Note: These are layman terms used to explain real life significance which are huge, so here making understand that simple an example.

Somebody said to me that they can’t understand where math and calculus fit into their lives. Here is a brief introduction to help clarify some concepts. The following curves are drawn with chatGPT help as my pen for screen is lost somewhere.

Let’s see the prompt:

“Consider the plot of life success with time. Plot it as an increasing function with time and decreasing form 50 years of age.”

ChatGPT generated the following, which is mostly ok to illustrate: life successes on the y-axis and time, measured by age, on the x-axis. Life success can be depicted as one or a compound of the following:

1 — Salary

2 — Health

3 — Family

4 — Business, and more likes

This graph illustrates that life success increases until the age of 50, after which it declines, in this case. Everyone’s life is not the same, some attains success when they are 80 year old, as well.

Note: This graph is not perfect but what I could draw as of now. Another graph with same prompt on Claude is:

Here is life’s pains graph. Its a sample, the graph can be multimodal as well.

Here is another kind of Pains graph drawn with Claude AI

and “What does integral describe” .

The integral of a curve in this context represents the cumulative life successover a specified age range, from 20 to 50. Here’s what it signifies:

Cumulative Measure:

• The integral sums up the “life success” values over time. It provides a total measure of success accumulated across the age range.

Area Under the Curve:

• Mathematically, it is the area under the success curve between the ages of 20 and 50. This area can be interpreted as the total “life achievements” or “progress” during this period.

This area represents the cumulative impact of your most productive years. It’s not just about peak success at 50, but the sustained high performance throughout your career. A larger area indicates more consistent success over time, which often translates to greater lifetime achievements, wealth accumulation, and professional legacy.

Significance in Life Success:

• If the curve represents accomplishments, wealth, or milestones achieved at each point in life, the integral tells us the total of these aspects accumulated over the specified period.

In simpler terms, it quantifies how much success has been accumulated from age 20 to 50. This could be seen as a way to evaluate the “productive years” of life.

Differentiate the curve above from 70 to 80

If we differentiate the life graph in the context of the given success curve, here’s what it conceptually means:

Rate of Change of Success:

• Differentiating the life graph measures the rate at which life success changes over time.
• A positive derivative indicates that success is increasing at that point in time (growth phase).
• A negative derivative indicates that success is decreasing at that point in time (decline phase).

Phases of Life:

• Early Years (0–50): The derivative is positive, indicating growth. Success accelerates as life progresses through education, career, and personal achievements.
• Middle to Later Years (50+): The derivative transitions to negative, reflecting a decline in success due to aging, reduced activity, or other challenges.

Turning Points in the first graph above:

• Peak Success (50 years): The derivative is zero at the peak of the curve, representing a transition point where success stops increasing and begins to decline.
• Inflection Points: Changes in the concavity of the graph (e.g., acceleration or deceleration of growth/decline) can be identified through second derivatives.

Significance of Derivative Analysis:

Understanding Growth and Decline:

• When the derivative is high and positive, success is rapidly increasing (e.g., in the 20–40 range).
• When the derivative is negative and steep, success is rapidly declining (e.g., after 70).

Life Optimization:

• Derivatives help identify periods of maximum growth (e.g., ages where success accelerates) or decline (e.g., where efforts should shift to maintain stability).

Real-Life Implications:

• Positive Derivative: Indicates progress — career advancements, personal achievements, etc.
• Negative Derivative: Suggests setbacks — loss of productivity, health challenges, or waning achievements.
• Zero Derivative: Denotes stability, where success is neither growing nor declining.

Summary

Area of curve is cummulative sum of individual successes.

Differentiating the life graph gives insight into the dynamics of life success— highlighting periods of growth, stability, and decline. It provides a mathematical way to analyze life’s trajectory and identify critical moments for reflection, adjustment, or celebration.

Photo and Content Credits: OpenAI and Claude AI