Question

Let \(A(c)\) denote the area of the region bounded from above by the line \(y=x+1\), from the left by the \(y\) -axis, from below by the \(x\) -axis, and from the right by the line \(x=c .\) Such a function is called an accumulation function. (See Figure 13.) Find (a) \(A(1)\) (b) \(A(2)\) (c) \(A(0)\) (d) \(A(c)\) (e) Sketch the graph of \(A(c)\). (f) What are the domain and range of \(A\) ?

Short answer

A(1)=1, A(2)=3, A(0)=0; A(c)=\frac{1}{2}(c^2+c). Domain: [0, ∞), Range: [0, ∞).

Step-by-step solution

Understanding the area of region

First, let's visualize the region described. The line \(y = x + 1\) forms a sloped upper boundary, y-axis \((x=0)\), the x-axis \((y=0)\), and a vertical line \(x=c\). The area of interest is a right-angled triangle with base on the x-axis from \(x=0\) to \(x=c\) and the hypotenuse from \((0,1)\) to \((c, c+1)\).

Calculate the height of the triangle

The height of the triangle is determined by the line \(y = x + 1\) at \(x = c\). Thus, the height is \(c + 1\) units.

Write down the formula for the area of a triangle

The area \(A\) of a triangle with base \(b\) and height \(h\) is given by \(A = \frac{1}{2} b h\).

Calculate the specific area function \(A(c)\)

Substitute the base \(c\) and height \(c+1\) into the formula: \[ A(c) = \frac{1}{2} c (c + 1) = \frac{1}{2} (c^2 + c) \]. This is the general formula for the area given any \(c\).

Calculate specific areas A(1), A(2), and A(0)

- For \(A(1)\), substitute \(c=1\) into the formula: \(A(1) = \frac{1}{2} (1^2 + 1) = 1\).- For \(A(2)\), substitute \(c=2\): \(A(2) = \frac{1}{2} (2^2 + 2) = 3\).- For \(A(0)\), substitute \(c=0\): \(A(0) = \frac{1}{2} (0^2 + 0) = 0\).

Graphing the area function \(A(c)\)

The graph of \(A(c) = \frac{1}{2} (c^2 + c)\) is a quadratic function (a parabola) starting at the origin (0,0), opening upwards. Its vertex is not on the x-axis as it's a quadratic function; however, it resembles an incomplete parabola starting from the origin.

Determine the domain and range of A

The function \(A(c) = \frac{1}{2} (c^2 + c)\) is defined for all non-negative \(c\) (since the region is divided by the y-axis at \(x=0\)). Therefore, the domain is \([0, \infty)\). The range, as \(c\) increases, \(A\) increases, making the range \([0, \infty)\).

Key concepts

Area Calculation

To understand the area calculation in this exercise, we visualize a geometric shape formed by prescribed boundaries: the line \(y = x + 1\), the y-axis, the x-axis, and a vertical line at \(x = c\).

This creates a triangular region. The task is to calculate the area of this right-angled triangle. The important elements are:

• Base: The segment along the x-axis from \(x=0\) to \(x=c\)
• Height: The vertical distance from the x-axis to the line \(y = x + 1\) at \(x = c\), which is \(c + 1\)

Given the formula for the area of a triangle is \(A = \frac{1}{2} \times \, \text{base} \times \, \text{height}\), substituting in the base \(c\) and height \(c+1\) gives \(A(c) = \frac{1}{2}c(c+1)\).

This formula efficiently calculates the accumulated area for any value of \(c\).

Quadratic Function

The formula \(A(c) = \frac{1}{2}(c^2 + c)\) derived from the area calculation represents a quadratic function. Here's how it works:

• The formula involves a term \(c^2\), which is characteristic of quadratic functions.
• This gives us a parabolic graph opening upwards because the coefficient of \(c^2\) is positive.
• The function starts at the origin \((0,0)\) which means when \(c = 0\), the area is 0.

The graph of this function, \(y = \frac{1}{2}(c^2 + c)\), describes how the area grows as \(c\) increases. This growth pattern shows exponential characteristics, commonly observed in quadratic functions where the rate of increase itself increases.

Understanding this visual and mathematical representation helps grasp how the accumulation function behaves.

Domain and Range

Identifying the domain and range of \(A(c)\) is crucial:

• Domain: Because the region we are calculating the area for is bound by the y-axis at \(x=0\), \(c\) must be non-negative. Therefore, \(c\) can be any non-negative number, meaning the domain is \([0, \infty)\).
• Range: Starting at 0 when \(c = 0\), the area exponentially increases as \(c\) increases. This allows \(A\) to take on all non-negative values, establishing the range as \([0, \infty)\).

In simple terms, any value of \(c\) starting from 0 and moving towards infinity is valid, and the area ranges similarly. This aligns with the characteristics of a mathematical function that performs unboundedly as \(c\) grows, typical for parabolic or quadratic functions.