = f(x)=\sqrt{x}=x^{\frac{1}{2}}$. Using the power rule $y^{\prime}=f^{\prime}(x)=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}$.

Step2: Evaluate $f(a)$ and $f^{\prime}(a)$

When $a = 4$, $f(4)=\sqrt{4}=2$, and $f^{\prime}(4)=\frac{1}{2\sqrt{4}}=\frac{1}{4}$. Also, $x = 4.5$, so $x - a=4.5 - 4 = 0.5$.

Step3: Apply the linear - approximation formula

$L(4.5)\approx f(4)+f^{\prime}(4)(4.5 - 4)=2+\frac{1}{4}(0.5)=2 + 0.125=2.125$.

Answer:

$2.125$