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Chapter 2: Problem 65

Describe the difference between the explicit form of a function and an implicit equation. Give an example of each.

Short Answer

Expert verified
Explicit form of a function is when 'y' is isolated on one side of the equation, like in \( y = x^2 + 3x - 4 \). Implicit form is when the dependent 'y' and independent 'x' variables intermingle in the equation, like in \( x^2 + y^2 = 4 \).

Step by step solution

01

Define explicit form of a function

The explicit form of a function is when the dependent variable (usually denoted as 'y') is isolated on one side of the equation. It explicitly shows how to calculate the dependent variable in terms of the independent variable(s). To illustrate this, an example will be considered.
02

Provide an example of explicit form

An example of a function in explicit form is \( y = x^2 + 3x - 4 \). In this function, 'y' is isolated on one side of the equation, and 'x' (the independent variable) is on the other side of the equation. The value of 'y' can be determined for any given value of 'x'.
03

Define implicit form of a function

The implicit form of a function is when the dependent variable is not isolated on one side of the equation. Instead, the dependent and independent variables are intermingled throughout the equation. To capture this concept, an example would be considered.
04

Provide an example of implicit form

An example of a function in implicit form is \( x^2 + y^2 = 4 \). In this function, both 'x' and 'y' are on the same side of the equation, and there isn't an explicit formula for 'y' in terms of 'x'.

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Most popular questions from this chapter

Using Relationships In Exercises \(103-106,\) use the given information to find \(f^{\prime}(2) .\) $$ \begin{array}{l}{g(2)=3 \quad \text { and } \quad g^{\prime}(2)=-2} \\\ {h(2)=-1 \quad \text { and } \quad h^{\prime}(2)=4}\end{array} $$ $$ f(x)=2 g(x)+h(x) $$

Linear and Quadratic Approximations The linear and quadratic approximations of a function \(f\) at \(x=a\) are $$\begin{array}{l}{P_{1}(x)=f^{\prime}(a)(x-a)+f(a) \text { and }} \\\ {P_{2}(x)=\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}+f^{\prime}(a)(x-a)+f(a)}\end{array}$$ In Exercises 123 and \(124,\) (a) find the specified linear and quadratic approximations of \(f,(b)\) use a graphing utility to graph \(f\) and the approximations, (c) determine whether \(P_{1}\) or \(P_{2}\) is the better approximation, and (d) state how the accuracy changes as you move farther from \(x=a\) . $$ f(x)=\tan x ; \quad a=\frac{\pi}{4} $$

True or False? In Exercises \(93-96\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a function has derivatives from both the right and the left at a point, then it is differentiable at that point.

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

Electricity The combined electrical resistance \(R\) of two resistors \(R_{1}\) and \(R_{2},\) connected in parallel, is given by $$\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}$$ where \(R, R_{1},\) and \(R_{2}\) are measured in ohms. \(R_{1}\) and \(R_{2}\) are increasing at rates of 1 and 1.5 ohms per second, respectively. At what rate is \(R\) changing when \(R_{1}=50\) ohms and \(R_{2}=75\) ohms?
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