Problem 5 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Identify the intervals, if any, on which \(f(x)\) is increasing and intervals, if any, on which \(f^{\prime}\) is increasing. a. \(f(x)=x^{2}-1 \quad-2 \leq x \leq 2\) b. \(f(x)=x^{2}-x \quad-2 \leq x \leq 2\) c. \(f(x)=x^{3}-x^{2} \quad-2 \leq x \leq 2\) d. \(f(x)=\frac{x}{x^{2}+1} \quad-2 \leq x \leq 2\) e. \(f(x)=e^{-x} \quad-2 \leq x \leq 2\) f. \(f(x)=x e^{-x} \quad 0 \leq x \leq 3\) g. \(f(x)=e^{-x^{2}} \quad-2 \leq x \leq 2\) h. \(\quad f(x)=e^{-2 x}-e^{-x} \quad 0 \leq x \leq 4\) i. \(f(x)=\ln x\) \(0

Step-by-Step Solution

Answer

Identify intervals by solving derivatives for increase/decrease.

Step 1: Calculate the Derivative of f(x) for Each Function

For each given function, calculate the derivative, which is denoted by \( f'(x) \). The derivative will help us identify where the function is increasing by analyzing sign changes in \( f'(x) \).

Step 2: Find Critical Points and Analyze f'(x)

Calculate where the derivative \( f'(x) \) is equal to zero or undefined. These points are known as critical points and are potential indicators of where \( f(x) \) could change from increasing to decreasing or vice versa. Additionally, observe intervals where \( f'(x) \) is positive to determine intervals where \( f(x) \) is increasing.

Step 3: Determine Intervals of Increasing Decreasing

Perform a test on intervals determined by the critical points to see whether \( f(x) \) is increasing or decreasing. Use test points within these intervals and plug them back into \( f'(x) \). If \( f'(x) > 0 \) at the test point, \( f(x) \) is increasing on that interval; if \( f'(x) < 0 \), it is decreasing.

Step 4: Examine Sign Changes in f'(x) for Increasing

Look further into \( f''(x) \), the derivative of \( f'(x) \), to evaluate where \( f'(x) \) is increasing or decreasing. By determining sign changes in \( f''(x) \), you can state the intervals where \( f'(x) \) itself is increasing.

Key Concepts

Derivatives

The concept of derivatives is central to calculus and involves measuring how a function changes at any given point. Essentially, a derivative represents the rate of change or the slope of a function at a specific point. It is expressed as \(f'(x)\), the first derivative.To find the derivative, you use a process called differentiation. This process applies rules to functions, such as the power rule, product rule, and chain rule, to find \(f'(x)\). For a simple function like \(f(x) = x^2 - 1\), differentiation gives \(f'(x) = 2x\). This indicates how fast \(f(x)\) is changing at every point along the curve.
Derivatives are used not only to find out if functions are increasing or decreasing but also in a variety of fields like physics, economics, and engineering.By studying the sign of \(f'(x)\), you can gain valuable insights into the behavior of the function. If \(f'(x) > 0\), the function \(f(x)\) is said to be increasing, while \(f'(x) < 0\) indicates that it is decreasing.

Critical Points

Critical points are where the derivative of a function is zero or undefined. These points are crucial because they can indicate where a function changes from increasing to decreasing (or vice versa) or where a local minimum or maximum could occur.To find these points, you set the derivative \(f'(x)\) equal to zero and solve for \(x\). For example, if \(f'(x) = 2x\), setting \(2x = 0\) gives \(x = 0\). This is a critical point, and further analysis can tell whether it's a point of increase or decrease.
If the second derivative \(f''(x)\), which measures the curvature of the function, is positive at a critical point, the function has a local minimum there. Conversely, if \(f''(x) < 0\), there is a local maximum.Critical points also help partition the function into different intervals for deeper analysis, revealing more about the function’s overall behavior and trends.

Increasing Intervals

Increasing intervals are segments of the function's domain where the function is continuously growing. To determine these intervals, you need to examine the sign of the derivative \(f'(x)\) across the domain.When \(f'(x)\) is positive over an interval, the function is increasing over that interval. After finding the critical points, test intervals around these points by plugging test values back into \(f'(x)\).
For instance, consider \(f(x) = x^2 - 1\). Its derivative, \(f'(x) = 2x\), is positive when \(x > 0\), indicating the function is increasing from \(x=0\) to the boundary of the domain.These intervals inform us of where the function's output consistently rises as \(x\) increases, which is vital for understanding the function’s behavior over specified domains.

Mathematical Analysis

Mathematical analysis involves delving deeper into the characteristics and behaviors of functions. It encompasses the study of limits, derivatives, integrals, and infinite series.In this context, analysis means considering various elements of the function, such as derivatives and critical points, to interpret its behavior fully. It includes solving for derivatives, finding when they are zero or undefined, and using these results to identify intervals of increase or decrease. Analysis extends to investigating \(f'(x)\) and \(f''(x)\). By examining \(f''(x)\), or the second derivative, you can assess the concavity of \(f(x)\).
Analysis often reveals intricate details about the nature of functions, making it foundational in mathematics for solving real-world problems.