The linear function ƒ(𝓍) = 3 ― 𝓍 is decreasing on the interval [0, 3]. Is its area function for ƒ (with left endpoint 0) increasing or decreasing on the interval [0, 3]? Draw a picture and explain. 

Verified step by step guidance

1. Step 1: Understand the problem. The linear function ƒ(𝓍) = 3 - 𝓍 is given, and we are tasked to determine whether its area function (integral) is increasing or decreasing on the interval [0, 3]. The area function represents the accumulated area under the curve of ƒ(𝓍) starting from the left endpoint 0.
2. Step 2: Recall the relationship between a function and its area function. The area function is the integral of ƒ(𝓍) with respect to 𝓍. Mathematically, the area function A(𝓍) is defined as: $A_x={\int }_{0}f\left(x\right)dx$. The derivative of the area function, A'(𝓍), is equal to ƒ(𝓍).
3. Step 3: Analyze the behavior of ƒ(𝓍) on the interval [0, 3]. The function ƒ(𝓍) = 3 - 𝓍 is linear with a negative slope (-1), meaning it decreases as 𝓍 increases. Specifically, ƒ(𝓍) starts at 3 when 𝓍 = 0 and decreases to 0 when 𝓍 = 3.
4. Step 4: Determine the behavior of the area function A(𝓍). Since A'(𝓍) = ƒ(𝓍), the area function A(𝓍) increases wherever ƒ(𝓍) is positive. On the interval [0, 3], ƒ(𝓍) is positive (above the x-axis), so the area function A(𝓍) is increasing throughout this interval.
5. Step 5: Visualize the problem. Draw the graph of ƒ(𝓍) = 3 - 𝓍, which is a straight line decreasing from (0, 3) to (3, 0). Shade the area under the curve from 𝓍 = 0 to a variable endpoint 𝓍. As 𝓍 moves from 0 to 3, the shaded area grows, confirming that the area function A(𝓍) is increasing on [0, 3].

Key Concepts

Decreasing Functions

A function is considered decreasing on an interval if, for any two points x1 and x2 within that interval, where x1 < x2, the function value at x1 is greater than the function value at x2 (ƒ(x1) > ƒ(x2)). In this case, the linear function ƒ(𝓍) = 3 - 𝓍 decreases as x increases, indicating that as we move from 0 to 3, the output values of the function get smaller.

Area Function

The area function A(x) associated with a function ƒ(𝓍) represents the accumulated area under the curve of ƒ from a starting point (in this case, 0) to a variable endpoint x. Mathematically, it is defined as A(x) = ∫[0,x] ƒ(t) dt. The behavior of the area function depends on the values of the original function; if ƒ is decreasing, the area function will reflect this change.

Relationship Between Function and Area Function

The relationship between a function and its area function is governed by the Fundamental Theorem of Calculus. If the original function is decreasing, the area function will increase at a decreasing rate. This means that while the area function A(x) is increasing as x moves from 0 to 3, the rate of increase diminishes because the heights of the rectangles (representing area) are getting smaller as the function value decreases.