/*! 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 3 Find \(d y / d x\). $$ x=\si... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 3

Find \(d y / d x\). $$ x=\sin ^{2} \theta, y=\cos ^{2} \theta $$

Short Answer

Expert verified
The derivative \(d y / d x\) is \(-1\).

Step by step solution

01

Differentiate \(x=\sin^2\theta\) and \(y=\cos^2\theta\) with respect to \(\theta\)

This will be done using the chain rule.The derivative of \(x=\sin^2\theta\) with respect to \(\theta\) is \(d x / d \theta = 2\sin(\theta) \cos(\theta)\), by differentiating the outer function first then multiplying by the derivative of the inner function.The derivative of \(y=\cos^2\theta\) with respect to \(\theta\) is \(d y / d \theta = -2\cos(\theta) \sin(\theta)\)
02

Find \(d y / d x\)

The derivative \(d y / d x\) can be found using the relation \((d y / d x)=(d y / d \theta) / (d x / d \theta)\). Using the results from Step 1, we get \(d y / d x = -2\cos(\theta) \sin(\theta) / 2\sin(\theta) \cos(\theta)\)
03

Simplify

Simplify the expression obtained in Step 2 to give your final result. Since the numerator and denominator are identical, their quotient is simply \(-1\).

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Most popular questions from this chapter

Use a graphing utility to graph the polar equation over the given interval. Use the integration capabilities of the graphing utility to approximate the length of the curve accurate to two decimal places. $$ r=2 \sin (2 \cos \theta), \quad 0 \leq \theta \leq \pi $$

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=\sqrt[4]{t}, \quad y=3-t $$

Conjecture (a) Use a graphing utility to graph the curves represented by the two sets of parametric equations. \(x=4 \cos t \quad x=4 \cos (-t)\) \(y=3 \sin t \quad y=3 \sin (-t)\) (b) Describe the change in the graph when the sign of the parameter is changed. (c) Make a conjecture about the change in the graph of parametric equations when the sign of the parameter is changed. (d) Test your conjecture with another set of parametric equations.

Write a short paragraph describing how the graphs of curves represented by different sets of parametric equations can differ even though eliminating the parameter from each yields the same rectangular equation.

Graphical Reasoning In Exercises 1-4, use a graphing utility to graph the polar equation when (a) \(e=1,\) (b) \(e=0.5\) and \((\mathrm{c}) e=1.5 .\) Identify the conic. \(r=\frac{2 e}{1-e \sin \theta}\)
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