Marsh Park
07/04/2023 · Senior High School
Consider the function \( y=\sqrt{9 x^{2}+25} \). Using the values \( x=4 \) and \( \Delta x=-0.5 \), calculate \( \triangle y-d y \). Round your answer to three decimal places if necessary. Answer How to enter your answer (opens in new window) \[ y-d y=1 \]
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To calculate \( \triangle y \) and \( d y \) for the function \( y = \sqrt{9x^2 + 25} \) at \( x = 4 \) with \( \Delta x = -0.5 \), we will follow these steps:
1. **Calculate \( y \) at \( x = 4 \)**:
\[
y(4) = \sqrt{9(4^2) + 25} = \sqrt{9 \cdot 16 + 25} = \sqrt{144 + 25} = \sqrt{169} = 13
\]
2. **Calculate \( y \) at \( x = 4 + \Delta x = 4 - 0.5 = 3.5 \)**:
\[
y(3.5) = \sqrt{9(3.5^2) + 25} = \sqrt{9 \cdot 12.25 + 25} = \sqrt{110.25 + 25} = \sqrt{135.25} \approx 11.628
\]
3. **Calculate \( \triangle y \)**:
\[
\triangle y = y(3.5) - y(4) = 11.628 - 13 = -1.372
\]
4. **Calculate \( dy \)** using the derivative of \( y \):
- First, find the derivative \( \frac{dy}{dx} \):
\[
y = (9x^2 + 25)^{1/2}
\]
Using the chain rule:
\[
\frac{dy}{dx} = \frac{1}{2}(9x^2 + 25)^{-1/2} \cdot (18x) = \frac{9x}{\sqrt{9x^2 + 25}}
\]
- Now, evaluate \( \frac{dy}{dx} \) at \( x = 4 \):
\[
\frac{dy}{dx} \bigg|_{x=4} = \frac{9 \cdot 4}{\sqrt{9 \cdot 16 + 25}} = \frac{36}{13}
\]
- Now calculate \( dy \):
\[
dy = \frac{dy}{dx} \cdot \Delta x = \frac{36}{13} \cdot (-0.5) = -\frac{18}{13} \approx -1.385
\]
5. **Calculate \( \triangle y - dy \)**:
\[
\triangle y - dy = -1.372 - (-1.385) = -1.372 + 1.385 = 0.013
\]
Thus, the final answer is:
\[
\triangle y - dy \approx 0.013
\]
Rounded to three decimal places, the answer is:
\[
\boxed{0.013}
\]
Quick Answer
\[
\triangle y - dy \approx 0.013
\]
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