Textbook Question
Tangent Lines
In Exercises 35–38, graph the curves over the given intervals, together with their tangent lines at the given values of x. Label each curve and tangent line with its equation.
y = sin x, −3π/2 ≤ x ≤ 2π
x = −π, 0, 3π/2
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 3.53
In Exercises 53 and 54, find dr/ds.
r cos 2s + sin²s = π
Verified step by step guidance1
Identify the given equation: \( r \cos(2s) + \sin^2(s) = \pi \). We need to find \( \frac{dr}{ds} \).
Differentiate both sides of the equation with respect to \( s \). Use the product rule for \( r \cos(2s) \) and the chain rule for \( \sin^2(s) \).
For the term \( r \cos(2s) \), apply the product rule: \( \frac{d}{ds}(r \cos(2s)) = \frac{dr}{ds} \cos(2s) + r \frac{d}{ds}(\cos(2s)) \).
Differentiate \( \cos(2s) \) using the chain rule: \( \frac{d}{ds}(\cos(2s)) = -2\sin(2s) \).
Differentiate \( \sin^2(s) \) using the chain rule: \( \frac{d}{ds}(\sin^2(s)) = 2\sin(s)\cos(s) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Implicit Differentiation
Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. In this case, we have an equation involving both r and s, and we need to find the derivative dr/ds. By differentiating both sides of the equation with respect to s, we can apply the chain rule to find the relationship between the derivatives.
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Finding The Implicit Derivative
Chain Rule
The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. When differentiating an expression like r(s), where r is a function of s, the chain rule states that the derivative of r with respect to s is the product of the derivative of r with respect to its argument and the derivative of that argument with respect to s. This is crucial for finding dr/ds in the given equation.
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Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. In the given equation, we encounter cos(2s) and sin²(s), which can be simplified using identities such as the double angle formula for cosine and the Pythagorean identity. Understanding these identities is essential for simplifying the equation and effectively finding the derivative.
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Related Practice
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