Second Derivatives, Inflection Second Derivatives, Inflection Points and ConcavityPoints and Concavity

(A) Important Terms(A) Important Terms Recall the following terms as they were presented in a previous lesson:Recall the following terms as they were presented in a previous lesson: turning pointturning point: points where the direction of the function changes: points where the direction of the function changes maximummaximum: the highest point on a function: the highest point on a function minimumminimum: the lowest point on a function: the lowest point on a function local vs absolutelocal vs absolute: a max can be a highest point in the entire domain : a max can be a highest point in the entire domain

(absolute) or only over a specified region within the domain (local). Likewise (absolute) or only over a specified region within the domain (local). Likewise for a minimum.for a minimum.

increaseincrease: the part of the domain (the interval) where the function values are : the part of the domain (the interval) where the function values are getting larger as the independent variable gets higher; if f(xgetting larger as the independent variable gets higher; if f(x11) < f(x) < f(x22) when x) when x11 < x< x22; the graph of the function is going up to the right (or down to the left); the graph of the function is going up to the right (or down to the left)

decreasedecrease: the part of the domain (the interval) where the function values are : the part of the domain (the interval) where the function values are getting smaller as the independent variable gets higher; if f(xgetting smaller as the independent variable gets higher; if f(x11) > f(x) > f(x22) when ) when xx11 < x < x22; the graph of the function is going up to the left (or down to the right); the graph of the function is going up to the left (or down to the right)

""end behaviourend behaviour": describing the function values (or appearance of the graph) ": describing the function values (or appearance of the graph) as as xx values getting infinitely large positively or infinitely large negatively or values getting infinitely large positively or infinitely large negatively or approaching an asymptoteapproaching an asymptote

(B) New Term – Graphs Showing (B) New Term – Graphs Showing ConcavityConcavity

(B) New Term – Concave Up(B) New Term – Concave Up

Concavity is best “defined” Concavity is best “defined” with graphswith graphs

(i) “(i) “concave upconcave up” means in ” means in simple terms that the “direction simple terms that the “direction of opening” is upward or the of opening” is upward or the curve is “cupped upward”curve is “cupped upward”

An alternative way to describe An alternative way to describe it is to visualize where you it is to visualize where you would draw the tangent lines would draw the tangent lines you would have to draw the you would have to draw the tangent lines “underneath” the tangent lines “underneath” the curvecurve

(B) New Term – Concave down(B) New Term – Concave down

Concavity is best “defined” Concavity is best “defined” with graphswith graphs

(ii) “(ii) “concave downconcave down” ” means in simple terms that the means in simple terms that the “direction of opening” is “direction of opening” is downward or the curve is downward or the curve is “cupped downward”“cupped downward”

An alternative way to describe An alternative way to describe it is to visualize where you it is to visualize where you would draw the tangent lines would draw the tangent lines you would have to draw the you would have to draw the tangent lines “above” the curvetangent lines “above” the curve

(B) New Term – Concavity(B) New Term – Concavity

In keeping with the idea of concavity and the In keeping with the idea of concavity and the drawn tangent lines, if a curve is concave up drawn tangent lines, if a curve is concave up and we were to draw a number of tangent lines and we were to draw a number of tangent lines and determine their slopes, we would see that and determine their slopes, we would see that the values of the tangent slopes increases the values of the tangent slopes increases (become more positive) as our x-value at which (become more positive) as our x-value at which we drew the tangent slopes increasewe drew the tangent slopes increase

This idea of the “increase of the tangent slope is This idea of the “increase of the tangent slope is illustrated on the next slides:illustrated on the next slides:

(B) New Term – Concave Up(B) New Term – Concave Up

(B) New Term – Concave Down(B) New Term – Concave Down

(C) Functions and Their Derivatives(C) Functions and Their Derivatives

In order to “see” the connection between a In order to “see” the connection between a graph of a function and the graph of its graph of a function and the graph of its derivative, we will use graphing technology to derivative, we will use graphing technology to generate graphs of functions and simultaneously generate graphs of functions and simultaneously generate a graph of its derivativegenerate a graph of its derivative

Then we will connect concepts like max/min, Then we will connect concepts like max/min, increase/decrease, concavities on the original increase/decrease, concavities on the original function to what we see on the graph of its function to what we see on the graph of its derivativederivative

(D) Example #1(D) Example #1

(D) Example #1(D) Example #1 Points to note:Points to note:

(1) the fcn has a minimum at x=2 and (1) the fcn has a minimum at x=2 and the derivative has an x-intercept at x=2the derivative has an x-intercept at x=2

(2) the fcn decreases on (-(2) the fcn decreases on (-∞,2) and the ∞,2) and the derivative has negative values on derivative has negative values on (-(-∞,2)∞,2)

(3) the fcn increases on (3) the fcn increases on ((2,+∞) and the 2,+∞) and the derivative has positive values on derivative has positive values on ((2,+∞)2,+∞)

(4) the fcn changes from decrease to (4) the fcn changes from decrease to increase at the min while the derivative increase at the min while the derivative values change from negative to positivevalues change from negative to positive

(5) the function is concave up and the (5) the function is concave up and the derivative fcn is an increasing fcnderivative fcn is an increasing fcn

(6) the second derivative graph is (6) the second derivative graph is positive on the entire domainpositive on the entire domain

(E) Example #2(E) Example #2

(E) Example #2(E) Example #2 f(x) has a max. at x = -3.1 and f `(x) has an x-f(x) has a max. at x = -3.1 and f `(x) has an x-

intercept at x = -3.1intercept at x = -3.1 f(x) has a min. at x = -0.2 and f `(x) has a root at f(x) has a min. at x = -0.2 and f `(x) has a root at

–0.2–0.2 f(x) increases on (-f(x) increases on (-, -3.1) & (-0.2, , -3.1) & (-0.2, ) and on ) and on

the same intervals, f `(x) has positive valuesthe same intervals, f `(x) has positive values f(x) decreases on (-3.1, -0.2) and on the same f(x) decreases on (-3.1, -0.2) and on the same

interval, f `(x) has negative valuesinterval, f `(x) has negative values At the max (x = -3.1), the fcn changes from At the max (x = -3.1), the fcn changes from

being an increasing fcn to a decreasing fcn being an increasing fcn to a decreasing fcn the derivative changes from positive values to the derivative changes from positive values to negative values negative values

At a the min (x = -0.2), the fcn changes from At a the min (x = -0.2), the fcn changes from decreasing to increasing decreasing to increasing the derivative the derivative changes from negative to positivechanges from negative to positive

f(x) is concave down on (-f(x) is concave down on (-, -1.67) while f `(x) , -1.67) while f `(x) decreases on (-decreases on (-, -1.67) and the 2, -1.67) and the 2ndnd derivative derivative is negative on (-is negative on (-, -1.67), -1.67)

f(x) is concave up on (-1.67, f(x) is concave up on (-1.67, ) while f `(x) ) while f `(x) increases on (-1.67, increases on (-1.67, ) and the 2) and the 2ndnd derivative is derivative is positive on (-1.67, positive on (-1.67, ))

The concavity of f(x) changes from CD to CU at The concavity of f(x) changes from CD to CU at x = -1.67, while the derivative has a min. at x = -x = -1.67, while the derivative has a min. at x = -1.671.67

(F) Second Derivative – A (F) Second Derivative – A SummarySummary

If f ``(x) >0, then f(x) is concave upIf f ``(x) >0, then f(x) is concave up If f `(x) < 0, then f(x) is concave downIf f `(x) < 0, then f(x) is concave down If f ``(x) = 0, then f(x) is neither concave nor concave down, but has If f ``(x) = 0, then f(x) is neither concave nor concave down, but has

an inflection points where the concavity is then changing directionsan inflection points where the concavity is then changing directions

The second derivative also gives information about the “extreme The second derivative also gives information about the “extreme points” or “critical points” or max/mins on the original function:points” or “critical points” or max/mins on the original function:

If f `(x) = 0 and f ``(x) > 0, then the critical point is a minimum If f `(x) = 0 and f ``(x) > 0, then the critical point is a minimum point (picture y = xpoint (picture y = x22 at x = 0) at x = 0)

If f `(x) = 0 and f ``(x) < 0, then the critical point is a maximum If f `(x) = 0 and f ``(x) < 0, then the critical point is a maximum point (picture y = -xpoint (picture y = -x22 at x = 0) at x = 0)

These last two points form the basis of the “Second Derivative Test” These last two points form the basis of the “Second Derivative Test” which allows us to test for maximum and minimum valueswhich allows us to test for maximum and minimum values

(G) Examples - Algebraically(G) Examples - Algebraically Find where the curve y = xFind where the curve y = x33 - 3x - 3x22 - 9x - 5 is concave up and concave down. - 9x - 5 is concave up and concave down.

Find and classify all extreme points. Then use this info to sketch the curve.Find and classify all extreme points. Then use this info to sketch the curve.

f(x) = xf(x) = x33 – 3x – 3x22 - 9x – 5 - 9x – 5 f `(x) = 3xf `(x) = 3x22 – 6x - 9 = 3(x – 6x - 9 = 3(x22 – 2x – 3) = 3(x – 3)(x + 1) – 2x – 3) = 3(x – 3)(x + 1) So f(x) has critical points (or local/global extrema) at x = -1 and x = 3So f(x) has critical points (or local/global extrema) at x = -1 and x = 3

f ``(x) = 6x – 6 = 6(x – 1)f ``(x) = 6x – 6 = 6(x – 1) So at x = 1, f ``(x) = 0 and we have a change of concavitySo at x = 1, f ``(x) = 0 and we have a change of concavity Then f ``(-1) = -12 Then f ``(-1) = -12 the curve is concave down, so x = -1 must represent the curve is concave down, so x = -1 must represent

a maximum pointa maximum point Also f `(3) = +12 Also f `(3) = +12 the curve is concave up, so x = 3 must represent a the curve is concave up, so x = 3 must represent a

minimum pointminimum point Then f(3) = -33, f(-1) = 0 as the ordered pairs for the function whose graph Then f(3) = -33, f(-1) = 0 as the ordered pairs for the function whose graph

is shown on the next slide:is shown on the next slide:

(H) Examples(H) Examples

ex 1. Find and classify all local extrema of f(x) = 3xex 1. Find and classify all local extrema of f(x) = 3x55 - - 25x25x33 + 60x. Sketch the curve + 60x. Sketch the curve

ex 2. Find and classify all local extrema of f(x) = 3xex 2. Find and classify all local extrema of f(x) = 3x44 - - 16x16x33 + 18x + 18x22 + 2. Sketch the curve + 2. Sketch the curve

ex 3. Find where the curve y = xex 3. Find where the curve y = x33 - 3x - 3x22 is concave up and is concave up and concave down. Then use this info to sketch the curveconcave down. Then use this info to sketch the curve

ex 4. For the function f(x) = xex 4. For the function f(x) = x(1/3)(1/3)*(x + 3)*(x + 3)(2/3)(2/3) find (a) find (a) intervals of increase and decrease, (b) local max/min (c) intervals of increase and decrease, (b) local max/min (c) intervals of concavity, (d) inflection point, (e) sketch the intervals of concavity, (d) inflection point, (e) sketch the graphgraph

(H) Examples(H) Examples

ex 4. For the function f(x) = xex 4. For the function f(x) = x(1/3)(1/3)*(x + 3)*(x + 3)(2/3)(2/3) find (a) intervals of find (a) intervals of increase and decrease, (b) local max/min (c) intervals of concavity, increase and decrease, (b) local max/min (c) intervals of concavity, (d) inflection point, (e) sketch the graph(d) inflection point, (e) sketch the graph

ex 5. For the function f(x) = xeex 5. For the function f(x) = xexx find (a) intervals of increase and find (a) intervals of increase and decrease, (b) local max/min (c) intervals of concavity, (d) inflection decrease, (b) local max/min (c) intervals of concavity, (d) inflection point, (e) sketch the graphpoint, (e) sketch the graph

ex 6. For the function f(x) = 2sin(x) + sin2(x), find (a) intervals of ex 6. For the function f(x) = 2sin(x) + sin2(x), find (a) intervals of increase and decrease, (b) local max/min (c) intervals of concavity, increase and decrease, (b) local max/min (c) intervals of concavity, (d) inflection point, (e) sketch the graph(d) inflection point, (e) sketch the graph

ex 7. For the function f(x) = ln(x) ex 7. For the function f(x) = ln(x) ÷ ÷ x, x, find (a) intervals of increase find (a) intervals of increase and decrease, (b) local max/min (c) intervals of concavity, (d) and decrease, (b) local max/min (c) intervals of concavity, (d) inflection point, (e) sketch the graphinflection point, (e) sketch the graph

(I) Internet Links(I) Internet Links We will work on the following problems in class: We will work on the following problems in class:

Graphing Using First and Second Derivatives from UC Graphing Using First and Second Derivatives from UC DavisDavis

Visual Calculus - Graphs and Derivatives from UTKVisual Calculus - Graphs and Derivatives from UTK

Calculus I (Math 2413) - Applications of Derivatives - Calculus I (Math 2413) - Applications of Derivatives - The Shape of a Graph, Part II Using the Second The Shape of a Graph, Part II Using the Second Derivative - from Paul DawkinsDerivative - from Paul Dawkins

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