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Notes for Introductory Calculus (Math 120)

Brenton LeMesurier

Section 4.5 Derivatives and the Shape of a Graph

References.

We have seen a number of connections between derivatives and the shape of a graph, and these can be useful in both directions: using derivatives to understand how the graph of a function will look, and using the graph to summarize and identify useful information about a function, such as the locations of its extrema.
For example, we have seen that positive derivatives are associated with function values that increase as the argument increases, zeros of the derivative are associated with minimum and maximum values, and the second derivative is associated with whether the curve is bending up or down.

\(\mathbf {f'}\) Tells Where a Curve Increases and Where it Decreases.

First let's make precise some intuitive concepts:

Definition 4.5.1. Increasing/Decreasing.

Function \(f\) is increasing on an interval if for any numbers \(c < d\) in that interval, \(f(c) < f(d)\text{.}\)
Likewise \(f\) is decreasing on an interval if \(f(d) < f(c)\) for all such \(c < d\text{.}\)
The proofs of (a)-(c) are based on \(f(b)-f(a) = f'(c)(b-a)\text{,}\) as given by the Mean Value Theorem.
Item (d) is shown by examples like \(f(x)=x^3\text{.}\)
Make sketches illustrating each of the above four statements.
Find where the function \(f(x)=3x^{4}-4x^{3}-12x^{2}+5\) is increasing, and where it is decreasing.

Which Critical Points Are Local Minima? Local Maxima?

Increasing/decreasing behavior can only change at critical points, and how it changes (if at all) determines whether a critical point is a local minimum or local maximum (or neither):
All of these results are intuitive when one sketches the situations described, and are proved by the increasing/decreasing properties given by the signs of the derivatives.
Make sketches illustrating all three cases above.
Find the local minimum and maximum points of
\begin{equation*} f(x)=3x^{4}-4x^{3}-12x^{2}+5. \end{equation*}
Find all local minima and maxima of
\begin{equation*} g(x) = x + 2\sin x, \; \text{ domain } 0 \leq x \leq 2\pi. \end{equation*}

Using \(\mathbf f'\) at Endpoints.

At endpoints we only need to look to one side:
Make sketches illustrating all four cases.
If the function \(f(x)=3x^{4}-4x^{3}-12x^{2}+5\) has domain \([-2,3]\text{,}\) check the endpoints for local minima and maxima, and then determine the global extrema.

\(\mathbf {f''}\) Tells Which way a Curve is Bending: Concavity.

Though we can usually classify critical points using just the first derivative, the second derivative gives an alternative that is sometime more convenient, and can improve the visual accuracy of a sketch graph.

Definition 4.5.12. Concavity.

If the graph of \(f\) lies above all its tangents on an interval \(I\) it is called concave upward on that interval.
If the graph of \(f\) lies below all its tangents on an interval \(I\) it is called concave downward on that interval.
Illustrate each of the above with sketches of simple functions like \(f(x)=x^2\) and \(g(x)=\sin(x)\text{.}\)
Concavity can be used in place of increasing/decreasing behavior to check a critical number \(c\) for a local minimum and maximum. This is sometimes easier, as you need only think about function values at one argument \(c\text{,}\) not at all nearby ones.
However, it only works where \(f'(c) = 0\text{,}\) not where \(f'(c)\) does not exist.
Part (a) is true because when \(f'(c)=0\text{,}\) the tangent there is the horizontal line \(y=f(c)\text{,}\) and \(f''(c)>0\) makes \(f\) concave up, so that the graph lies above this horizontal tangent line: nearby values of \(f(x)\) are greater than \(f(c)\text{,}\) which is a local minimum.
Part (b) is the same but upside down.
Illustrate each part of the above theorem.

Where Concavity Changes: Inflections.

We have seen unusual cases like \(f(x) = x^3\) where a critical point is not a local extremum. This is related to concavity flipping at such points:

Definition 4.5.17.

A point \(P\) on a curve is called an inflection point if the curve changes from being concave up to being concave down (with continuity at that point.)
This is akin to a local extremum being a point where the curve changes from increasing to decreasing. Inflections can occur at points where \(f''=0\text{,}\) and also at points where \(f''\) does not exist. Locating inflections along with local extrema can help get the overall shape in a sketch of a function.

A Curve Sketching Strategy.

A key to sketching a curve is to find “interesting” points (domain end-points, axis intercepts, vertical asymptotes, critical points, inflection points) and then find out about behavior in each interval between these, such as whether the curve in increasing or decreasing, concave up or concave down.
Note that within each interval between two consecutive “interesting” points, there is no change between increasing and decreasing, and no change in concavity.
It helps to summarize this information in a table, with the top row listing in increasing order the “interesting” values of the argument (\(x\)-values), a column in between each for the intervals in between, and all other useful information gathered in rows below.
As a variant, you can draw the number line for the domain, mark the interesting values on there, and summarize other information below it.
To sketch \(y=f(x)\text{,}\)
  1. Compute the first and second derivatives, \(y'=f'(x)\) and \(y''=f''(x)\text{.}\)
  2. Note any endpoints of the domain and points where the function is undefined.
  3. If feasible, find the \(x\)-intercepts, points where \(y=0\text{.}\)
  4. Find the critical points, where \(y'\) is zero or does not exist (possible local extrema).
  5. Find the points where \(y''\) is zero or DNE (possible inflections).
  6. Evaluate the function at all these \(x\) values, and summarize on a table with a column for each of these \(x\) values (\(a\text{,}\) \(b\text{,}\) \(c\) etc.) and a column in between each:
    \(x\) \(2\) \(\qquad\qquad\) \(4\) \(\qquad\qquad\) \(7\) \(\qquad\qquad\) \(\cdots\)
    \(y\) \(5\) \(6\) \(4\) \(\cdots\)
    \(y'\)
    \(y''\)
  7. Determine the sign of \(y'\) and of \(y''\text{,}\) meaning positive, negative or zero, and add this information in the next rows of the table as \(+\text{,}\) \(-\) or \(0\text{.}\) (One way to do this is to evaluate \(y'\) and \(y''\) at one \(x\) value in each interval between “interesting” values.)
    \(x\) \(2\) \(\qquad\qquad\) \(4\) \(\qquad\qquad\) \(7\) \(\qquad\qquad\) \(\cdots\)
    \(y\) \(5\) \(6\) \(4\) \(\cdots\)
    \(y'\) \(+\) \(+\) 0 \(-\) \(-\) \(-\)
    \(y''\) \(-\) \(-\) \(-\) \(-\) 0 \(+\)
  8. At the bottom of the table, you might want to draw a little fragment of curve in each column with the correct increasing/decreasing behavior and correct concavity, with a dot on the curve where each “interesting value” occurs. Alternatively do this directly as you sketch the graph. (In this table, \(x=2\) is the left end-point, \(x=4\) a critical number and \(x=7\) gives an inflection.)
  9. Sketch the graph using the points in the top two lines and (if drawn) the shapes drawn at the bottom of the table.

Exercises Exercises

Study Calculus Volume 1, Section 4.5 2 ; in particular the Problem Solving Strategy, the First Derivative Test, the Second Derivative Test all Examples and Checkpoints, and a selection from Exercises 194–200, 201–205, 206–210, 211–215, 216–220, 221–223 and 224–230.
Some suggested selections are Exercises 199, 201, 203, 213, 215, 217, 223, 225, 229.
openstax.org/books/calculus-volume-1/pages/4-5-derivatives-and-the-shape-of-a-graph
openstax.org/books/calculus-volume-1/pages/4-5-derivatives-and-the-shape-of-a-graph
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