Problem 31 - Calculus for Biology and Medicine

Question

Determine whether the functions have absolute maxima and minima, and, if so, find their coordinates. Find inflection points. Find the intervals on which the function is increasing, on which it is decreasing, on which it is concave up, and on which it is concave down. Sketch the graph of each function. $$ y=x+\cos x, x \in \mathbf{R} $$

Step-by-Step Solution

Answer

The function has no absolute extrema. Inflection points: \(x = \frac{\pi}{2} + k\pi\). Increasing on \((-\frac{\pi}{2} + 2k\pi, \frac{\pi}{2} + 2k\pi)\), decreasing on \((\frac{\pi}{2} + 2k\pi, \frac{3\pi}{2} + 2k\pi)\).

Step 1: Find the First Derivative

To determine where the function is increasing or decreasing, we first find the first derivative of the function. Given the function \(y = x + \cos x\), we differentiate to get:\[y' = \frac{d}{dx}(x + \cos x) = 1 - \sin x\]

Step 2: Solve First Derivative for Critical Points

The critical points are where the first derivative is zero or undefined. Here, the first derivative is:\[1 - \sin x = 0\]Solving for \(x\), we have:\[\sin x = 1\]Since \(x \in \mathbb{R}\), the critical points are:\[x = \frac{\pi}{2} + 2k\pi, \text{where } k \in \mathbb{Z}\]

Step 3: Determine Increasing and Decreasing Intervals

The sign of \(1 - \sin x\) will determine the increasing or decreasing nature of the function. Since \(\sin x\) oscillates between -1 and 1, \(1 - \sin x > 0\) when \(-1 \leq \sin x < 1\), and \(1 - \sin x < 0\) when \(\sin x > 1\). Thus, the function is increasing on intervals:\[x \in \left(-\frac{\pi}{2} + 2k\pi, \frac{\pi}{2} + 2k\pi\right)\]and decreasing on intervals:\[x \in \left(\frac{\pi}{2} + 2k\pi, \frac{3\pi}{2} + 2k\pi\right)\]for integer \(k\).

Step 4: Find the Second Derivative

To determine intervals of concavity, find the second derivative:\[y'' = \frac{d}{dx}(1 - \sin x) = -\cos x\]

Step 5: Analyze Second Derivative for Concavity

The intervals of concavity are determined by the sign of the second derivative \(-\cos x\). Since \(\cos x\) oscillates between -1 and 1:\[-\cos x > 0 \text{ (concave up) when } \cos x < 0\] and\[-\cos x < 0 \text{ (concave down) when } \cos x > 0\].These intervals are:\[-\frac{\pi}{2} + 2k\pi < x < \frac{\pi}{2} + 2k\pi \text{ for concave down},\]and\[\frac{\pi}{2} + 2k\pi < x < \frac{3\pi}{2} + 2k\pi \text{ for concave up}\]for integer \(k\).

Step 6: Identify Inflection Points

Inflection points occur where the concavity changes, which happens where \(-\cos x = 0\). Solving:\[-\cos x = 0 \Rightarrow \cos x = 0\]The solutions are:\[x = \frac{\pi}{2} + k\pi, \text{ where } k \in \mathbb{Z}\]These are the locations of the inflection points.

Step 7: Examine Absolute Maxima and Minima

Given the periodic nature and unbounded increase of \(x\), the function \(y = x + \cos x\) does not have absolute maxima or minima across its domain.

Key Concepts

Derivative

In calculus, the derivative of a function is a fundamental concept that represents the rate of change of the function with respect to a variable. When we talk about finding the derivative, we're essentially trying to understand how the function behaves as its input values change. For the function given as \(y = x + \cos x\), taking the derivative helps us understand where the function is increasing or decreasing.

By computing the first derivative, \( y' = 1 - \sin x\), we gain insight into the slope of the tangent at any point \(x\). This derivative shows the change in \(y\) with respect to \(x\). A positive derivative results in an increasing function, and a negative derivative results in a decreasing function.

Understanding the derivative is crucial for studying functions in depth and is widely applicable not only in calculus but also in various areas like biology and medicine, where change rates are pivotal.

Critical Points

Critical points of a function can be found by solving \( y' = 0 \) or if \( y' \) is undefined. They allow us to find where the function may have local maxima or minima. In our problem, critical points arise when \( 1 - \sin x = 0 \), simplifying to \( \sin x = 1 \).

The periodic nature of \( \sin x \) means that the critical points are at \( x = \frac{\pi}{2} + 2k\pi \), where \( k \) is an integer. These are the points where the function changes its direction – switching from increasing to decreasing, or vice versa.

In biology and medicine, identifying critical points can help determine key phases or changes in biological models, like when a population might reach maximum growth before stabilizing.

Concave Up and Down

Concavity describes how the direction of a curve changes, which is determined by the second derivative. Concave up resembles a U-shape and indicates that the tangent line lies below the curve, while concave down resembles an n-shape and indicates that the tangent line lies above.

For the function \( y = x + \cos x \), the second derivative is \( y'' = -\cos x \). Analyzing its sign tells us about the concavity:

• \( -\cos x > 0 \) suggests the function is concave up. This happens when \( \cos x < 0 \), between intervals like \( \frac{\pi}{2} + 2k\pi < x < \frac{3\pi}{2} + 2k\pi \).
• \( -\cos x < 0 \) suggests the function is concave down. This occurs when \( \cos x > 0 \), within intervals such as \( -\frac{\pi}{2} + 2k\pi < x < \frac{\pi}{2} + 2k\pi \).

These aspects of concavity are very relevant in biological models, where knowing the concavity gives insights into biological processes, such as blood flow dynamics or the bending of DNA.

Inflection Points

Inflection points occur where the function changes concavity, from concave up to concave down or vice versa. For the function \( y = x + \cos x \), the inflection points are determined by setting the second derivative equal to zero, \( -\cos x = 0 \).

Solving gives \( \cos x = 0 \), leading to points \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer. These inflection points mark where the curve switches its bending direction.

In the context of biology and medicine, understanding inflection points can be used for modeling biological systems under stress, such as predicting changes in metabolic pathways or response curves to medications. Recognizing where these changes occur enables more precise fine-tuning of models to real-world data.