Rates of Change

%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

Slope and Rate of Change

t	speed
1	4
4 .	10
8	12

#rate of change from 1 to 4?

#rate of change from 4 to 8?

Slope

\[(x_1, y_1) \quad (x_2, y_2)\]

# slope of line through (5, 2) and (8, 4)

Equation of Line

# equation of line through (5, 2) and (8,4)

Secant

def f(x): return x**2
x = np.linspace(0, 3, 100)

plt.figure(figsize = (15, 5))

plt.plot(x, f(x))

plt.plot([1, 3], [f(1), f(3)], '-o', label = '(0, 3)')

plt.plot([1, 2], [f(1), f(2)], '-o', label = '(0, 2)')

plt.plot([1, 1.1], [f(1), f(1.1)], '-o', label = '(0, 1)')

plt.legend()

<matplotlib.legend.Legend at 0x122a25190>

Definition of Derivative

\[f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\]

Use this definition (and show work) to determine the derivative of:

• \(f(x) = 7x + 6\)
• \(f(x) = 4x^2 - 3x + 8\)

Writing a Derivative Function

The function below will compute the derivative of a function f at some point x. We are really approximating by letting h be a very small value.

def ddf(f, x, h = 0.0000001):
    derivative = (f(x + h) - f(x))/h
    return np.round(derivative, 4)

#example functions
def f(x):
    return x**2

#value of f at x=3
f(3)

9

#derivative of f at x=3
ddf(f, 3)

6.0

a = 3
print(f'The slope of the tangent line to f at x = {a} is {ddf(f, a)}')

The slope of the tangent line to f at x = 3 is 6.0

%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np

#plot the function and derivative together
x = np.linspace(-4, 4, 1000)
plt.plot(x, f(x), label = '$f$')
plt.plot(3, f(3), 'o',label = "$f(3)$")
plt.legend(fontsize = 20)
plt.title('Write an equation for the line through the point (3, 3)\ntangent to $f$.');

#problem 2
def f(x): return -x**2 + 1
x = np.linspace(-4, 4, 1000)
plt.plot(x, f(x), label = '$f$')
plt.plot(-1, f(-1), 'o',label = "$f(3)$")
plt.legend(fontsize = 20)
plt.title('Write an equation for the line through the point (-1, -1)\ntangent to $f$.');

Describe a general procedure for writing an equation for the line tangent to the graph of \(f\) at some point \(x = a\). This should culminate in an expression for the line in terms of \(f\) and \(f'\).

Power Rule

In general, we can compute the derivative of a given polynomial function using the power rule that states:

\[f(x) = x^n \quad \rightarrow \quad f'(x) = nx^{n-1}\]

Use the power rule to evaluate the derivative of the following functions:

• \(f(t) = 2t^5 - 7t^3 + 4t^2 - 6t + 1\)
• \(g(t) = 3t^4 - t^3 + 7t^2 - t + 17\)

Problem: Sequences

In derivatives, we find the following two patterns:

• \(f(x) = mx + b \rightarrow \quad f'(x) = m\)
• \(f(x) = ax^2 + bx + c \rightarrow \quad f'(x) = 2ax + b\)

Rephrase these symbolic statements in your own words. Should we have expected these results? What do they have to do with the sequences:

• 1, 3, 5, 7, 9, 11
• 1, 4, 9, 16, 25, 36