Lesson 15 Graphical Analysis
15.2 Lecture Notes
15.2.1 Finding Critical Numbers
Critical numbers occur where the first or second derivative is zero or undefined.
15.2.2 Classifying Maximums, Minimums, and Inflection Points
15.2.2.1 First Derivative Test (for relative extrema):
- If \(f'(x)\) changes from positive to negative at \(x = c\), then \(f(c)\) is a local maximum.
- If \(f'(x)\) changes from negative to positive at \(x = c\), then \(f(c)\) is a local minimum.
15.2.3 Extreme Value Theorem (EVT)
The Extreme Value Theorem states:
If a function \(f(x)\) is continuous on a closed interval \([a, b]\), then \(f(x)\) has both an absolute maximum and an absolute minimum on that interval.
15.2.4 Intervals of Increase, Decrease, and Concavity
15.3 Examples
15.3.1 Maximum and Minimum Values
Given the function \(g(x) = 8x^3 + 72x^2 + 192\), find the first derivative, \(g'(x)\). Notice that \(g'(x) = 0\) when \(x = -2\), i.e., \(g'(-2) = 0\). Use the second derivative test to determine whether there is a local minimum or maximum at \(x = -2\). Find the second derivative, \(g''(x)\). Evaluate \(g''(-2)\). Based on the sign of this number, is the graph of \(g(x)\) concave up or down at \(x = -2\)?
The function \(f(x) = 2x^3 - 36x^2 + 210x + 5\) has one local minimum and one local maximum.
Consider the function \(f(x) = 5 - 7x^2,\) with \(-5 \leq x \leq 1\). Find the absolute maximum and minimum.
Consider the function \(f(x) = 2x^3 + 12x^2 - 192x + 8\), with $ -12 x $. Find the absolute maximum and minimum.
15.3.2 Applied Problems
A rectangular storage area is to be constructed along a side of a building. A security fence is required along the remaining three sides of the area. You have 865 ft of fencing available. What is the maximum storage area?
A rancher wants to fence in an area of 1,000,000 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. Find the shortest length of fence that the rancher can use.
A box with a square base and open top must have a volume of 143748 \(cm^3\). We want to find the dimensions of the box that minimize the amount of material used.
A small object is moving along a vertical line. Its location function is \(t^3 - 24t^2 + 45t + 3\) with \(0 \leq t \leq 11\).
- At what time(s) is the object’s velocity greatest?
- At what time(s) is the object’s speed greatest?
A ball is tossed into the air on the planet Bellona. The height of the ball in meters above the ground \(t\) seconds after being thrown is: \[ s(t) = \frac{7}{2}t^2 - t^3. \]
- Find the velocity and acceleration functions.
- Over what interval(s) of time is the ball moving upward? (Give interval notation)
- After initially being thrown, when does the ball return to the ground?
- What is the velocity of the ball when it returns to the ground?
15.3.3 Increasing and Decreasing
- Let \(f(x) = x^3 + 3x^2 - 240x + 18\).
- Find the first derivative \(f'(x)\).
- Find the second derivative \(f''(x)\).
- Find the interval(s) where \(f\) is increasing (include endpoints).
- Find the interval(s) where \(f\) is decreasing (include endpoints).
- Find the interval(s) where \(f\) is concave downward (include endpoints).
- Find the interval(s) where \(f\) is concave upward (include endpoints).
- Let \(f(x) = \frac{x^4 - 32x^2}{7}\).
- Find the first derivative \(f'(x)\).
- Find the second derivative \(f''(x)\).
- Find the interval(s) where \(f\) is increasing (include endpoints).
- Find the interval(s) where \(f\) is decreasing (include endpoints).
- Find the interval(s) where \(f\) is concave downward (include endpoints).
- Find the interval(s) where \(f\) is concave upward (include endpoints).
- Consider the function \(f(x) = 5(x-2)^{2/3}\).
- Find the critical number \(A\).
- On each interval below, state if \(f(x)\) is increasing or decreasing.
- On each interval below, state if \(f(x)\) is concave up (type “CU”) or concave down (type “CD”).
- Consider the function \(f(x) = x^2 e^{9x}\).
- Find the critical numbers.
- Determine the intervals of increasing and decreasing.
- Consider the function \(f(x) = x^2 e^{8x}\). Find the intervals of concavity.
15.3.4 Curve Sketching
- Consider the function $f(x) = 2x^3 + 6x^2 - 18x - 54. Find
- the first derivative,
- all critical values,
- intervals of increasing and decreasing,
- maximum and minimum values,
- the second derivative,
- inflection points,
- intervals of concavity.
- Consider the function \(f(x) = \frac{x^2 - 1}{x -3}\). Find
- the first derivative,
- all critical values,
- intervals of increasing and decreasing,
- maximum and minimum values,
- the second derivative,
- inflection points,
- intervals of concavity.
- Let \(f(x) = x^3 -6x^2 - 1\). Find
- the first derivative,
- all critical values,
- intervals of increasing and decreasing,
- maximum and minimum values,
- the second derivative,
- inflection points,
- intervals of concavity.
- Let \(f(x) = x^3 + 9x^2 + 27x + 4\). Find
- the first derivative,
- all critical values,
- intervals of increasing and decreasing,
- maximum and minimum values,
- the second derivative,
- inflection points,
- intervals of concavity.
15.4 Practice Problems
Practice the techniques discussed in class and in the online videos by solving the following examples.
Find the location of all relative and absolute extrema for \[f(x) = 2x^2 - 2x + 3\] on \([0, 3]\).
Find the location of all relative and absolute extrema for \[f(x) = \sqrt{x}(x+1)\] on \([0, \infty)\).
Find the location of all relative and absolute extrema for \[f(x) = \frac{x^2 + 1}{x^2 - 1}\] on \([-2, 2]\) with \(x \neq \pm 1\).
Find the location of all relative and absolute extrema for \[f(x) = (x+1)^{2/5}\] on \([-2, 0]\).
Find and classify the critical points and locate all inflection points for \[f(x) = 2x^2 - 2x + 3.\]
Find and classify the critical points and locate all inflection points for \[f(x) = -x^3 + 3x.\]
Sketch the graph of \[h(x) = -2x^3 - 3x^2 + 36x.\]
Sketch the graph of \[f(t) = \frac{t^2 - 1}{t^2 + 1}.\]
(Applied) Maximize \[P = xy\] with \[x + 2y = 40.\]
(Applied) The cost of controlling emissions at a firm rises rapidly as the amount of emissions reduced increases. Here is a possible model:
\[ C(q) = 4,000 + 100 q^2 \]
where \(q\) is the reduction in emissions (in pounds of pollutant per day) and \(C\) is the daily cost to the firm (in dollars) of this reduction. What level of reduction corresponds to the lowest average cost per pound of pollutant, and what would be the resulting average cost to the nearest dollar?
15.5 Self Assessment
Time yourself and try to solve the following questions within twenty minutes.
Find the location of all relative and absolute extrema for \[f(x) = 2x^3 + 3x^2\] on \([-2, \infty)\).
Find and classify the critical points and locate all inflection points for \[f(x) = x e^{-x^2}.\]
Sketch the graph of \[g(x) = \frac{x^3}{x^2 - 3}.\]
Hercules Films is also deciding on the price of the video release of its film Bride of the Son of Frankenstein. Again, marketing estimates that at a price of \(p\) dollars, it can sell \[ q = 200,000 - 10,000 p \] copies, but each copy costs $4 to make. What price will give the greatest profit?
Let \[ f(x) = \frac{N}{1 + A e^{-k x}} \] for constants \(N\), \(A\), and \(k\) (\(A\) and \(k\) positive). Show that \(f\) has a single point of inflection at \[x = \frac{\ln A}{k}.\]
15.6 Lesson Checklist
This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an \(X\) in the appropriate box beside the skill below.
\[\begin{align*} &\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\ &\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\ &\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} \end{align*}\]
| Skill | D | CON | COM |
|---|---|---|---|
| Find and classify critical numbers of the first derivative. |