We think you have liked this presentation. If you wish to download it, please recommend it to your friends in any social system. Share buttons are a little bit lower. Thank you!
Presentation is loading. Please wait.
Published byKellie Garrison
Modified about 1 year ago
Maxima and Minima An old friend with a new twist!
Basic Conditions…. The slope of the tangent plane must be zero! We can build a tangent plane out of the sum of two independent vectors so … f(x,y) is at a maximum (or min) if : at the same time!
Critical Points A point is critical if… f x and f y = 0 One of f x or f y (or both) fails to exist Example: Find critical points on the surface Tangent plane x+y+z = 9
Challenge… Where will the function have critical points? Sketch this.
Saddle Points… Sometimes a critical point is not a max or a min. This is analogous to inflection points. Such points are called saddle points pringle potato chip points
The 2 nd Derivative Test… If the 2 nd partial derivatives are continuous on a disk with center (a,b) and define:
Sample Questions… Try 15.7: 2, 3, 7, 13,14,37, 47 Use Maple! 15.7 #17
Chapter 16 Section 16.5 Local Extreme Values. Critical (or Stationary) Points A critical or stationary point is a point (i.e. values for the independent.
Local Extrema Do you remember in Calculus 1 & 2, we sometimes found points where both f'(x) and f"(x) were zero? What did this mean?? What is the corresponding.
Relative Extrema. Objective To find the coordinates of the relative extrema of a function. To find the coordinates of the relative extrema of a function.
Section 15.7 Maximum and Minimum Values. MAXIMA AND MINIMA A function of two variables has a local maximum at (a, b) if f (x, y) ≤ f (a, b) when (x, y)
Unit 11 – Derivative Graphs Section 11.1 – First Derivative Graphs First Derivative Slope of the Tangent Line.
Maxima and Minima of Functions Maxima and minima of functions occur where there is a change from increasing to decreasing, or vice versa.
DO NOW: Find where the function f(x) = 3x 4 – 4x 3 – 12x is increasing and decreasing.
Second Order Partial Derivatives Since derivatives of functions are themselves functions, they can be differentiated. Remember for 1 independent variable,
Warm Up. 5.3C – Second Derivative test Review One way to find local mins and maxs is to make a sign chart with the critical values. There is a theorem.
Chapter 14 – Partial Derivatives 14.7 Maximum and Minimum Values 1 Objectives: Use directional derivatives to locate maxima and minima of multivariable.
Semester 1 Review Unit vector =. A line that contains the point P(x 0,y 0,z 0 ) and direction vector : parametric symmetric.
Derivatives - Equation of the Tangent Line Now that we can find the slope of the tangent line of a function at a given point, we need to find the equation.
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;
Maximum ??? Minimum??? How can we tell? and decreasing just to the right of c, then f has a local minimum at c If f is increasing just to the left of a.
Tangents and Normals We’re going to talk about tangents and normals to 3-D surfaces such as x 2 + y 2 + z 2 = 4 It’s useful to think of these surfaces.
Functions of Several Variables 13 Copyright © Cengage Learning. All rights reserved.
2.3 Curve Sketching (Introduction). We have four main steps for sketching curves: 1.Starting with f(x), compute f’(x) and f’’(x). 2.Locate all relative.
Using Derivatives to Sketch the Graph of a Function Lesson 4.3.
Stationary/Turning Points How do we find them?. What are they? Turning points are points where a graph is changing direction Stationary points are.
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.
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,
Objectives Use the first-order derivative to find the stationary points of a function. Use the second-order derivative to classify the stationary points.
MAT 213 Brief Calculus Section 4.2 Relative and Absolute Extreme Points.
SECT 3-8B RELATING GRAPHS Handout: Relating Graphs.
Extreme Values Let f (x,y) be defined on a region R containing P(x 0,y 0 ): P is a relative max of f if f (x,y) ≤ f (x 0,y 0 ) for all (x,y) on an open.
Point Value : 20 Time limit : 2 min #1 Find. #1 Point Value : 30 Time limit : 2.5 min #2 Find.
Analyzing Multivariable Change: Optimization Chapter 8.1 Extreme Points and Saddle Points.
Functions of Several Variables Copyright © Cengage Learning. All rights reserved.
Take it to the Extrema Maximum and Minimum Value Problems By: Rakesh Biswas.
4.3 – Derivatives and the shapes of curves The Mean Value Theorem: If f(x) is defined and continuous on the interval [a,b] and differentiable on (a,b),
Concavity and the Second Derivative Test Determine the intervals on which the graphs of functions are concave upward or concave downward. Find the.
Relative Extrema of Two Variable Functions. “Understanding Variables”
Ch. 5 – Applications of Derivatives 5.3 – Connecting f’ and f’’ with the Graph of f.
1 Concavity and the Second Derivative Test Section 3.4.
3.2 The Second-Derivative Test 1 What does the sign of the second derivative tell us about the graph? Suppose the second derivative is positive. But, is.
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 ),
1 f ’’(x) > 0 for all x in I f(x) concave Up Concavity Test Sec 4.3: Concavity and the Second Derivative Test the curve lies above the tangentsthe curve.
Copyright © Cengage Learning. All rights reserved. 14 Partial Derivatives.
How derivatives affect the shape of a graph ( Section 4.3) Alex Karassev.
Section 3.1 Maximum and Minimum Values Math 1231: Single-Variable Calculus.
3.3 Relationship between First Derivative, Second Derivative and the Shape of a Graph.
1 Example 4 Sketch the graph of the function k(x) = (x 2 -4) 4/5. Solution Observe that k is an even function, and its graph is symmetric with respect.
1 Maxima and Minima The derivative measures the slope of the tangent to a curve. At the maximum or minimum points, the tangent is horizontal and has slope.
AP Calculus Unit 4 Day 5 Finish Concavity Mean Value Theorem Curve Sketching.
ESSENTIAL CALCULUS CH11 Partial derivatives. In this Chapter: 11.1 Functions of Several Variables 11.2 Limits and Continuity 11.3 Partial Derivatives.
MA Day 34- February 22, 2013 Review for test #2 Chapter 11: Differential Multivariable Calculus.
Applications of Differentiation Calculus Chapter 3.
TS: Explicitly assessing information and drawing conclusions Increasing & Decreasing Functions.
Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points.
Curve Sketching Today we will look at the graph of a function, and then the graph of the derivative of a function and draw conclusions about important.
© 2017 SlidePlayer.com Inc. All rights reserved.