Problem 63 - Biocalculus Calculus for the Life Sciences

Question

(a) Show that \(1 \leqslant \sqrt{1+x^{3}} \leqslant 1+x^{3}\) for \(x \geqslant 0\) . (b) Show that 1\(\leqslant \int_{0}^{1} \sqrt{1+x^{3}} d x \leqslant 1.25\)

Step-by-Step Solution

Answer

The inequalities and integral bounds are proven for the given expressions.

Step 1: Show the lower bound inequality

We start by providing the argument that helps to show the left inequality \( 1 \leq \sqrt{1 + x^3} \). Since \( x \) is non-negative, this implies that \( x^3 \geq 0 \). Thus, \( 1 + x^3 \geq 1 \). Since the square root function is increasing, taking the square root of both sides gives \( \sqrt{1 + x^3} \geq \sqrt{1} = 1 \). Therefore, the lower bound inequality is satisfied.

Step 2: Show the upper bound inequality

For the upper bound \( \sqrt{1 + x^3} \leq 1 + x^3 \), note that since \( x \geq 0 \), we have \( x^3 \geq 0 \). Hence, \( 1 \leq 1 + x^3 \), and both sides are greater than 1. By squaring \( \sqrt{1 + x^3} \leq 1 + x^3 \), we get \( 1 + x^3 \leq (1 + x^3)^2 \), which is valid as \( x^3 \geq 0 \). Thus, the right-side inequality holds.

Step 3: Show the lower bound for the integral

Now we need to show \( 1 \leq \int_{0}^{1} \sqrt{1+x^3} \, dx \). Since \( \sqrt{1+x^3} \geq 1 \) for all \( x \geq 0 \), the integral \( \int_{0}^{1} \sqrt{1+x^3} \, dx \geq \int_{0}^{1} 1 \, dx = 1 \). This verifies the lower bound for the integral.

Step 4: Show the upper bound for the integral

To show \( \int_{0}^{1} \sqrt{1+x^3} \, dx \leq 1.25 \), we use the upper bound \( \sqrt{1+x^3} \leq 1 + x^3 \). Thus, \( \int_{0}^{1} \sqrt{1+x^3} \, dx \leq \int_{0}^{1} (1 + x^3) \, dx \). Evaluating the integrals gives:\[\int_{0}^{1} (1 + x^3) \, dx = \left[ x + \frac{x^4}{4} \right]_{0}^{1} = \left(1 + \frac{1}{4}\right) = 1.25\]Thus, the upper bound for the integral is shown.

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics that helps us find quantities like areas under curves, volumes, and more, using integrals. The integration process accumulates small slices, typically over a particular range, to compute a total. It's practical in scenarios where physically measuring something is difficult.
In the context of the exercise, we deal with a definite integral, which represents the area under the curve of the function from one point to another. Here, we calculate the definite integral of \( \sqrt{1+x^3} \) from 0 to 1.

• This gives us an idea of the values that the integral of the function will cover.
• It helps us understand the magnitude of change in the function over the interval \([0, 1]\).

To show bounds, we must calculate integrals of related functions as done in Steps 3 and 4 in the solution, using known bounds on \(\sqrt{1+x^3}\) to find bounds of the integral.

Square Root Function

The square root function is fundamental in mathematics, characterized by the expression \( \sqrt{x} \). It behaves by taking any non-negative number \( x \) and returning another non-negative number whose square equals \( x \).
The square root function is increasing, meaning as \( x \) increases, so does \( \sqrt{x} \).
In the exercise, we are considering \( \sqrt{1+x^3} \). Understanding these properties allows us to analyze and establish inequalities: knowing that the square root function is increasing tells us that if \( 1 + x^3 \geq 1 \), then \( \sqrt{1 + x^3} \geq \sqrt{1} = 1 \).

• Important for understanding why the bounds exist.
• Assists correct evaluation of function behaviors over specified intervals.

These characteristics are crucial to establish proofs like those in the exercise.

Bounds of Integration

Bounds of integration specify the interval over which we perform an integration. In other words, it's the range of the \( x \)-values we consider. This range can fundamentally affect the value of an integral.
For the given problem, we integrate from 0 to 1.

• The lower bound is 0, representing the starting point of integration.
• The upper bound is 1, signifying where the integration stops.

Knowing the bounds of integration is imperative because it confines our problem's scope, ensuring we are considering the correct part of the function that satisfies the mathematical inequalities laid out.
These bounds also help in evaluating inequalities for integrals as seen in the steps provided where the integral was broken down over \([0, 1]\).

Non-negative Functions

A non-negative function means that for every input, \( x \), the output is zero or positive. When dealing with inequalities, verifying that a function is non-negative is essential. In this exercise, \( \sqrt{1+x^3} \) is a non-negative function because it outputs non-negative values for non-negative inputs.
Some crucial points to remember:

• Non-negative functions ensure integrals evaluate to non-negative numbers.
• They confirm the validity of inequalities.
• They are useful for proving bounds, as seen with the function of this exercise over specified intervals.

Thus, understanding the nature of non-negative functions helps establish bounds as done with Step 1 and Step 3, where it's shown that the function remains non-negative ensuring valid inequality conclusions.