Problem 64 - Calculus: Early Transcendentals

Question

If \( \displaystyle f(x) = \int^x_0 (1 - t^2) e^{t^2} \,dt \), on what interval is \( f \) increasing?

Step-by-Step Solution

Answer

\(-1 < x < 1\)

Step 1: Understand the Function

The function given is \( f(x) = \int_0^x (1 - t^2) e^{t^2} \,dt \). To determine where \( f \) is increasing, we need to consider the derivative of \( f(x) \). According to the Fundamental Theorem of Calculus, the derivative of a definite integral with a variable upper bound is the integrand evaluated at that bound.

Step 2: Find the Derivative

Apply the Fundamental Theorem of Calculus to find the derivative of \( f(x) \). The derivative is \( f'(x) = (1 - x^2)e^{x^2} \). The sign of \( f'(x) \) determines whether \( f \) is increasing or decreasing.

Step 3: Determine When \( f'(x) > 0 \)

For \( f(x) \) to be increasing, \( f'(x) = (1 - x^2)e^{x^2} \) must be positive. Since the exponential function \( e^{x^2} \) is always positive, the sign of \( f'(x) \) depends on \( 1 - x^2 \). Therefore, \( 1 - x^2 > 0 \) leads to \( x^2 < 1 \), which simplifies to \(-1 < x < 1\).

Step 4: Conclude the Interval

Based on the inequality obtained in the previous step, \( f(x) \) is increasing on the interval \(-1 < x < 1\). This is because within this interval, the expression \( (1 - x^2)e^{x^2} \) is positive, thus \( f'(x) > 0 \).

Key Concepts

Increasing Functions

Understanding which intervals make a function increasing is essential in calculus. An increasing function is one where, as you move from left to right along the x-axis, the value of the function also goes up. But how can you determine this behavior mathematically? The secret lies in the derivative:

• If the derivative of a function, denoted as \(f'(x)\), is greater than zero, the function is increasing within that interval.
• If \(f'(x) < 0\), the function will be decreasing in that region.

In the given problem, \(f(x) = \int_0^x (1 - t^2)e^{t^2} \,dt\), to find where \(f\) is increasing, we determine where \(f'(x) = (1 - x^2)e^{x^2}\) remains positive. Specifically, solving the inequality \(1 - x^2 > 0\) (which simplifies to \(-1 < x < 1\)) tells us the intervals where \(f(x)\) climbs upwards.

Derivative of an Integral

The Fundamental Theorem of Calculus is a powerful tool that bridges the concept of differentiation and integration. It provides a way to find the derivative of an integral function. According to this theorem:

• The derivative of the integral of a function with respect to a variable, especially when that variable is the upper limit of the integral, is simply the original function.

In our specific example, the function \( f(x) = \int_0^x (1 - t^2) e^{t^2} \,dt \) uses the Fundamental Theorem of Calculus. By differentiating it, we find:

• \(f'(x) = (1 - x^2)e^{x^2}\)

This derivative helps us identify whether the function is increasing or decreasing, by analyzing the sign of \(f'(x)\).

Exponential Function

The exponential function, denoted as \(e^{x^2}\) in this problem, plays a crucial role. Exponential functions have unique properties:

• They are always positive, regardless of the value they are raised to.
• They grow rapidly, increasing the speed of growth as the exponent becomes larger.

In the context of this problem, because \(e^{x^2}\) is always positive, it doesn't affect the sign of \(f'(x)\) when determining intervals of increase. Instead, we only need to consider the term \(1 - x^2\). This simplifies our task as we only have to ensure that \(1 - x^2 > 0\) for our derivative to be positive, meaning the function is increasing.

Inequality Solutions

Solving inequalities is a fundamental algebraic skill that is particularly useful when studying the behavior of functions in calculus. In our problem, we need to solve \(1 - x^2 > 0\) to determine where the function \(f(x)\) is increasing.

• Start by rearranging terms: \(1 > x^2\).
• Recognize that this means \(-1 < x < 1\).

This solution gives us the interval where the function's derivative is positive, indicating it is increasing. Solving such elementary inequalities is crucial for analyzing where functions take on certain behaviors and for unlocking deeper understanding in calculus problems. Always check and interpret these solutions in the context of the original problem to ensure accuracy.