Increasing / Decreasing Test

Slides:

Advertisements

Similar presentations

Maxima and Minima of Functions Maxima and minima of functions occur where there is a change from increasing to decreasing, or vice versa.

Advertisements

DO NOW: Find where the function f(x) = 3x4 – 4x3 – 12x2 + 5

Section 3.4 – Concavity and the Second Derivative Test

1 Concavity and the Second Derivative Test Section 3.4.

Objectives: 1.Be able to determine where a function is concave upward or concave downward with the use of calculus. 2.Be able to apply the second derivative.

Aim: Concavity & 2 nd Derivative Course: Calculus Do Now: Aim: The Scoop, the lump and the Second Derivative. Find the critical points for f(x) = sinxcosx;

Section 3.3 How Derivatives Affect the Shape of a Graph.

4.1 Extreme Values for a function Absolute Extreme Values (a)There is an absolute maximum value at x = c iff f(c)  f(x) for all x in the entire domain.

Relative Extrema.

Unit 11 – Derivative Graphs Section 11.1 – First Derivative Graphs First Derivative Slope of the Tangent Line.

1 Example 1 Sketch the graph of the function f(x) = x 3 -3x 2 -45x+47. Solution I. Intercepts The x-intercepts are the solutions to 0= f(x) = x 3 -3x 2.

Math – Getting Information from the Graph of a Function 1.

First and Second Derivative Test for Relative Extrema

Relationships of CREATED BY CANDACE SMALLEY

Section 5.1 – Increasing and Decreasing Functions The First Derivative Test (Max/Min) and its documentation 5.2.

5.3 A – Curve Sketching.

Section 5.2 – Applications of the Second Derivative.

The Shape of the Graph 3.3. Definition: Increasing Functions, Decreasing Functions Let f be a function defined on an interval I. Then, 1.f increases on.

CHAPTER Continuity Derivatives and the Shapes of Curves.

Increasing/ Decreasing

Section 3.1 Maximum and Minimum Values Math 1231: Single-Variable Calculus.

4.4 Concavity and Inflection Points Wed Oct 21 Do Now Find the 2nd derivative of each function 1) 2)

Using Derivatives to Sketch the Graph of a Function Lesson 4.3.

In the past, one of the important uses of derivatives was as an aid in curve sketching. We usually use a calculator of computer to draw complicated graphs,

Ch. 5 – Applications of Derivatives

Chapter 16B.5 Graphing Derivatives The derivative is the slope of the original function. The derivative is defined at the end points of a function on.

How derivatives affect the shape of a graph ( Section 4.3) Alex Karassev.

Sketching Functions We are now going to use the concepts in the previous sections to sketch a function, find all max and min ( relative and absolute ),

4.3 – Derivatives and the shapes of curves

First derivative: is positive Curve is rising. is negative Curve is falling. is zero Possible local maximum or minimum. Second derivative: is positive.

Ch. 5 – Applications of Derivatives

Maxima and Minima of Functions

Relative Extrema and More Analysis of Functions

4.3 Using Derivatives for Curve Sketching.

Graph of a Function Def. A function f (x) has a local maximum (relative max) at x = p if f (x) < f (p) for all points near p. Def. A function f (x) has.

Extreme Values of Functions

4.3 How Derivatives Affect the Shape of a Graph

Increasing and Decreasing Functions and the First Derivative Test

Lesson 13: Analyzing Other Types of Functions

Relationship between First Derivative, Second Derivative and the Shape of a Graph 3.3.

Lesson 13: Analyzing Other Types of Functions

Concavity and the Second Derivative Test

Concavity and Second Derivative Test

3.2 – Concavity and Points of Inflection

Using Derivatives For Curve Sketching

4.3 – Derivatives and the shapes of curves

Second Derivative Test

1 2 Sec 4.3: Concavity and the Second Derivative Test

Application of Derivative in Analyzing the Properties of Functions

MATH 1311 Section 1.3.

5.3 Using Derivatives for Curve Sketching

For each table, decide if y’is positive or negative and if y’’ is positive or negative

5.2 Section 5.1 – Increasing and Decreasing Functions

Concavity and the Second Derivative Test

58 – First Derivative Graphs Calculator Required

4.3 Connecting f’ and f’’ with the graph of f

Concavity and the Second Derivative Test

MATH 1311 Section 1.3.

For each table, decide if y’is positive or negative and if y’’ is positive or negative

Warm Up Cinco Chapter 3.4 Concavity and the Second Derivative Test

Derivatives and Graphing

Packet #14 First Derivatives and Graphs

1 2 Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH

4.2 Critical Points, Local Maxima and Local Minima

4.4 Concavity and the Second Derivative Test

Analyzing f(x) and f’(x) /

Concavity & the 2nd Derivative Test

Math 1304 Calculus I 4.03 – Curve Shape.

- Derivatives and the shapes of graphs - Curve Sketching

Relationship between First Derivative, Second Derivative and the Shape of a Graph 3.3.

Presentation transcript:

Increasing / Decreasing Test If f’(x) > 0 on an interval, then f is increasing on that interval. If f’(x) < 0 on an interval, then f is decreasing on that interval. ex: Find where the function f(x) = 3x4 – 4x3 – 12x2 + 5 is increasing and decreasing… f ’(x) = 12x3 – 12x2 – 24x Factor: 12x(x – 2)(x + 1) Critical Numbers: x = -1, 0, 2 Test: ( - , -1) (-1, 0) (0, 2) (2, )

The First Derivative Test For c being a critical number of a continuous function f… If f ’ changes from positive to negative at c, then f has a local maximum at c. If f ’ changes from negative to positive at c, then f has a local minimum at c. If f ’ does not change sign at c, then f has no local maximum or minimum at c. c2 c1 f ’ FROM – TO + = MIN FROM + TO – = MAX

ex: Critical Numbers: x = -1, 0, 2 Test: ( - , -1) (-1, 0) (0, 2) (2, ) MIN @ -1 MAX @ 0 MIN @ 2 Concavity Test If f ’’(x) > 0 in [a,b], then f is concave up on [a,b] If f ’’(x) < 0 in [a,b], then f is concave down on [a,b]

The Second Derivative Test For f ’’ being continuous near c… If f ’(c) = 0 and f ’’(c) > 0 then f has a local minimum at c. If f ’(c) = 0 and f ’’(c) < 0 then f has a local maximum at c. f ’ f ’’ c1 c2 f

ex: For the curve f(x) = x4 - 4x3 use calculus to find local maxima and minima, concavity intervals and points of inflection… Critical #’s = 0, 3 [ Test f ’ ] (- , 0) f’(-1) = No Max / Min @ 0 f’(1) = (0, 3) (3, ) f’(4) = + NegPos = Min @3 Local Min

+ + Test second derivative for concavity intervals: [ Test f ’’ ] 2nd derivative test inconclusive @ f ’ critical # 0 [ Test f ’’ ] (- , 0) f ’’(-1) = + f ’ critical # 3 = (36)(1) = 36, so f ’’ is + = MIN @ 3 CC Inflection point @ 0 f ’’(1) = (0, 2) CC Inflection point @ 2 (2, ) f ’’(5) = + CC Summary Local min at (3, -27) No local max CC up from (– , 0) U (2, ) CC down from (0, 2)

f(x) = x4 - 4x3

nDeriv For f(x) = x2/3(6 – x)1/3 graph the function and its first and second derivatives…. Type f(x) into Y1 Go to Y2 line Go to MATH [8] (nDeriv) / Press ENTER Set nDeriv arguments to (Y1, X, X) *Y1 in VARS ~ Y-VARS ~ FUNCTION ~ #2 Repeat on Y3 line using (Y2, X, X) for nDeriv Graph