Connecting F and F with the Graph of F

CalculusNotes 4.2/4.3

Connecting f’ and f” with the graph of f

First Derivative Test for Local Extremes

ONCE I KNOW WHERE THE CRITICAL POINTS OCCUR, HOW DO I FIGURE OUT IF THEY ARE REALLY MAXIMUMS OR MINIMUMS?

** - Precalculus flash-back

If, as a function’s x-values are increasing (over some interval), the function’s y-values are decreasing, we say that the function is decreasing on that interval (of x-values).

If, as a function’s x-values are increasing (over some interval), the function’s y-values are increasing, we say that the function is increasing on that interval (of x-values).

Refer back to the graph from earlier:

For a non-endpoint to be a maximum value, the function must increase up to that point and then decrease afterwards (for a left endpoint, it would decrease after only; for a right endpoint, it would increase before only).

For a non-endpoint to be a minimum value, the function must decrease up to that point and then increase afterwards (for a left endpoint, it would increase after only; for a right endpoint, it would decrease before only).

IMPORTANT CALCULUS THINGS TO LEARN / KNOW / REMEMBER:

Assuming that the function f is continuous over [a, b] and differentiable on (a, b):

(1) If > 0 on some interval (a, b), then the original function is increasing on the closed interval [a, b].

(2) If < 0 on some interval (a, b), then the original function is decreasing on the closed interval [a, b].

If you know a function is increasing over an interval, you know that on that interval.

If you know a function is decreasing over an interval, you know that on that interval.

If a function’s derivative is positive over an interval, then you know the function must be increasing on that same interval.

If a function’s derivative is negative over an interval, then you know the function must be decreasing on that same interval.

EXAMPLES: For the functions from before, decide if the critical points are local maximums, local minimums, or neither.

(C)

We found the derivative function to be and the critical points to be. We are now going to create afirst derivative sign chart to organize our information:

Step #1: Draw a number line that contains the domain of the original function (all real numbers). Label the front of the number line with “”. You could also use or if you wanted. Put a mark on the number line for each of your critical points:

Step #2: Notice that the critical points divide the number line into three areas: x < -2, , and x > 2. Now we will substitute a number from each interval into the first derivative function to see if we get back a positive or negative answer. You should get the following:

Step #3: Using (1) and (2) at the top of this page, an interval marked with a “+” indicates the original function is increasing and a “-“ indicates the original function is decreasing. I will use arrows on the sign chart to indicate this:

Step #4: Now interpret the first derivative sign chart to decide if the critical points are maximums or minimums:

For x = -2: since the first derivative changes from positive to negative at that point, then x = -2 is a local maximum.

For x = +2: since the first derivative changes from negative to positive at that point, then x = +2 is a local minimum.

Notice by using the sign chart, you could also answer the following questions easily.

Does have an absolute maximum or minimum?

When is increasing?

When is decreasing?

How would you answer this question: What are the local minimum and maximum values?

COMMON QUESTIONS:

What do I do if the domain starts or stops somewhere?

Adjust your number line accordingly. Remember, endpoints can be critical points as well.

Do I have to write all of that stuff like you did in step #4?

YES. The AP Exam people have made it very clear that sign charts ARE NOT ACCEPTED as reasons or explanations. You must interpret the data in the chart as I did in step #4.

Do I have to use a sign chart?

NO, but I would do it for now. You can decide later, after we cover some other topics, whether it is useful or not.

How do I know if you want an x = answer or a y = answer?

If I ask you “where does the maximum/minimum occur?” I want an x-value answer.

If I ask you “what is the maximum/minimum (value)?” I want a y-value answer.

If the textbook says “find the maximum/minimum”, they usually want a y-value.

YOU TRY: For the examples from earlier, decide if the critical points are local maximums, local minimums, or neither. Then find the interval(s) that the functions are increasing and/or decreasing.

(F) (G)

(H) & (I): Use the following graphs to answer these questions about the original function.

(i) When is increasing? Decreasing? Neither?

(ii) How many critical points doeshave?

(iii) Label the critical points from part (ii) as local maximums, local minimums, or neither.

(H) (I)

YOU TRY Answers:

(A) Letter A is a local minimum; B is a local maximum; C is NOTHING; D is a local & absolute minimum; E is a local maximum; F is a local minimum; G is a local & absolute maximum

(B) A = 3; B = 1; D = 2; E = 2; F = 1, G = 3(C) x = 2, -2(D) x = -2, 0, ½

(E) [Note: domain is and derivative is]. x = -5, 0, 5

(F)

x = -2 and x = ½ are local minimums and x = 0 is a local maximum;

increasing on intervals (-2,0) U and decreasing on intervals U (0, ½)

(G)

x = -5 and x = 5 are local minimums and x = 0 is a local maximum;

increasing on interval (-5, 0) and decreasing on interval (0, 5)

(H)

(i) increasing: U ; decreasing ; neither: -2, ½, 2

(ii) 3 critical points (-2, ½, 2)

(iii) -2 is neither, ½ is a local maximum, 2 is a local minimum

(I)

(i) increasing: (-3, -2) U (-1, 0) U ; decreasing (-2, -1) U (0, 2); neither -2, -1, 0, 2, 4

(ii) 6 critical points (-3, -2, -1, 0, 2, 4)

(iii) -3 is a local minimum, -2 is a local maximum, -1 is a local minimum, 0 is a local maximum, 2 is a local minimum, 4 is neither

Points of inflection as places where concavity changes.

II. Derivatives: Derivative as a function

Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.

Textbook: Sections 4.1-4.3 (pages 177-206)

Monotonicity and Concavity

We say that a function is monotonic over an interval if it only increases or only decreases over that interval. For example, the function is monotonic for its entire domain because it only increases. Functions will be monotonic between their critical points as well.

Concavity has to do with where points are located relative to their tangent lines. If the points of a function lie above the tangent lines, then we say the function, on that interval, is concave up. If the points of a function lie below the tangent lines, then we say the function, on that interval, is concave down.

EXAMPLES:

A concave up regionA concave down regionChanging from concave down to

concave up at the origin

The concavity of a curve is related to its 2nd derivative values.

If over an interval, then the original function is concave up over that interval.

If over an interval, then the original function is concave down over that interval.

If the original function is concave up over an interval, then for that interval.

If the original function is concave down over an interval, then for that interval.

A point where a curve changes concavity is called a POINT OF INFLECTION.

To organize our information about the second derivative, we will use a 2nd derivative sign chart.

The points that go on the 2nd derivative sign chart are any places whereor does not exist.

To get the “+” and “-“ signs for the chart, substitute interval values into the 2nd derivative. I like to draw U-shaped symbols around the signs to indicate whether the original is concave up or concave down. If there is a sign-change at either of these points, as long as the original function exists at that point, it is a point of inflection.

EXAMPLE: Find where is concave up, concave down, and has inflection points.

First find the 2nd derivative: . Notice that y’’ = 0 at x = - 2. Now create a 2nd derivative sign chart:

Substitute values from each interval into the SECOND DERIVATIVE to get your signs. Draw U-shapes to indicate concavity:

The function is concave up on the interval because

The function is concave down on the interval because

Since changes from positive to negative at x = -2, then it is a point of inflection. To get its y-value, substitute

x = -2 into the ORIGINAL function. The coordinate of the point of inflection is .

YOU TRY: Find the interval(s) where the following functions are concave up, concave down, and where the points of inflection are located (if any).

(A) (B)(C)

There is an additional use for the 2nd derivative called the 2nd Derivative Test for Extrema. Here is how it works:

Given that and the 2nd derivative exists in an interval around x = c, then:

If, then x = c is a relative minimum on the original function.
If, then x = c is a relative maximum on the original function.
If , this test fails, and you must use the first derivative test to draw your conclusion.

This new test means there are two ways to determine if a value is a maximum or minimum: sign change in the 1st derivative or look at the value of the 2nd derivative at that point.

YOU TRY:

(D) Given that are critical points of [letter (C) on the previous page], use the 2nd derivative test for Extrema to determine if these points are local maximums or local minimums.

Use the graph below to answer the questions.

(E) If the graph above is :

(1) Find where (3) Find where (5) Find any extreme values.

(2) Find where (4) Find where (6) Find any points of inflection

(F) If the graph above is

(7) Find where is increasing(9) Find where is concave up(11) Find where any extreme

values of occur

(8) Find where is decreasing(10) Find where is concave down(12) Find where any points of

inflection occur on

(G) If the graph above is

(13) Find where is concave up(14) Find where is concave down(15) Find where any points of

inflection occur on

YOU TRY answers:

(A) Concave up: ; Concave down: ; x=1 is a point of inflection.

(B) Concave up: ; Concave down: ; x = 0 is a point of inflection

(D) x = -2 is a local minimum because it occurs in a concave up region; x = 0 is a local maximum because it occurs in a concave down region; x = 2 is a local minimum because it occurs in a concave up region.

(E)

(1) (3) (5) Points:

Local min local max local min

(2) (4) (6) Points: (-2,0); (0,0); (3,0)

(F)

(7) (9) (11) x = -4; x = -2; x = 0; x = 3 Local max local min local max local min

(8) (10) (12) x = -3; x = -1; x = 2

(G)

(13) (14) (15) x = -4; x = -2; x = 0; x = 3

CLASSWORK / HOMEWORK: Section 4.3 ( 8,10-11, 13, 16, 19, 21b, 22b, 39c, 40c, 41-42, 45-47, 51)

* Note any given information about a graph being symmetric to the y-axis (or even, or know that) or being symmetric to the origin (or odd, or know that ). *

Use to find:

Any x-intercepts (zeros) by setting y = 0. Write as (x, 0).
Any y-intercepts by setting x = 0. Write as (0, y).
Any horizontal asymptotes by evaluating Lines will be y = c.
Any vertical asymptotes by evaluating. Lines will be x = c.
Any domain endpoints (add to your critical point list)

Use to find:

All other critical points (=0, or is undefined). Go ahead and note if points are vertical asymptotes. Then:

Use a sign chart to find intervals where the function is increasing (= “+”) and decreasing (= “-“).
At each critical point between a sign change and each endpoint, find (x, ).
Use the second derivative test later (on points where = 0) to determine local maximums and local minimums.
For points that are = undefined, take limits on each side to determine if the point is a corner or a cusp.
For endpoints, use the sign chart to determine if they are local maximums or minimums.

Use to find:

All points where =0 or = undefined to setup a sign chart.
Use the sign chart to find intervals where the function is concave up ( > 0) and concave down ( < 0).
At each point between a sign change (points of inflection), find (x, ) if possible.
Use the second derivative test to determine if the critical points (where = 0) are local maximums ( < 0) or local minimums (> 0).

Plot all points that you have coordinates for, draw dashed lines for each asymptote, use the sign charts, and use given knowledge about symmetry to sketch the graph of.