Problem 4 Let \(f(x)=c,\) where \(c\) is a... [FREE SOLUTION]

Chapter 5: Problem 4

Let \(f(x)=c,\) where \(c\) is a positive constant. Explain why an area function of \(f\) is an increasing function.

Short Answer

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Question: Explain why the area function of the given function f(x) = c, where c is a positive constant, is an increasing function. Answer: The area function of f(x) = c is an increasing function because as the x-value (b) increases, the area under the curve also increases. Since f(x) = c is a horizontal line, the area under the curve is a rectangle with height c and width (b-a). As b increases, the width of the rectangle increases, resulting in a larger area under the curve.

Step by step solution

Define an Area Function

An area function, denoted as A(x), is a function that represents the accumulated area under a curve from the graph of a given function, f(x), to the x-axis over a specific interval. In other words, A(x) gives the total area under the curve of f(x) from a starting point to x.

Consider the Given Function f(x) = c

The provided function is f(x) = c, where c is a positive constant. This means that the function is a horizontal line at y = c, where c is a positive value. The graph of this function is always above the x-axis.

Determine the Area under the Curve for f(x) = c

To determine the area function of f(x) = c, we start by finding the area under the curve from a fixed point, say x = a, to an arbitrary point, x = b, where a < b. Since f(x) = c is a straight horizontal line, the area under the curve is a rectangle with height c and width (b-a). Therefore, the area A(b) can be determined by the product of the height and width: A(b) = c(b-a).

Check If the Area Function is Increasing

We need to examine if A(b) increases as b increases. Take two points on the x-axis, b1 and b2, such that b1 < b2. We have: A(b1) = c(b1-a), A(b2) = c(b2-a). Since b2 > b1, it follows that b2 - a > b1 - a. We multiply both sides by the positive constant c: c(b2-a) > c(b1-a). This inequality implies that the area function A(x) increases as the x-value (b) increases. Hence, the area function A(x) is an increasing function for f(x) = c, where c is a positive constant.

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91影视

Let \(f(x)=c,\) where \(c\) is a positive constant. Explain why an area function of \(f\) is an increasing function.

Short Answer

Step by step solution

Define an Area Function

Consider the Given Function f(x) = c

Determine the Area under the Curve for f(x) = c

Check If the Area Function is Increasing

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