To solve this problem, we need to match each function $f(x)$ with its corresponding area function $A(x) = \int_a^x f(t) \, dt$. Here's a step-by-step approach:

1. Analyze the Characteristics of Each $f(x)$:
   The functions $f(x)$ given in graphs (a) to (d) display different behaviors (e.g., positive, negative, zero crossings). The behavior of $A(x)$ will depend on the integral of these functions over $[a, x]$.

2. Identify the Properties of $A(x)$: The area function $A(x)$ is determined by the accumulation of the area under $f(x)$. Key properties to consider:

   • When $f(x) > 0$, $A(x)$ will be increasing.
   • When $f(x) < 0$, $A(x)$ will be decreasing.
   • When $f(x) = 0$, $A(x)$ will be constant.

3. Match Each $f(x)$ to the Corresponding $A(x)$: Let's examine each graph one-by-one.

   • Graph (a) of $f(x)$: $f(x)$ is negative, constant over an interval, and then goes positive.

     • Corresponding $A(x)$ would start decreasing and then increase when $f(x)$ becomes positive. This matches with graph (C) in the $A(x)$ section.

   • Graph (b) of $f(x)$: $f(x)$ is positive and increases, implying a continually increasing $A(x)$ with an increasing slope.

     • This matches with graph (B), which shows a curve increasing at an accelerating rate.

   • Graph (c) of $f(x)$: $f(x)$ is positive, constant over an interval, and then negative.

     • The corresponding $A(x)$ will increase initially (due to the positive portion) and then start decreasing. This matches with graph (D).

   • Graph (d) of $f(x)$: $f(x)$ is positive, with an interval of constant zero in the middle.

     • The corresponding $A(x)$ would increase initially, remain constant when $f(x) = 0$, and then increase again. This behavior is consistent with graph (A).

Solution:

• $a = \text{D}$
• $b = \text{B}$
• $c = \text{C}$
• $d = \text{A}$

Let me know if you'd like further details on the reasoning or have any questions.

Questions for Further Exploration

1. How does the Fundamental Theorem of Calculus apply to finding area functions?
2. What changes in the area function $A(x)$ if $f(x)$ has discontinuities?
3. How does the behavior of $A(x)$ change if $f(x)$ has a sharp corner or cusp?
4. Can we determine the exact shape of $A(x)$ just from the sign of $f(x)$?
5. How does the rate of change of $f(x)$ impact the concavity of $A(x)$?

Tip

When analyzing area functions, always look for intervals where the function $f(x)$ is positive, negative, or zero; these directly influence whether $A(x)$ is increasing, decreasing, or constant in those intervals.