Problem 69 - Calculus: Early Transcendental Functions

Question

Find first and second derivatives of \(g(x)=\int_{0}^{x}\left(\int_{0}^{u} f(t) d t\right) d u,\) where \(f\) is a continuous function. Identify the graphical feature of \(y=g(x)\) that corresponds to a zero of \(f(x)\)

Step-by-Step Solution

Answer

The first derivative is \(g'(x)= \int_{0}^{x} f(t) dt\); the second derivative \(g''(x) = f(x)\). In context, a zero of \(f(x)\) corresponds to a point where \(y=g(x)\) becomes a horizontal straight line since there is no change in the area being accumulated.

Step 1: Find the first derivative \(g'(x)\)

We start by applying the Fundamental Theorem of Calculus Part 1, which states the derivative of the integral of a function from \(a\) to \(x\) is just the function evaluated at \(x\). Thus, we have that \(g'(x)= \left(\int_{0}^{x} f(t) dt\right)\)

Step 2: Find the second derivative \(g''(x)\)

To find the second derivative, we again use the Fundamental Theorem of Calculus Part 1 to take the derivative of the first derivative and obtain that \(g''(x) = f(x)\).

Step 3: Determine the graphical feature of \(y=g(x)\)

In calculus, when we have \(f(x) = 0\), it typically corresponds to a critical point of the function. In the case of the second derivative, zeroes often signify points of inflection. However, because our function graph is defined by integral accumulation, zero for \(f(x)\) means that the rate of change of \(y=g(x)\) becomes constant. This appears as a straight horizontal line in the graph of \(y=g(x)\).

Key Concepts

First Derivative

Finding the first derivative is an essential step in understanding how a function behaves as it changes. In this exercise, we have the function:\[ g(x) = \int_{0}^{x} \left( \int_{0}^{u} f(t) dt \right) du \]To find the first derivative, \( g'(x) \), we apply the Fundamental Theorem of Calculus. This theorem helps us take the derivative of an integral when the limits are variable. The theorem tells us that the derivative of an integral from a constant to \( x \) of a function is that function evaluated at \( x \). Thus:\[ g'(x) = \int_{0}^{x} f(t) dt \]This first derivative, \( g'(x) \), now tells us the function behavior—in this case, how the area under the curve of \( f(t) \) from 0 to \( x \) changes as \( x \) changes. This description aligns with how we interpret derivatives, as rates of change, or slopes of tangent lines.

Second Derivative

The second derivative gives us more detailed information about the function's behavior, particularly regarding its concavity. From the previous step, we know the first derivative is:\[ g'(x) = \int_{0}^{x} f(t) dt \]To find the second derivative, \( g''(x) \), we take the derivative of \( g'(x) \). Using the Fundamental Theorem of Calculus again, the derivative of this integral gives us:\[ g''(x) = f(x) \]This tells us that the second derivative is directly influenced by the original function \( f(x) \). When \( g''(x) \) is positive, \( g(x) \) is concave up, meaning it is shaped like a cup. Conversely, when \( g''(x) \) is negative, \( g(x) \) is concave down, like a cap. Understanding this helps in sketching and interpreting function graphs.

Graphical Features of Functions

The zeros of a function's derivatives are crucial in understanding its graphical behavior. The zero of \( f(x) \) in this exercise provides us with interesting insights. When \( f(x) = 0 \), the second derivative \( g''(x) = f(x) \) also becomes zero.Here's what these zero points imply in terms of the graphical features:

• Zero of \( f(x) \): This translates to \( g''(x) = 0 \), indicating a potential point of inflection for \( g(x) \) where the concavity may change.
• Constant Rate of Change: Since \( g''(x) = 0 \), \( g'(x) = \int_{0}^{x} f(t) dt \) reaches a constant rate. As a result, the function \( g(x) \) appears as a straight horizontal line at this point.

In graphical terms, whether \( g(x) \) continues as concave up or down beyond the zero depends on the sign change around \( x \). These insights are vital for sketching \( g(x) \) accurately and understanding its behavior without needing to plot every value.