Derivatives - Graphically and Numerically

Learning Objectives

• Procedural Skills:

  • Given a function and a point, sketch the corresponding tangent line.
  • Use the tangent line to estimate the value of the derivative at a point.
  • Recognize keywords in a context that correspond to the derivative.
  • Understand the general procedure to sketch the derivative of the plot of a function.

• Interpretation Skills:

  • Compare and contrast secant lines and tangent lines.
  • Understand the relationship between the tangent line of a function at a point and the derivative of a function at that point.
  • Explain the difference between the derivative at a point and the average rate of change (AROC) over an interval.

Instantaneous Rate of Change and the Derivative at a Point

• Definition of Instantaneous Rate of Change:

  • The instantaneous rate of change of a function $f(x)$ at point $x = a$ is defined as the limit of the average rate of change (AROC) of $f(x)$ over the interval $[a, a + h]$ as $h$ approaches zero ($h \rightarrow 0$ or $\Delta x \rightarrow 0$):     $f'(a) := \lim_{h \rightarrow 0} \frac{f(a + h) - f(a)}{h}$
  • The notation $f'(a)$ is read as "the derivative of $f(x)$ at the point $x = a$".

• Relationship Between Secant Lines, Tangent Lines, and Derivatives:

  • A secant line connects two points on the graph of $f(x)$ and its slope represents the average rate of change (AROC) over that interval.
  • Considering intervals such as $[0, 6]$, $[2, 6]$, and $[5, 6]$, as the interval width $h$ becomes smaller, the corresponding secant lines rotate and converge toward a limiting line known as the tangent line at $x = 6$.

• The solid line shown in the graph represents the tangent line of $f(x)$ at $x = 6$
• The slope of the tangent line at $x = a$ equals the instantaneous rate of change of $f(x)$ at $x = a$, which is precisely the derivative $f'(a)$.
• Consequently, the derivative of a function at a given point can be estimated geometrically by drawing the tangent line at that point and computing its slope.

Estimating Derivatives and Contextual Applications

• Example: Bacteria Population Growth Rate:
  • Let $g(t)$ represent the population of bacteria $t\,\text{minutes}$ after an infection starts.

• Estimating $g'(5)$:
  • To sketch the tangent line at $t = 5$, select a small interval around $t = 5$ and construct the line tangent to $g(t)$ at that point.

- Identify two coordinates on the sketched tangent line: $(4, 0)$ and $(7, 450)$.
- Calculate the slope of the tangent line:

      $g'(5) \approx \frac{450 - 0}{7 - 4} = \frac{450}{3} = 150$ - Including physical units in the derivative calculation:       $g'(5) \approx \frac{(450 - 0)\,\text{bacteria}}{(7 - 4)\,\text{minutes}} = 150\,\text{bacteria}\,\text{minute}^{-1}$ - Practical interpretation: Exactly $5\,\text{minutes}$ into the infection, the bacteria population is increasing (indicated by the positive sign) at an instantaneous rate of $150\,\text{bacteria}\,\text{minute}^{-1}$.

• Graph Analysis Exercise:
  • Consider the plot of $f(x)$ shown below:

• Tangent Line Construction:
  • Draw tangent lines to $f(x)$ at $x = -2$, $x = 0$, and $x = 1.1$
• Estimation of $f'(0)$:
  • At $x = 0$, the curve passes through $(0, -6)$ going downwards from left to right. Estimating the slope of the tangent line at $x = 0$ gives $f'(0) \approx -5$
• Ordering the Derivatives:
  • At $x = -2$, the function is near a local peak, so the slope $f'(-2)$ is slightly positive or near zero.
  • At $x = 0$, the function is decreasing moderately, so $f'(0)$ is negative.
  • At $x = 1.1$, the curve is decreasing steeply near the inflection point, so $f'(1.1)$ has the largest negative magnitude.
  • Ordering from least to greatest:       $f'(1.1) \le f'(0) \le f'(-2)$

The Derivative Function and Graphical Principles

• Concept of the Derivative Function $f'(x)$:

  • Evaluating the derivative of a function $f(x)$ at every point across its domain creates a new function called the derivative function $f'(x)$.
  • To construct the graph of $f'(x)$ from $f(x)$:
  • Sample multiple $x$-values in the domain of $f(x)$.
  • Determine or estimate the numerical derivative (slope of tangent line) at each point.
  • Plot points with coordinates $(x, f'(x))$.
  • Connect the plotted points smoothly, assuming $f'(x)$ has no jumps or gaps.

• Behavior of the Derivative relative to $f(x)$:

  • As $f(x)$ "flattens out" (approaches a horizontal tangent), $f'(x)$ approaches $0$
  • On intervals where $f(x)$ is increasing (rising from left to right), $f'(x) > 0$ (above the $x$-axis).
  • On intervals where $f(x)$ is decreasing (falling from left to right), $f'(x) < 0$ (below the $x$-axis).

• Guidelines for Sketching $f'(x)$ Given $f(x)$:

  • Locate all points where the derivative is zero ($f'(x) = 0$). These occur at local maximums, local minimums, or horizontal inflection points of $f(x)$.
  • Partition the domain into intervals based on where $f(x)$ is increasing or decreasing to assign positive or negative values to $f'(x)$.
  • Estimate the instantaneous rate of change at representative key points to establish specific heights for the $f'(x)$ curve.

• Generalizations Connecting $f(x)$ and $f'(x)$:

  • If $f' > 0$ everywhere on an interval, then $f$ is increasing over that interval.
  • If $f' < 0$ everywhere on an interval, then $f$ is decreasing over that interval.
  • If $f' = 0$ everywhere on an interval, then $f$ is constant over that interval.

Graphical Derivative Practice Examples

• Example: Plotting the Derivative of a Cubic-like Curve:

• Analysis of $f(x)$,:

  • Local maximum occurs at $x \approx 1.5$, so $f'(1.5) = 0$
  • Local minimum occurs at $x \approx 5.5$, so $f'(5.5) = 0$
  • Interval $(-\infty, 1.5)$: $f(x)$ is increasing, so $f'(x) > 0$
  • Interval $(1.5, 5.5)$: $f(x)$ is decreasing, so $f'(x) < 0$
  • Interval $(5.5, \infty)$: $f(x)$ is increasing, so $f'(x) > 0$

• Sketching $f'(x)$: The derivative graph is a parabola opening upwards, intersecting the $x$-axis at $x \approx 1.5$ and $x \approx 5.5$, reaching a negative minimum between $x = 1.5$ and $x = 5.5$

  • Example: Sketching Derivatives for Specific Function Types:

• Function (a): Parabola opening upwards.

- Minimum vertex at $x = 3$ where $f'(3) = 0$
- For $x < 3$, $f(x)$ decreases, so $f'(x) < 0$
- For $x > 3$, $f(x)$ increases, so $f'(x) > 0$
- The derivative graph $f'(x)$ is a straight line with a positive slope passing through $(3, 0)$.

• Function (b): Cubic function.

- Local maximum near $x = 1.5$ and local minimum near $x = 5.2$
- $f'(x) = 0$ at $x \approx 1.5$ and $x \approx 5.2$
- Derivative graph $f'(x)$ is an upward-opening parabola with roots at $x \approx 1.5$ and $x \approx 5.2$

• Function (c): Linear function.

- Linear function passing through $(0, 5)$ and $(8, 29)$.
- Constant slope calculation:

      $m = \frac{29 - 5}{8 - 0} = \frac{24}{8} = 3$ - The derivative function $f'(x) = 3$ is a constant horizontal line at $y = 3$

Numerical and Tabular Derivative Evaluations

• Evaluating Derivative Signs for $f(x) = x^2$:

  • Consider the parabola $f(x) = x^2$:
  • (a) $f'(1)$: Positive ($f'(1) = 2(1) = 2 > 0$), because $f(x)$ is increasing at $x = 1$
  • (b) $f'(-1)$: Negative ($f'(-1) = 2(-1) = -2 < 0$), because $f(x)$ is decreasing at $x = -1$
  • (d) $f'(0)$: Zero ($f'(0) = 2(0) = 0$), because $x = 0$ is the global minimum of the parabola where the tangent line is horizontal.

• Estimating Derivatives from Tabular Data:

  • Farm Land Area Data (1980–2000):

• Problem: Estimate the instantaneous rate of change of farm land area in the year $1990$ ($t = 1990$).
• Estimation Methods:
  • Left Average Rate of Change ($[1985, 1990]$):       $\text{AROC}_{[1985, 1990]} = \frac{987 - 1012}{1990 - 1985} = \frac{-25}{5} = -5\,\text{million acres}\,\text{year}^{-1}$
  • Right Average Rate of Change ($[1990, 1995]$):       $\text{AROC}_{[1990, 1995]} = \frac{963 - 987}{1995 - 1990} = \frac{-24}{5} = -4.8\,\text{million acres}\,\text{year}^{-1}$
  • Symmetric Difference Quotient / Average of AROCs ($[1985, 1995]$):       $f'(1990) \approx \frac{963 - 1012}{1995 - 1985} = \frac{-49}{10} = -4.9\,\text{million acres}\,\text{year}^{-1}$
  • Conclusion: In $1990$, farm land was decreasing at an instantaneous rate of approximately $4.9\,\text{million acres}\,\text{year}^{-1}$.

Reasoning Exercises and Concepts

1. What is the derivative of a linear function? Why?

   • Answer: The derivative of a linear function $f(x) = mx + b$ is equal to its slope $m$ ($f'(x) = m$).
   • Explanation: A linear function has a constant rate of change across its entire domain. The tangent line at any point on a line is the line itself, so its slope is always $m$

2. What is the derivative of any constant function? Why?

   • Answer: The derivative of a constant function $f(x) = c$ is $0$ ($f'(x) = 0$).
   • Explanation: A constant function produces a horizontal line on a graph. Horizontal lines have zero rise over run, meaning their slope is zero everywhere.

3. How is the derivative of a function $f(x)$ at $x = a$ related to the AROC of $f(x)$ containing the point $x = a$?

   • Answer: The derivative $f'(a)$ is defined as the limit of the average rate of change (AROC) over an interval $[a, a + h]$ as the interval width $h$ approaches zero ($h \rightarrow 0$). Graphically, $f'(a)$ is the slope of the tangent line, which is the limit of the slopes of the secant lines through $(a, f(a))$.

4. True or False: If $f'(x) = 0$ at the point $x = a$, then $f(x)$ must have either a maximum or a minimum at that point.

   • Answer: False.
   • Explanation: A derivative of zero ($f'(a) = 0$) indicates a horizontal tangent line, but this can also occur at a stationary inflection point (such as $f(x) = x^3$ at $x = 0$), where the function has neither a local maximum nor a local minimum.

5. True or False: If a function $f(x)$ attains a maximum or a minimum at $x = a$, then $f'(a) = 0$

   • Answer: False.
   • Explanation: A local maximum or minimum can occur at a sharp corner or cusp where the derivative does not exist (is undefined), such as $f(x) = |x|$ at $x = 0$. Fermat's Theorem states that if $f(x)$ is differentiable at a local extremum $x = a$, then $f'(a) = 0$; however, without the guarantee of differentiability, the general statement is false.