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Chapter 2: Problem 90

True or False? In Exercises \(87-92,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ y=x / \pi, \text { then } d y / d x=1 / \pi $$

Short Answer

Expert verified
The statement 'If \(y = x / \pi\), then \(dy/ dx = 1 / \pi\)' is True.

Step by step solution

01

Differentiate the given equation

Differentiate the function \(y = x / \pi \)with respect to \(x\). By using the rule that the derivative of \(x\) with respect to \(x\) is \(1\), and the derivative of a constant remains the same, differentiate the equation to get \(dy/dx = 1 / \pi \)
02

Comparing the derived derivative with the proposed derivative

After differentiating the equation given, we find that the derivative \(dy/dx = 1 / \pi \). Comparing this with the proposed derivative \(dy/dx = 1 / \pi \), we can see they are equal.

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Key Concepts

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

Differentiation Rules
Differentiation is a cornerstone of calculus, used to find the rate of change of a function. When differentiating, specific rules apply to different types of functions.
  • For a constant multiplied by a function, like \(y = rac{x}{\pi}\), apply the constant factor rule: differentiate \(x\) as usual and multiply by the constant. Here, \(\pi\) is a constant, meaning \(\frac{1}{\pi}\) stays as it is.
  • The basic rule for derivatives, \(\frac{d}{dx}(x) = 1\), states that the derivative of \(x\) with respect to \(x\) is \(1\). So, for \(y = \frac{x}{\pi}\), the derivative \(\frac{dy}{dx}\) is simply \(\frac{1}{\pi} \).
These differentiation rules are especially handy as they allow you to break down more complex problems into smaller, more manageable parts. Understanding these rules helps in solving various calculus problems effectively.
Constant Function
A constant function is one of the simplest types of functions in calculus. It doesn't change its value as the input changes.
  • If \(y = C\), where \(C\) is a constant, then \(\frac{dy}{dx} = 0\) because there's no change in \(y\).
  • In the exercise \(y = \frac{x}{\pi}\), the constant is \(\frac{1}{\pi}\). Even though \(y\) is a function of \(x\), \(\pi\) remains constant.
It's crucial to distinguish between a constant multiplier within a function, like \(\frac{1}{\pi}x\), and a standalone constant function. Recognizing these can help break down derivative calculations correctly.
Rate of Change
The rate of change reflects how a quantity changes with respect to another, most commonly how a function's output changes with its input.
  • In calculus, this is represented by the derivative. For instance, when you differentiate \(y = \frac{x}{\pi}\), the result \(\frac{dy}{dx} = \frac{1}{\pi}\) shows the rate at which \(y\) changes as \(x\) increases by a unit.
  • A constant rate of change means the relationship between the variables is linear. Hence, the graph of \(y = \frac{x}{\pi}\) is a straight line with slope \(\frac{1}{\pi}\).
Understanding the rate of change is vital for interpreting graphs and real-life situations, like speed being the rate of change of distance with time.

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Most popular questions from this chapter

True or False? In Exercises \(125-128\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y\) is a differentiable function of \(u,\) and \(u\) is a differentiable function of \(x,\) then \(y\) is a differentiable function of \(x .\)

Proof. Prove the following differentiation rules. $$ \begin{array}{l}{\text { (a) } \frac{d}{d x}[\sec x]=\sec x \tan x} \\ {\text { (b) } \frac{d}{d x}[\csc x]=-\csc x \cot x} \\ {\text { (c) } \frac{d}{d x}[\cot x]=-\csc ^{2} x}\end{array} $$

Area The included angle of the two sides of constant equal length \(s\) of an isosceles triangle is \(\theta\) . (a) Show that the area of the triangle is given by \(A=\frac{1}{2} s^{2} \sin \theta .\) (b) the angle \(\theta\) is increasing at the rate of \(\frac{1}{2}\) radian per minute. Find the rates of change of the area when \(\theta=\pi / 6\) and \(\theta=\pi / 3 .\) (c) Explain why the rate of change of the area of the triangle is not constant even though \(d \theta / d t\) is constant.

Finding a Second Derivative In Exercises \(91-98\) , find the second derivative of the function. $$ f(x)=\sec x $$

Conjecture Consider the functions \(f(x)=x^{2}\) and \(g(x)=x^{3} .\) (a) Graph \(f\) and \(f^{\prime}\) on the same set of axes. (b) Graph \(g\) and \(g^{\prime}\) 'on the same set of axes. (c) Identify a pattern between \(f\) and \(g\) and their respective derivatives. Use the pattern to make a conjecture about \(h^{\prime}(x)\) if \(h(x)=x^{n},\) where \(n\) is an integer and \(n \geq 2\) . (d) Find \(f^{\prime}(x)\) if \(f(x)=x^{4}\) . Compare the result with the conjecture in part (c). Is this a proof of your conjecture? Explain.
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