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dy M Find an expression for for the implicitly defined curve x2 - y³ = 0. Show all the steps in your process. da . Where did you use the Chain Rule, and why? dy . What restriction (on x or y) does have? What point (x, y) does this correspond da to?

dy M Find an expression for for the implicitly defined curve x2 - y³ = 0. Show all the steps in your process. da . Where did you use the Chain Rule, and why? dy . What restriction (on x or y) does have? What point (x, y) does this correspond da to?

Calculus: Early Transcendentals
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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**Calculus Problem Solving: Implicit Differentiation Tutorial** 1. **Find an expression for \(\frac{dy}{dx}\), for the implicitly defined curve \(x^2 - y^3 = 0\). Show all the steps in your process.** 2. **Where did you use the Chain Rule, and why?** 3. **What restriction (on \(x\) or \(y\)) does \(\frac{dy}{dx}\) have? What point \((x, y)\) does this correspond to?** 4. **Find the equation of the tangent line corresponding to \(x = 1\). (Hint: What \(y\) value does this point have?) Explain how you get this equation, how you choose your computations.** 5. **Solve for \(y\) as an explicit function of \(x\). (Use fractional powers to represent roots and simplify.)** 6. **Use this form to check that the results you got using implicit differentiation for the questions above are reasonable, showing your process.** 7. **Use the curve sketching process on the explicit form (where \(y\) is a function of \(x\)).** 8. **What do you observe about the point you found, which corresponds to a restriction of \(\frac{dy}{dx}\)? Explain how this is related to differentiability and why we need to consider domain exclusions of the derivative as critical points when searching for extrema.** --- **Application - Graphic Design** You are designing a poster which has a border around the printed portion. The border is 2 inches wide on the left and right sides, and 3 inches wide on the top and bottom. The entire poster will be a total of 300 square inches, including the borders and printed portion combined.
Transcribed Image Text:**Calculus Problem Solving: Implicit Differentiation Tutorial** 1. **Find an expression for \(\frac{dy}{dx}\), for the implicitly defined curve \(x^2 - y^3 = 0\). Show all the steps in your process.** 2. **Where did you use the Chain Rule, and why?** 3. **What restriction (on \(x\) or \(y\)) does \(\frac{dy}{dx}\) have? What point \((x, y)\) does this correspond to?** 4. **Find the equation of the tangent line corresponding to \(x = 1\). (Hint: What \(y\) value does this point have?) Explain how you get this equation, how you choose your computations.** 5. **Solve for \(y\) as an explicit function of \(x\). (Use fractional powers to represent roots and simplify.)** 6. **Use this form to check that the results you got using implicit differentiation for the questions above are reasonable, showing your process.** 7. **Use the curve sketching process on the explicit form (where \(y\) is a function of \(x\)).** 8. **What do you observe about the point you found, which corresponds to a restriction of \(\frac{dy}{dx}\)? Explain how this is related to differentiability and why we need to consider domain exclusions of the derivative as critical points when searching for extrema.** --- **Application - Graphic Design** You are designing a poster which has a border around the printed portion. The border is 2 inches wide on the left and right sides, and 3 inches wide on the top and bottom. The entire poster will be a total of 300 square inches, including the borders and printed portion combined.
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