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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 21
Chapter 3, Problem 21

In Exercises 19–22, find the values of the derivatives.


dr/dθ |θ₌₀ if r = 2/√(4 – θ)

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First, identify the function given: \( r(\theta) = \frac{2}{\sqrt{4 - \theta}} \). We need to find the derivative \( \frac{dr}{d\theta} \) and evaluate it at \( \theta = 0 \).
To differentiate \( r(\theta) \), recognize it as a composition of functions. Use the chain rule: if \( r(\theta) = f(g(\theta)) \), then \( \frac{dr}{d\theta} = f'(g(\theta)) \cdot g'(\theta) \).
Rewrite \( r(\theta) = (4 - \theta)^{-1/2} \) to make differentiation easier. Now, apply the chain rule: let \( u = 4 - \theta \), so \( r = u^{-1/2} \).
Differentiate \( r = u^{-1/2} \) with respect to \( u \): \( \frac{dr}{du} = -\frac{1}{2}u^{-3/2} \). Then, differentiate \( u = 4 - \theta \) with respect to \( \theta \): \( \frac{du}{d\theta} = -1 \).
Combine the derivatives using the chain rule: \( \frac{dr}{d\theta} = \frac{dr}{du} \cdot \frac{du}{d\theta} = -\frac{1}{2}(4 - \theta)^{-3/2} \cdot (-1) \). Evaluate this expression at \( \theta = 0 \) to find the derivative at that point.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivatives

A derivative represents the rate of change of a function with respect to a variable. In calculus, it is a fundamental concept that allows us to determine how a function behaves as its input changes. The notation dr/dθ indicates the derivative of the function r with respect to the variable θ, which is essential for understanding how r varies as θ changes.
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Derivatives

Chain Rule

The chain rule is a formula for computing the derivative of a composite function. It states that if a function y is composed of two functions u and v, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is particularly useful when differentiating functions that are expressed in terms of other functions, such as r(θ).
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Intro to the Chain Rule

Evaluating Derivatives at a Point

Evaluating a derivative at a specific point involves substituting the value of the variable into the derivative function. In this case, we need to find dr/dθ at θ = 0. This process provides the instantaneous rate of change of the function r at that particular value of θ, which is crucial for understanding the behavior of the function at that point.
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Critical Points
Related Practice
Textbook Question

Economics


Marginal revenue

Suppose that the revenue from selling x washing machines is


r(x) = 20000(1 − 1/x) dollars.


b. Use the function r'(x) to estimate the increase in revenue that will result from increasing production from 100 machines a week to 101 machines a week.

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Textbook Question

Economics


Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x − 0.1x².


a. Find the average cost per machine of producing the first 100 washing machines.

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Textbook Question

Economics


Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x − 0.1x².


c. Show that the marginal cost when 100 washing machines are produced is approximately the cost of producing one more washing machine after the first 100 have been made, by calculating the latter cost directly.

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Textbook Question

Understanding Motion from Graphs


The accompanying figure shows the velocity v = f(t) of a particle moving on a horizontal coordinate line.


c. When does the particle move at its greatest speed?

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Textbook Question

Understanding Motion from Graphs


The accompanying figure shows the velocity v = f(t) of a particle moving on a horizontal coordinate line.


d. When does the particle stand still for more than an instant?

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Textbook Question

Understanding Motion from Graphs


The accompanying figure shows the velocity v = f(t) of a particle moving on a horizontal coordinate line.


b. When is the particle’s acceleration positive? Negative? Zero?

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