Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the interval(s) on which the graph of \(y=f(x), x \geq 0\), is (a) increasing, and (b) concave up. $$ f(x)=\int_{1}^{x} \frac{1}{\theta} d \theta, x>0 $$

Step-by-Step Solution

Answer

(a) Increasing for all \(x > 0\); (b) never concave up for \(x > 0\).

Step 1: Understanding Increasing Functions

A function is increasing on an interval if its derivative is positive over that interval. For the function \(f(x) = \int_{1}^{x} \frac{1}{\theta} d\theta\), apply the Fundamental Theorem of Calculus to find that \(f'(x) = \frac{1}{x}\).

Step 2: Determine When f(x) is Increasing

The derivative \(f'(x) = \frac{1}{x}\) is positive for \(x > 0\). Therefore, \(f(x)\) is increasing for all \(x > 0\).

Step 3: Understanding Concavity and Second Derivative

A function is concave up on an interval if its second derivative is positive over that interval. To find the second derivative, differentiate \(f'(x) = \frac{1}{x}\) with respect to \(x\).

Step 4: Calculate the Second Derivative

The second derivative \(f''(x) = -\frac{1}{x^2}\).

Step 5: Determine Concavity

The second derivative \(f''(x) = -\frac{1}{x^2}\) is negative for all \(x > 0\). Hence, \(f(x)\) is concave down for \(x > 0\) and never concave up in this domain.

Key Concepts

Increasing Functions

An increasing function is one where, as you move from left to right along the x-axis, the function's values also rise. In simpler terms, if you pick any two points where one is to the left of the other, the function at the right point should be higher. This behavior is detected by examining the derivative of the function.

For our function, given by the definite integral \( f(x) = \int_{1}^{x} \frac{1}{\theta} d\theta \), the Fundamental Theorem of Calculus helps us determine its behavior. According to this theorem, the derivative \( f'(x) \) is \( \frac{1}{x} \). The crucial part is that whenever this derivative is positive, the function \( f(x) \) will be increasing.

• Since \( f'(x) = \frac{1}{x} \) is positive for all \( x > 0 \), the function is increasing on the entire interval \( x > 0 \).

This means that on any segment of the x-axis beyond zero, the function values will continue to rise.

Concavity

Concavity deals with how the slope of a function is changing. Imagine the graph as the surface of a bowl or spoon; it can either be curving upwards or downwards. If it's curving upwards like a valley, we call it concave up. Conversely, if it's curving downwards like a mountain, it's concave down. Concavity is dictated by the second derivative of a function.

To determine concavity for our function, we need the second derivative. Starting from the first derivative \( f'(x) = \frac{1}{x} \), we differentiate once more to get the second derivative, \( f''(x) = -\frac{1}{x^2} \).

• Here, \( f''(x) \) is always negative for \( x > 0 \).

A negative second derivative means the function is concave down, signifying that the graph is curved downward like the top of a hill, and thus never curving upward in this interval. This means \( f(x) \) cannot be concave up for any \( x > 0 \).

Derivatives

Derivatives play a fundamental role in understanding the behavior of a function. They tell us how the function's rate of change behaves. Essentially, the derivative at a point on a graph describes the slope of the tangent to the curve at that point.

For the given function \( f(x) = \int_{1}^{x} \frac{1}{\theta} d\theta \), the Fundamental Theorem of Calculus allows us to calculate its derivative, known as \( f'(x) \). This derivative is \( \frac{1}{x} \), which indicates the slope of the tangent. The slope tells us about the increasing or decreasing nature of the function.

• When \( f'(x) = \frac{1}{x} > 0 \) for all \( x > 0 \), it assures us that \( f(x) \) is strictly increasing across its domain.

To explore further, calculating the second derivative tells us about concavity. Thus, derivatives offer comprehensive insights into both the slope and shape of a function's graph, influencing how we interpret its growth and curvature.