Lab 05 - Differentiation and Tangent Lines.ipynb

Kernel: SageMath 9.2

Lab 05 - Differentiation and Tangent Lines

Overview

In this lab, we will learn how to use SageMath to find derivatives and the equation of the tangent line to a curve at a given point.

Important SageMath Commands Introduced in this Lab

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Related Course Material

Sections 3.1 and 3.2. 

Recall the point-slope form of the equation of the line: $$y - y_1 = m(x-x_1),$$ where $(x_1, y_1)$ is a point on the line and $m$ is the slope of the line. Next, since $(x_1, f(x_1))$ is on the tangent line, we can substitute $y_1 = f(x_1)$ and move it to the other side. Therefore, we get: $$y = m(x-x_1) + f(x_1).$$ Finally, we know that the derivative evaluated at $x_1$ is the same as the slpoe of the tangent line to the graph of $y = f(x)$ at $x_1$. Thus, we get the following formula for the equation of the tangent line to the graph of $y = f(x)$ at $x_1$: $$y = f'(x_1)(x-x_1) + f(x_1).$$

Example 1

In the previous lab, we learned how to use limits and the difference quotient to calculate the derivative of $f(x)$. In SageMath, a more direct way is to use the $\textbf{diff}(f(x),x)$ command to calculate $f'(x)$. This command can also be used to find higher order derivatives. The command $\textbf{diff}(f(x), x, n)$ will calculate $f^{(n)}(x),$ the $n^\text{th}$ derivative of $f(x)$.

Let $f(x) = x^{10}$. Use SageMath to find the following:

1. $f'(x)$

2. $f''(x)$

3. $f'''(x)$

4. $f^{(10)}(x)$

Suppose we want to calculate $f'(3)$. One thought might be to try $\textbf{diff}(f(3),x)$.

diff(f(3),x)

Note that this output is wrong. If we use this command, SageMath first calculates $f(3)$, and then takes the derivative of this constant which resulted in 0. Instead, we could do one of the following:

1. We can let $df(x) = f'(x)$ and then evaluate $df(3)$.

2. We can have SageMath calculate $f'(x)$, and then use our command $(\textit{expression})\textbf{(x=3)}$ to evaluate the function $f'(x)$ at $x=3.$

df(x) = diff(f(x),x)
df(3)

(diff(f(x),x))(x=3)

Example 2

Find $f'(x)$, $f''(10)$, and $f^{(10)}(\pi)$ for the following functions:

1. $f(x) = x^3 + 2x - 5$

2. $f(x) = x\cos(x)^2$

3. $f(x) = \sin(\cos(\tan(x)))$

Example 3

Consider the function $f(x) = x^2$. We will use SageMath to find the equation of the tangent line of $f(x)$ at $x = 1$ and to plot both the function and the tangent line. In order to find the equation of the tangent line, we need to find both $f(1)$ and $f'(1)$.

def f(x):
    return x^2
f(1)

df(x) = diff(f(x),x)
df(1)

Recall from the notes at the beginning of this lab that the equation of the tangent line of $f(x)$ at $x = 1$ is $y = f'(1)(x-1) + f(1).$ We use SageMath to find this line.

y = df(1)*(x-1) + f(1)
y

Therefore, $y = 2x - 1$ is the equation of the tangent line of $f(x)$ at $x = 1$. We now plot both $f(x)$ and the tangent line on the same graph. Choose a domain which has the $x$-value $x = 1$ in the center.

plot([f(x),y], xmin = 0, xmax = 2, color=['blue', 'red'], linestyle=['-', '--'])

Example 4

Repeat Example 3 with the following functions:

1. $f(x) = x^3 + 2x^2 + 1$ at $x = -1$

2. $f(x) = 2^x$ at $x = 2$

3. $f(x) = \cos(x)$ at $x = \frac{\pi}{4}$