Textbook Question
Find the slope of the curve x³y³ + y² = x + y at the points (1, 1) and (1, -1).
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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 3.8.20
Heating a plate When a circular plate of metal is heated in an oven, its radius increases at the rate of 0.01 cm/min. At what rate is the plate’s area increasing when the radius is 50 cm?
Verified step by step guidance1
First, identify the relationship between the radius of the circle and its area. The area A of a circle is given by the formula: , where r is the radius.
Next, differentiate the area formula with respect to time t to find the rate of change of the area. Use the chain rule for differentiation: .
Calculate by differentiating the area formula with respect to r: .
Substitute the given rate of change of the radius cm/min and the radius r = 50 cm into the differentiated formula: .
Simplify the expression to find the rate at which the area is increasing. This will give you the final answer in cm²/min.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Related Rates
Related rates involve finding the rate at which one quantity changes with respect to another. In this problem, we need to determine how the area of the circle changes as its radius changes over time. This requires understanding how to differentiate equations that relate these quantities with respect to time.
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04:16
Intro To Related Rates
Differentiation
Differentiation is the process of finding the derivative of a function, which represents the rate of change of a quantity. Here, we differentiate the area of a circle, A = πr², with respect to time to find how the area changes as the radius changes. This involves applying the chain rule to account for the rate of change of the radius.
Recommended video:
05:53Finding Differentials
Chain Rule
The chain rule is a fundamental technique in calculus used to differentiate composite functions. In this context, it helps us differentiate the area function A = πr² with respect to time by considering the rate of change of the radius. The chain rule allows us to express the derivative of the area in terms of the derivative of the radius.
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05:02Intro to the Chain Rule
Related Practice
Textbook Question
In Exercises 41–58, find dy/dt.
y = sin²(πt − 2)
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Textbook Question
Using the Alternative Formula for Derivatives
Use the formula
f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)
to find the derivative of the functions in Exercises 23–26.
g(x) = x / (x − 1)
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Textbook Question
Find the derivatives of the functions in Exercises 1–42.
s = (sec t + tan t)⁵
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Textbook Question
Linearization for Approximation
In Exercises 7–12, find a linearization at a suitably chosen integer near a at which the given function and its derivative are easy to evaluate.
f(x) = ∛x, a = 8.5
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Textbook Question
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
r = 12/θ − 4/θ³ + 1/θ⁴
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