**Mathematical Problem and Solution Explanation** **Problem Statement:** Given the equation: \[ Ax^2 - 5xy = B \cos y + 10 \] where \( A \) and \( B \) are constants. 1. **Find** \(\frac{dy}{dx}\): \[ \frac{dy}{dx} = \frac{5y - 2Ax}{B \sin y - 5x} \] 2. **Suppose** \((9, 0)\) is on the curve. Find the equation that \( A \) and \( B \) must satisfy. 3. **Suppose** the tangent line to the curve at point \((9, 0)\) is \( y = 4x - 36 \). Find \( A \) and \( B \). **Explanation:** - The equation given involves implicit differentiation to find \(\frac{dy}{dx}\). - The point \((9, 0)\) lies on this curve, which should be used to substitute and solve for \( A \) and \( B \). - The slope of the tangent line is \(4\). Use this slope in the derivative expression to find specific values for \( A \) and \( B \). **Steps to Solve:** 1. **Find \(\frac{dy}{dx}\):** Implicitly differentiate both sides of the given equation with respect to \( x \), using product rule and chain rule where necessary. 2. **Substitute the Point:** Plug \((x, y) = (9, 0)\) into the differentiated equation to solve for \( A \) and \( B \). 3. **Match the Slope:** Since the slope of the tangent (from the derivative) must equal the given slope \(4\), set them equal and solve for unknowns. This problem demonstrates methods in calculus such as implicit differentiation and understanding how the slope of a tangent relates to derivatives.
**Mathematical Problem and Solution Explanation** **Problem Statement:** Given the equation: \[ Ax^2 - 5xy = B \cos y + 10 \] where \( A \) and \( B \) are constants. 1. **Find** \(\frac{dy}{dx}\): \[ \frac{dy}{dx} = \frac{5y - 2Ax}{B \sin y - 5x} \] 2. **Suppose** \((9, 0)\) is on the curve. Find the equation that \( A \) and \( B \) must satisfy. 3. **Suppose** the tangent line to the curve at point \((9, 0)\) is \( y = 4x - 36 \). Find \( A \) and \( B \). **Explanation:** - The equation given involves implicit differentiation to find \(\frac{dy}{dx}\). - The point \((9, 0)\) lies on this curve, which should be used to substitute and solve for \( A \) and \( B \). - The slope of the tangent line is \(4\). Use this slope in the derivative expression to find specific values for \( A \) and \( B \). **Steps to Solve:** 1. **Find \(\frac{dy}{dx}\):** Implicitly differentiate both sides of the given equation with respect to \( x \), using product rule and chain rule where necessary. 2. **Substitute the Point:** Plug \((x, y) = (9, 0)\) into the differentiated equation to solve for \( A \) and \( B \). 3. **Match the Slope:** Since the slope of the tangent (from the derivative) must equal the given slope \(4\), set them equal and solve for unknowns. This problem demonstrates methods in calculus such as implicit differentiation and understanding how the slope of a tangent relates to derivatives.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Mathematical Problem and Solution Explanation**
**Problem Statement:**
Given the equation:
\[ Ax^2 - 5xy = B \cos y + 10 \]
where \( A \) and \( B \) are constants.
1. **Find** \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{5y - 2Ax}{B \sin y - 5x}
\]
2. **Suppose** \((9, 0)\) is on the curve. Find the equation that \( A \) and \( B \) must satisfy.
3. **Suppose** the tangent line to the curve at point \((9, 0)\) is \( y = 4x - 36 \). Find \( A \) and \( B \).
**Explanation:**
- The equation given involves implicit differentiation to find \(\frac{dy}{dx}\).
- The point \((9, 0)\) lies on this curve, which should be used to substitute and solve for \( A \) and \( B \).
- The slope of the tangent line is \(4\). Use this slope in the derivative expression to find specific values for \( A \) and \( B \).
**Steps to Solve:**
1. **Find \(\frac{dy}{dx}\):**
Implicitly differentiate both sides of the given equation with respect to \( x \), using product rule and chain rule where necessary.
2. **Substitute the Point:**
Plug \((x, y) = (9, 0)\) into the differentiated equation to solve for \( A \) and \( B \).
3. **Match the Slope:**
Since the slope of the tangent (from the derivative) must equal the given slope \(4\), set them equal and solve for unknowns.
This problem demonstrates methods in calculus such as implicit differentiation and understanding how the slope of a tangent relates to derivatives.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F40d81c3d-217c-474a-b6c2-74f4c4a8c8c0%2F0fc639b3-85ea-492d-a7fa-b7ee834fdaf7%2Fp5kp2jc.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematical Problem and Solution Explanation**
**Problem Statement:**
Given the equation:
\[ Ax^2 - 5xy = B \cos y + 10 \]
where \( A \) and \( B \) are constants.
1. **Find** \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{5y - 2Ax}{B \sin y - 5x}
\]
2. **Suppose** \((9, 0)\) is on the curve. Find the equation that \( A \) and \( B \) must satisfy.
3. **Suppose** the tangent line to the curve at point \((9, 0)\) is \( y = 4x - 36 \). Find \( A \) and \( B \).
**Explanation:**
- The equation given involves implicit differentiation to find \(\frac{dy}{dx}\).
- The point \((9, 0)\) lies on this curve, which should be used to substitute and solve for \( A \) and \( B \).
- The slope of the tangent line is \(4\). Use this slope in the derivative expression to find specific values for \( A \) and \( B \).
**Steps to Solve:**
1. **Find \(\frac{dy}{dx}\):**
Implicitly differentiate both sides of the given equation with respect to \( x \), using product rule and chain rule where necessary.
2. **Substitute the Point:**
Plug \((x, y) = (9, 0)\) into the differentiated equation to solve for \( A \) and \( B \).
3. **Match the Slope:**
Since the slope of the tangent (from the derivative) must equal the given slope \(4\), set them equal and solve for unknowns.
This problem demonstrates methods in calculus such as implicit differentiation and understanding how the slope of a tangent relates to derivatives.
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