Problem 2 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Find the intervals on which \(f\) is increasing. Identify the local minima and local maxima. Plot the local minima and local maxima on a graph, and sketch a candidate graph of the function. a. \(f(x)=x^{2}-3 x+7\) b. \(f(x)=-x^{2}+5 x+16\) c. \(f(x)=3 x^{4}+8 x^{3}-6 x^{2}-24 x+17\) d. \(f(x)=x^{5}-5 x+2\) e. \(f(x)=x^{4}+4 x+8\) f. \(f(x)=x \ln x\) g. \(f(x)=x e^{-x}\) h. \(f(x)=x^{2} e^{-x}\)

Step-by-Step Solution

Answer

Find derivative, solve \( f'(x) = 0 \), and analyze sign changes for intervals and extrema.

Step 1: Take the Derivative

To find the intervals on which the function is increasing or decreasing, we first need to calculate the derivative of the function, denoted as \( f'(x) \). The derivative gives us the slope of the function at any point.

Step 2: Solve for Critical Points

Find the critical points by setting \( f'(x) = 0 \) and solving for \( x \). These points are where the function's slope changes sign and are potential locations for local minima and maxima.

Step 3: Determine Intervals of Increase/Decrease

Use a sign chart for \( f'(x) \) around the critical points to determine the intervals on which the function is increasing (where \( f'(x) > 0 \)) or decreasing (where \( f'(x) < 0 \)).

Step 4: Identify Local Minima and Maxima

Evaluate \( f(x) \) at the critical points from Step 2. Use the First Derivative Test: If \( f'(x) \) changes from positive to negative at a critical point, it's a local maximum. If \( f'(x) \) changes from negative to positive, it's a local minimum.

Step 5: Plot Local Minima and Maxima

Plot the points found in Step 4 on a graph. These are the local minima and maxima of the function.

Step 6: Sketch a Candidate Graph

Using the information gathered from the previous steps, sketch a candidate graph of the function, highlighting the intervals of increase and decrease as well as the local minima and maxima.

Key Concepts

Derivative

In calculus, the derivative of a function is a fundamental concept that represents the rate at which the function's value changes. Think of it as the slope of the tangent line at any point on the function's graph. Calculating the derivative, denoted as \( f'(x) \), helps us analyze how the function behaves. It tells us whether the function is increasing or decreasing, and helps locate critical points. Here, derivatives are calculated for functions like \(f(x)=x^{2}-3x+7\), and each derivative gives us a new function showing how the original function's slope varies across different \(x\).
The process of finding a derivative generally involves applying rules like the power rule or the product rule. For example, the derivative of \( x^n \) is \( nx^{n-1} \). With this tool, we're equipped to explore the fine details of the function's graph.

Critical Points

Critical points are essential in identifying where a function's graph changes direction. To find these points, we set the derivative \( f'(x) \) to zero and solve for \(x\). These solutions reveal where the function can potentially reach a peak or a valley.
Critical points occur where the function's slope is zero, meaning no change, or when the derivative does not exist. For each function in the given exercise, solving \( f'(x) = 0 \) helps locate these points. Critical points indicate possible locations of local minima or maxima, which guide our understanding of the function's overall shape. Understanding these aspects is crucial for graph analysis and optimization problems.

Local Minima and Maxima

Local minima and maxima are specific points on a function's graph where it reaches either a lowest or highest point, relative to nearby values. Once we locate the critical points, we use the First Derivative Test to classify them. Here's how:

• If \( f'(x) \) changes from positive to negative at a critical point, the function reaches a local maximum there.
• If \( f'(x) \) changes from negative to positive, it's a local minimum.

For the exercises, determining whether points are local minima or maxima helps shape the graph's profile. These points define the peaks and valleys and make clear the intervals of increase or decrease. Evaluating the function at these points gives a precise measurement of their heights or depths.

Graphing Functions

Graphing functions involves visualizing all the information gathered from derivatives, critical points, and local extrema. It simplifies understanding the function's behavior across its domain.
After identifying local minima and maxima, these points are plotted on a coordinate plane. We connect them while respecting the intervals of increase and decrease identified earlier.
This exercise entails sketching candidate graphs for functions like \(f(x)=x^2-3x+7\), highlighting important features such as:

• Intervals where the function is increasing or decreasing.
• The positions of local maxima and minima.
• The overall shape, which can represent different types of curves.

Creating a graph brings abstract numerical calculations into a tangible form, facilitating a deeper comprehension of the function's nature.