/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Find \(d r / d \theta.\) $$\th... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 15

Find \(d r / d \theta.\) $$\theta^{1 / 2}+r^{1 / 2}=1$$

Short Answer

Expert verified
\( \frac{dr}{d\theta} = -\theta^{-1/2} + \theta^{1/2} \)

Step by step solution

01

Implicitly Differentiate the Equation

Start by differentiating both sides of the given equation with respect to \( \theta \). Remember that \( \theta \) is the variable and \( r \) is considered a function of \( \theta \). Given: \[ \theta^{1/2} + r^{1/2} = 1 \]Differentiating both sides, we have:\[ \frac{d}{d\theta}(\theta^{1/2}) + \frac{d}{d\theta}(r^{1/2}) = \frac{d}{d\theta}(1) \]
02

Differentiate Each Term

Differentiate each term on the left side.- For \( \theta^{1/2} \):\[ \frac{d}{d\theta}(\theta^{1/2}) = \frac{1}{2}\theta^{-1/2} \]- For \( r^{1/2} \) using the chain rule:\[ \frac{d}{d\theta}(r^{1/2}) = \frac{1}{2}r^{-1/2}\frac{dr}{d\theta} \]The derivative of a constant (right side) is zero:\[ \frac{d}{d\theta}(1) = 0 \]
03

Construct the Derived Equation

Combine the differentiated results into one equation:\[ \frac{1}{2}\theta^{-1/2} + \frac{1}{2}r^{-1/2}\frac{dr}{d\theta} = 0 \]
04

Solve for \( \frac{dr}{d\theta} \)

Isolate \( \frac{dr}{d\theta} \) in the equation:First, move \( \frac{1}{2}\theta^{-1/2} \) to the right side:\[ \frac{1}{2}r^{-1/2}\frac{dr}{d\theta} = -\frac{1}{2}\theta^{-1/2} \]Multiply through by 2 to clear the fractions:\[ r^{-1/2}\frac{dr}{d\theta} = -\theta^{-1/2} \]Finally, multiply by \( r^{1/2} \) to solve for \( \frac{dr}{d\theta} \):\[ \frac{dr}{d\theta} = -\theta^{-1/2}r^{1/2} \]
05

Express \( r^{1/2} \) in Terms of \( \theta \)

Since \( \theta^{1/2} + r^{1/2} = 1 \), you can express \( r^{1/2} \) as:\[ r^{1/2} = 1 - \theta^{1/2} \]
06

Substitute and Simplify

Substitute \( r^{1/2} = 1 - \theta^{1/2} \) into the expression for \( \frac{dr}{d\theta} \):\[ \frac{dr}{d\theta} = -\theta^{-1/2}(1 - \theta^{1/2}) \]Simplify:\[ \frac{dr}{d\theta} = -\theta^{-1/2} + \theta^{1/2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chain Rule
The Chain Rule is a powerful tool in calculus for finding derivatives of composite functions. It helps you differentiate when one function is nested within another. In this exercise, we use the chain rule to differentiate \( r^{1/2} \), where \( r \) is a function of \( \theta \).
If you have a composite function, say \( f(g(x)) \), the Chain Rule states:
  • \( \frac{d}{dx}[f(g(x))] = f'(g(x))g'(x) \)
In our example:
  • Let \( f(u) = u^{1/2} \) and \( u = r \).
  • So, \( f'(u) = \frac{1}{2}u^{-1/2} \).
  • Since \( u = r \), apply the chain rule, \( \frac{d}{d\theta}(r^{1/2}) = \frac{1}{2}r^{-1/2} \cdot \frac{dr}{d\theta} \).
This step ensures you handle the function's dependencies correctly.
Differentiation
Differentiation is the core process of calculus that allows us to find the rate at which a function changes. In this exercise, we started by implicitly differentiating the equation \( \theta^{1/2} + r^{1/2} = 1 \) with respect to \( \theta \).
Implicit differentiation is used when you have functions of multiple variables intertwined, like when \( r \) is considered a function of \( \theta \).
  • Differentiate each term independently.
  • For \( \theta^{1/2} \), the derivative using standard derivative rules is \( \frac{1}{2} \theta^{-1/2} \).
  • For constant terms like 1, the derivative is simply 0.
By using these rules, we obtain the derivatives needed to solve for \( \frac{dr}{d\theta} \).
Calculus Problem Solving
Solving problems in calculus often involves multiple steps and techniques. This exercise exemplifies a comprehensive calculus problem-solving approach.
Here are the major steps:
  • Start by clearly understanding the given relationship or equation, here \( \theta^{1/2} + r^{1/2} = 1 \).
  • Use implicit differentiation to find the derivatives if needed, especially when functions are inter-related.
  • Carefully isolate variables. For instance, our goal was to find \( \frac{dr}{d\theta} \).
  • Simplify your expressions. Express \( r^{1/2} \) in terms of \( \theta \): \( r^{1/2} = 1 - \theta^{1/2} \).
  • Substitute and solve for the derivative expression: \( \frac{dr}{d\theta} = -\theta^{-1/2} + \theta^{1/2} \).
This methodical approach ensures that you accurately solve calculus problems while gaining a deeper understanding of the concepts involved.

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Most popular questions from this chapter

Show that if it is possible to draw three normals from the point \((a, 0)\) to the parabola \(x=y^{2}\) shown in the accompanying diagram, then \(a\) must be greater than \(1 / 2\). One of the normals is the \(x\) -axis. For what value of \(a\) are the other two normals perpendicular?

Using the Chain Rule, show that the Power Rule \((d / d x) x^{n}=n x^{n-1}\) holds for the functions \(x^{n}\). $$x^{1 / 4}=\sqrt{\sqrt{x}}$$

Let \(p\) and \(q\) be integers with \(q>0 .\) If \(y=x^{p / q},\) differentiate the equivalent equation \(y^{q}=x^{p}\) implicitly and show that, for \(y \neq 0\) $$\frac{d}{d x} x^{p / q}=\frac{p}{q} x^{(p / q)-1}.$$

Give the position function \(s=f(t)\) of an object moving along the \(s\) -axis as a function of time \(t\). Graph \(f\) together with the velocity function \(v(t)=d s / d t=f^{\prime}(t)\) and the acceleration function \(a(t)=d^{2} s / d t^{2}=f^{\prime \prime}(t) .\) Comment on the object's behavior in relation to the signs and values of \(v\) and \(a .\) Include in your commentary such topics as the following: a. When is the object momentarily at rest? b. When does it move to the left (down) or to the right (up)? c. When does it change direction? d. When does it speed up and slow down? e. When is it moving fastest (highest speed)? Slowest? f. When is it farthest from the axis origin? $$s=t^{3}-6 t^{2}+7 t, \quad 0 \leq t \leq 4$$

Graph \(y=\tan x\) and its derivative together on \((-\pi / 2, \pi / 2) .\) Does the graph of the tangent function appear to have a smallest slope? A largest slope? Is the slope ever negative? Give reasons for your answers.
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