/*! 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 13 Find \(\frac{d y}{d x}\) if \(y=... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 13

Find \(\frac{d y}{d x}\) if \(y=\arccos x\).

Short Answer

Expert verified
The derivative \(\frac{dy}{dx}\) for \(y = \arccos x\) is \(-\frac{1}{\sqrt{1-x^2}}\).

Step by step solution

01

Understand the Derivative of Inverse Functions

We need to find the derivative of the inverse trigonometric function \(y = \arccos x\). First, recall that for \(y = \arccos x\), the derivative is given by the formula \(-\frac{1}{\sqrt{1-x^2}}\).
02

Apply the Formula

Using the derivative formula for \(\arccos x\), substitute it into the expression for \(\frac{d y}{d x}\). This gives us \(-\frac{1}{\sqrt{1-x^2}}\), which is already simplified as the result.

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Key Concepts

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

Understanding Arccosine
The arccosine function, denoted as \( \arccos x \), is the inverse of the cosine function. This function plays a crucial role in trigonometry as it helps find the angle whose cosine value is \( x \). Unlike the simple cosine function that maps angles to values, arccosine takes a value and returns an angle.
  • Domain of \( \arccos x \): \([-1, 1] \)
  • Range of \( \arccos x \): \([0, \pi] \)

The range restriction ensures that each output corresponds to a unique input, a necessary condition for inverse functions. The arccosine function tends to be more complex when dealing with differentiation due to its limited domain and range.
Derivative Formulas
Derivative formulas are critical tools in calculus that aid in finding the rate at which one variable changes with respect to another. When dealing with inverse trigonometric functions, these formulas become even more important since these functions do not have straightforward definitions like their trigonometric counterparts.
  • Derivative of \( \arccos x \): \(-\frac{1}{\sqrt{1-x^2}}\)

This formula arises from the foundational definition of derivatives and the specific properties of the inverse trigonometric functions, like \( \arcsin x \) and \( \arccos x \). When using this derivative, it's also essential to remember that it only holds for \( -1 < x < 1 \), where the expression under the square root doesn鈥檛 become negative.
Inverse Trigonometric Functions
Inverse trigonometric functions are essential in bridging the gap between algebra and trigonometry. They allow us to find specific angles when the value of a trigonometric function is known. In addition to \( \arccos x \), common inverse trigonometric functions include \( \arcsin x \), \( \arctan x \), and their counterparts.
  • Inverse functions reverse the action of the original trigonometric function.
  • They have specific derivatives that aid in calculus, such as in finding the slope of a tangent line.

Understanding these functions is vital because they often appear in numerous calculus problems, especially when differentiating or integrating. They provide a way to transition from a trigonometric evaluation to an analytical expression easily.

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Most popular questions from this chapter

Compute the directional derivative of \(f(x, y)\) at the point \(P\) in the direction of the point \(Q\). $$ f(x, y)=4 x y+y^{2}, P=(-1,1), Q=(3,2) $$

Find a linear approximation to $$\mathbf{f}(x, y)=\left[\begin{array}{c} \sqrt{2 x+y} \\ x-y^{2} \end{array}\right]$$ at \((1,2)\). Use your result to find an approximation for \(f(1.05,2.05)\), and compare the approximation with the value of \(f(1.05,2.05)\) that you get when you use a calculator.

Refer to the negative binomial host-parasitoid model. Problems \(7,8,11\), and 12 are best done with the help of a spreadsheet, but can also be done with a calculator. The negative binomial model is a discrete-generation host- parasitoid model of the form $$ \begin{array}{l} N_{t+1}=b N_{t}\left(1+\frac{a P_{t}}{k}\right)^{-k} \\ P_{t+1}=c N_{t}\left[1-\left(1+\frac{a P_{t}}{k}\right)^{ k}\right] \end{array} $$ for \(t=0,1,2, \ldots\) Evaluate the negative binomial model for the first 25 generations when \(a=0.02, c=3, k=0.75\), and \(b=0.5 .\) For the initial host density, choose \(N_{0}=100\), and for the initial parasitoid density, choose \(P_{0}=50\).

The functional responses of some predators are sigmoidal; that is, the number of prey attacked per predator as a function of prey density has a sigmoidal shape. If we denote the prey density by \(N\), the predator density by \(P\), the time available for searching for prey by \(T\), and the handling time of each prey item per predator by \(T_{h}\), then the number of prey encounters per predator as a function of \(N, T\), and \(T_{h}\) can be expressed as $$ f\left(N, T, T_{h}\right)=\frac{b^{2} N^{2} T}{1+c N+b T_{h} N^{2}} $$ where \(b\) and \(c\) are positive constants. (a) Investigate how an increase in the prey density \(N\) affects the function \(f\left(N, T, T_{h}\right)\) (b) Investigate how an increase in the time \(T\) available for search affects the function \(f\left(N, T, T_{h}\right)\). (c) Investigate how an increase in the handling time \(T_{h}\) affects the function \(f\left(N, T, T_{h}\right)\) (d) Graph \(f\left(N, T, T_{h}\right)\) as a function of \(N\) when \(T=2.4\) hours, \(T_{h}=0.2\) hours, \(b=0.8\), and \(c=0.5\)

Show that the equilibrium \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) of $$ \left[\begin{array}{c} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{ll} 2 & -4 \\ 5 & -6 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ is stable.
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