Working with area functions Consider the function ƒ and its graph.
(c) Sketch a graph of A, for 0 ≤ 𝓍 ≤ 10 , without a scale on the y-axis.

Verified step by step guidance

1. Understand that the function A(x) represents the area function defined as the integral of f(t) from 0 to x, i.e., \(A(x) = \int_0^x f(t) \, dt\). This means A(x) accumulates the net area under the curve of f(t) from 0 to x.
2. Identify the intervals where f(t) is positive and where it is negative by looking at the graph. When f(t) is above the t-axis, the area contributes positively to A(x), and when f(t) is below the t-axis, the area contributes negatively.
3. Start sketching A(x) at x=0 with A(0) = 0, since the integral from 0 to 0 is zero. As x increases, the slope of A(x) at any point x is given by f(x), because \(A'(x) = f(x)\).
4. Use the shape of f(t) to determine the slope of A(x): where f(t) is positive, A(x) is increasing; where f(t) is negative, A(x) is decreasing. Also, where f(t) has local maxima or minima, A(x) will have points where the slope changes accordingly.
5. Sketch A(x) by accumulating the net area: when f(t) is above the axis, A(x) rises; when f(t) is below, A(x) falls. The graph of A(x) will be smooth and continuous, reflecting the integral of the oscillating function f(t).

Key Concepts

Area Function and Definite Integral

An area function A(x) represents the accumulated area under the curve of a function f(t) from a fixed point (usually 0) to x. It is defined as A(x) = ∫₀ˣ f(t) dt, capturing the net area, which can be positive or negative depending on whether f(t) is above or below the t-axis.

Relationship Between a Function and Its Area Function

The derivative of the area function A(x) is the original function f(x), i.e., A'(x) = f(x). This means the slope of the graph of A at any point x equals the value of f at x, guiding how the area function increases or decreases based on f's sign and magnitude.

Sketching the Area Function Without a Y-Scale

When sketching A(x) without a y-scale, focus on qualitative features: where A(x) increases or decreases (based on f's sign), where it has local maxima or minima (where f crosses zero), and the concavity (related to f's slope). The graph of A(x) is smooth and accumulates area, reflecting the integral of f.