2.1 Average and Instantaneous Rate of Change

1. Average Rate of Change

The average rate of change of a function \( f(x) \) over an interval \([a, x]\) is calculated by the following formula:

\[ \frac{f(x) - f(a)}{x - a} \quad \text{or} \quad \frac{f(a + h) - f(a)}{(a + h) - a} \]

This rate of change represents the slope of the secant line connecting two points on the function: \((a, f(a))\) and \((x, f(x))\).

2. Instantaneous Rate of Change

The instantaneous rate of change of a function \( f(x) \) at a specific point \( x = a \) is determined by taking the limit of the average rate of change as the interval approaches zero:

\[ \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \quad \text{or} \quad \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \]

This limit defines the slope of the tangent line at the point \( x = a \), which is also the derivative of \( f(x) \) at \( x = a \).

In summary:

• The average rate of change calculates how much a function changes over a certain interval (represented by a secant line).
• The instantaneous rate of change measures the rate of change at a specific point (represented by a tangent line) and is obtained by taking the limit as the interval approaches zero.

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2.2 Defining the Derivative of a Function and Using Derivative Notation

Definition of the Derivative

The derivative of a function is an expression that calculates the instantaneous rate of change (or the slope of the tangent line) of the function at any given \( x \)-value. In other words, it provides the slope of the function at a particular point.

Mathematically, the derivative of a function \( f(x) \) at a point \( x = a \) can be defined using two equivalent limit forms:

1. Using a small increment \( h \):

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

2. Using the alternate form with a limit \( x \) approaching \( a \):

   \[ f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \]

Both of these definitions provide the instantaneous rate of change of \( f(x) \) at any given \( x \)-value, capturing how \( f(x) \) changes as \( x \) changes infinitesimally.

Notation for the Derivative

There are several standard notations for representing the derivative of a function, most commonly attributed to Lagrange and Leibniz.

• Lagrange Notation: \( f'(x) \) or \( y' \)
• Leibniz Notation: \( \frac{dy}{dx} \)

Each notation has its own context of use, with Leibniz's notation often preferred in calculus when dealing with variables explicitly, such as \( \frac{dy}{dx} \), where \( y \) is a function of \( x \).

Equation of the Tangent Line

The tangent line to the curve of \( f(x) \) at a point \( x = a \) represents the best linear approximation of \( f(x) \) near \( x = a \). The equation of this tangent line can be written in point-slope form as:

\[ y - y_1 = m(x - x_1) \]

where \( m \) is the slope of the line, and \((x_1, y_1)\) is a point on the line.

Since the slope \( m \) of the tangent line at \( x = a \) is given by \( f'(a) \), we can rewrite the equation of the tangent line at \( x = a \) as:

\[ y - f(a) = f'(a)(x - a) \]

Summary

• The derivative \( f'(x) \) represents the instantaneous rate of change of \( f(x) \) at any point \( x \).
• The notation for derivatives includes both Lagrange's \( f'(x) \) and Leibniz's \( \frac{dy}{dx} \).
• The tangent line to \( f(x) \) at \( x = a \) can be found using the point-slope form \( y - f(a) = f'(a)(x - a) \).

This foundational understanding of derivatives and tangent lines is essential in calculus, as it allows us to analyze how functions behave locally and understand rates of change within various contexts.

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2.3 Estimating Derivatives of a Function at a Point

Estimating Derivatives from Tables

Estimating \( f'(3) \)

Given the following table for \( f(x) \):

\( x \) (hours)	\( f(x) \) (miles)
0	-2
2	3
4	10
7	1
11	-3

To estimate \( f'(3) \), we approximate the derivative by calculating the average rate of change of \( f(x) \) around \( x = 3 \). Since 3 is between 2 and 4, we can use the points \( (2, 3) \) and \( (4, 10) \).

The difference quotient formula is:

\[ f'(3) \approx \frac{f(4) - f(2)}{4 - 2} \]

Substituting the values from the table:

\[ f'(3) \approx \frac{10 - 3}{4 - 2} = \frac{7}{2} = 3.5 \]

So, \( f'(3) \approx 3.5 \).

Estimating \( w'(100) \)

Given the following table for \( w(x) \):

\( x \) (seconds)	\( w(x) \) (gallons per second)
10	950
50	850
80	700
120	500
150	150

To estimate \( w'(100) \), we calculate the average rate of change of \( w(x) \) around \( x = 100 \). Since 100 is between 80 and 120, we use the points \( (80, 700) \) and \( (120, 500) \).

The difference quotient formula is:

\[ w'(100) \approx \frac{w(120) - w(80)}{120 - 80} \]

Substituting the values from the table:

\[ w'(100) \approx \frac{500 - 700}{120 - 80} = \frac{-200}{40} = -5 \]

So, \( w'(100) \approx -5 \).

These estimates provide an approximation of the rate of change at the specified points using the data from the tables.

Estimating the Derivative of a Function at a Point Using a Graph

Identify the Point

Start by identifying the point on the graph where you want to estimate the derivative. Let’s say this point is \( (a, f(a)) \).

Understand What the Derivative Represents

The derivative at a point, \( f'(a) \), represents the slope of the tangent line to the graph at that point. This gives the instantaneous rate of change of the function at \( x = a \).

Draw the Tangent Line

Using the graph, carefully draw a line that touches the curve only at point \( (a, f(a)) \) and does not cross the curve. This line should "follow" the curve as closely as possible at that point. This is the tangent line.

Choose Two Points on the Tangent Line

Once you’ve drawn the tangent line, pick two points on this line to estimate its slope. It’s best to choose points that are easy to read from the graph, such as points where the tangent line crosses grid lines.

Calculate the Slope

Use the slope formula to calculate the derivative:

\[ f'(a) \approx \frac{y_2 - y_1}{x_2 - x_1} \]

where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points you selected on the tangent line.

Interpret the Result

The value of \( f'(a) \) gives you an estimate of the instantaneous rate of change of \( f(x) \) at \( x = a \). Positive values indicate an increasing function, while negative values indicate a decreasing function at that point.

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2.4 Connecting Differentiability and Continuity

Differentiability

Differentiability at a point means that the derivative exists at that point. For a function to be differentiable:

• The graph of the function must be smooth and without breaks, corners, or vertical slopes.
• The concept of local linearity is essential, meaning that if you zoom in on the graph, it should start looking like a straight line at every point within the domain.

If a function is differentiable at a point, it is also continuous at that point. However, the reverse isn’t always true—a continuous function may not be differentiable. Below are cases where differentiability fails.

Cases Where Differentiability Fails

The derivative does not exist in the following scenarios, which prevent a function from being differentiable:

Discontinuity

When there is a break or gap in the function, the function is not continuous at that point. Since differentiability implies continuity, a function that is not continuous at a point cannot be differentiable there.

Corner or Cusp

A sharp turn or abrupt change in direction, known as a corner or cusp, causes differentiability to fail. At a corner or cusp, the slope of the function changes suddenly, meaning the derivative from the left doesn’t match the derivative from the right.

Vertical Tangent

If the tangent line at a point is vertical (approaching an infinite slope), the derivative at that point doesn’t exist. In this case, the function might be continuous, but the vertical slope prevents differentiability.

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