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Chapter 2: Problem 85

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

Short Answer

Expert verified
Explicit functions are those in which the dependent variable is given in terms of the independent variables, while implicit equations are equations that do not explicitly solve for one variable in terms of the others. For example, \(y=2x+3\) is an explicit function, while \(x^2+y^2=25\) is an implicit function.

Step by step solution

01

Understand Explicit Function

An explicit function is a function in which the dependent variable (generally denoted as \(y\)) is given in terms of the independent variable (normally denoted as \(x\)). In an explicit function, you can isolate \(y\) on one side of the equation easily. For example, the function \(y = 2x + 3\) is an explicit function because \(y\) is expressed solely in terms of \(x\).
02

Understand Implicit Function

An implicit function is one where it is not possible (or not easy) to express the dependent variable entirely in terms of the independent variable. Implicit functions are typically equations that involve both \(x\) and \(y\) on the same side. For example, the equation \(x^2 + y^2 = 25\) is implicit, as it is not immediately solved for \(y\).

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Most popular questions from this chapter

In Exercises \(81-88\), (a) find an equation of the tangent line to the graph of \(f\) at the indicated point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(\frac{\text { Function }}{y=4-x^{2}-\ln \left(\frac{1}{2} x+1\right)} \quad \frac{\text { Point }}{\left(0,4\right)}\)

Evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. \(f(x)=\frac{1}{\sqrt{x+4}}, \quad\left(0, \frac{1}{2}\right)\)

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