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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 70c
Chapter 3, Problem 70c

The following equations implicitly define one or more functions.
c. Use the functions found in part (b) to graph the given equation.
y² = x²(4 − x) / 4 + x (right strophoid)

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1
Identify the given implicit equation: \( y^2 = \frac{x^2(4 - x)}{4 + x} \). This is known as a right strophoid.
To graph the equation, first solve for \( y \) in terms of \( x \). This involves taking the square root of both sides: \( y = \pm \sqrt{\frac{x^2(4 - x)}{4 + x}} \).
Analyze the domain of the function. The expression under the square root, \( \frac{x^2(4 - x)}{4 + x} \), must be non-negative. Determine the values of \( x \) for which this is true.
Consider the behavior of the function as \( x \) approaches the critical points, such as where the denominator \( 4 + x = 0 \) or where the expression under the square root changes sign.
Plot the function using the derived expressions for \( y \) and the determined domain. Pay attention to symmetry and any asymptotic behavior to accurately represent the right strophoid.

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Key Concepts

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

Implicit Functions

Implicit functions are defined by equations where the dependent and independent variables are not isolated on one side. In the context of calculus, understanding how to derive and manipulate these functions is crucial for analyzing their behavior and graphing them. For example, the equation y² = x²(4 − x) / 4 + x defines y implicitly in terms of x.
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Finding The Implicit Derivative

Graphing Techniques

Graphing techniques involve methods for visually representing mathematical functions and equations. This includes understanding the shape, intercepts, and asymptotic behavior of the graph. For the given equation, one must identify key points and the overall structure of the right strophoid to accurately depict it on a coordinate plane.
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Graphing The Derivative

Strophoid Curves

Strophoid curves are a family of curves defined by specific mathematical properties, often related to the geometry of circles and lines. The right strophoid, in particular, has unique characteristics that can be derived from its defining equation. Recognizing these properties helps in understanding the shape and behavior of the graph produced by the equation y² = x²(4 − x) / 4 + x.
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Summary of Curve Sketching
Related Practice
Textbook Question

Find f′(x), f′′(x), and f′′′(x) for the following functions.

f(x) = (x2 - 7x - 8) / (x + 1)

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Textbook Question

The right-sided and left-sided derivatives of a function at a point aa are given by f+′(a)=limh→0+f(a+h)−f(a)hf_{+}^{\(\prime\)}\(\left\)(a\(\right\))={\(\displaystyle\]\lim\)_{h\(\to\)0^{+}}{\(\frac{f(a+h)-f(a)}{h}\)}} and f−′(a)=limh→0−f(a+h)−f(a)hf_{-}^{\(\prime\)}\(\left\)(a\(\right\))={\(\displaystyle\]\lim\)_{h\(\to\)0^{-}}{\(\frac{f(a+h)-f(a)}{h}\)}}, respectively, provided these limits exist. The derivative f′(a)f^{\(\prime\)}\(\left\)(a\(\right\)) exists if and only if f+′(a)=f−′(a)f_{+}^{\(\prime\)}\(\left\)(a\(\right\))=f_{-}^{\(\prime\)}\(\left\)(a\(\right\)).

Compute f+′(a)f_{+}^{\(\prime\)}\(\left\)(a\(\right\)) and f−′(a)f_{-}^{\(\prime\)}\(\left\)(a\(\right\)) at the given point aa.

f(x)=∣x−2∣f\(\left\)(x\(\right\))=\(\left\)|x-2\(\right\)|; a=2a=2

Textbook Question

Calculate the derivative of the following functions.

y = e^2x(2x-7)⁵

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Textbook Question

The following equations implicitly define one or more functions.

a. Find dy/dx using implicit differentiation.

y² = x²(4 − x) / 4 + x (right strophoid)

Textbook Question

The following equations implicitly define one or more functions.

b. Solve the given equation for y to identify the implicitly defined functions y=f₁(x), y = f₂(x), ….

y² = x²(4 − x) / 4 + x (right strophoid)

Textbook Question

First and second derivatives Find f′(x),f′′(x).

f(x) = x/x+2