Calculus of One Variable with SageMath - P2

 

Welcome back to our "Calculus of One Variable with SageMath" series! In this second part, we'll dive into key calculus concepts such as derivatives, hyperbolic functions, implicit derivatives, and the geometric meaning of the Mean Value Theorem. Using SageMath, we'll explore these concepts with both symbolic and graphical approaches, ensuring a deep understanding of the underlying mathematics.

1. Derivatives

1.1 Derivatives Using the First Principle

One of the fundamental ways to calculate the derivative of a function is using the first principle, or limit definition of the derivative. Let's compute the derivative of the function

By using SageMath, we can find the derivative symbolically and also compute its value at a specific point. However, remember that the first principle is particularly useful when working with basic definitions of derivatives and can help solidify your understanding of how derivatives are defined.

1.2 Hyperbolic Functions

Hyperbolic functions, such as sinh(x) and cosh(x), are analogs of the regular trigonometric functions but are defined using exponential functions. Let's plot these functions to better understand their behavior.

These plots give a clear visual of how these hyperbolic functions behave over a range of values. Notice that both  and  are continuous and smooth, just like their trigonometric counterparts.

1.3 Implicit Derivative

Sometimes we encounter functions that are defined implicitly rather than explicitly. For example, consider the equation:

To find the derivative ​, we can use implicit differentiation in SageMath.

We can also plot the curve and visualize the slope of the tangent at specific points.

This shows how implicit differentiation works and how to graphically represent the tangent line at a given point.

1.4 Geometric Meaning of the Mean Value Theorem

The Mean Value Theorem (MVT) states that for a continuous and differentiable function on a closed interval, there exists at least one point c where the derivative is equal to the average rate of change over the interval.

Let’s compute the value(s) of ccc for the function:

on the interval

 

By plotting the graph and finding the points where the derivative matches the slope of the secant line, we visually demonstrate the MVT.

1.5 Taylor’s Theorem

Taylor's theorem is a powerful tool for approximating functions near a point using polynomials. The n-th degree Taylor polynomial of a function  at a point a is given by:

Let’s compute and plot the Taylor series approximation for the function:

for n=10.

We can see how the Taylor polynomial approximates the function and the error involved for different degrees.

🔜 What's Next? Practice Exercises to Sharpen Your Skills!

In this section, we'll dive into a variety of practice exercises to strengthen your understanding of key calculus concepts, including derivatives, implicit differentiation, the Lagrange Mean Value Theorem, Taylor's polynomial approximations, and error analysis. Let’s break it down step by step!