DOCSLIB.ORG

Mean Value, Taylor, and All That

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mean Value, Taylor, and All That Mean Value, Taylor, and all that Ambar N. Sengupta Louisiana State University November 2009 Careful: Not proofread! Derivative Recall the definition of the derivative of a function f at a point p: f (w) − f (p) f 0(p) = lim (1) w!p w − p Derivative Thus, to say that f 0(p) = 3 means that if we take any neighborhood U of 3, say the interval (1; 5), then the ratio f (w) − f (p) w − p falls inside U when w is close enough to p, i.e. in some neighborhood of p. (Of course, we can’t let w be equal to p, because of the w − p in the denominator.) In particular, f (w) − f (p) > 0 if w is close enough to p, but 6= p. w − p Derivative So if f 0(p) = 3 then the ratio f (w) − f (p) w − p lies in (1; 5) when w is close enough to p, i.e. in some neighborhood of p, but not equal to p. Derivative So if f 0(p) = 3 then the ratio f (w) − f (p) w − p lies in (1; 5) when w is close enough to p, i.e. in some neighborhood of p, but not equal to p. In particular, f (w) − f (p) > 0 if w is close enough to p, but 6= p. w − p • when w > p, but near p, the value f (w) is > f (p). • when w < p, but near p, the value f (w) is < f (p). Derivative From f 0(p) = 3 we found that f (w) − f (p) > 0 if w is close enough to p, but 6= p. w − p Looking at this you see that : • when w < p, but near p, the value f (w) is < f (p). Derivative From f 0(p) = 3 we found that f (w) − f (p) > 0 if w is close enough to p, but 6= p. w − p Looking at this you see that : • when w > p, but near p, the value f (w) is > f (p). Derivative From f 0(p) = 3 we found that f (w) − f (p) > 0 if w is close enough to p, but 6= p. w − p Looking at this you see that : • when w > p, but near p, the value f (w) is > f (p). • when w < p, but near p, the value f (w) is < f (p). Thus: Theorem If f 0(p) > 0 then : the values of f to the right of p, but close to p, are > f (p), and the values of f to the left of p, but close to p, are < f (p). Positive Derivative and Increasing behavior Looking back at the argument, we see that the only thing about the value 3 for f 0(p) which made it all work is that it is > 0. the values of f to the right of p, but close to p, are > f (p), and the values of f to the left of p, but close to p, are < f (p). Positive Derivative and Increasing behavior Looking back at the argument, we see that the only thing about the value 3 for f 0(p) which made it all work is that it is > 0. Thus: Theorem If f 0(p) > 0 then : and the values of f to the left of p, but close to p, are < f (p). Positive Derivative and Increasing behavior Looking back at the argument, we see that the only thing about the value 3 for f 0(p) which made it all work is that it is > 0. Thus: Theorem If f 0(p) > 0 then : the values of f to the right of p, but close to p, are > f (p), the values of f to the left of p, but close to p, are < f (p). Positive Derivative and Increasing behavior Looking back at the argument, we see that the only thing about the value 3 for f 0(p) which made it all work is that it is > 0. Thus: Theorem If f 0(p) > 0 then : the values of f to the right of p, but close to p, are > f (p), and Positive Derivative and Increasing behavior Looking back at the argument, we see that the only thing about the value 3 for f 0(p) which made it all work is that it is > 0. Thus: Theorem If f 0(p) > 0 then : the values of f to the right of p, but close to p, are > f (p), and the values of f to the left of p, but close to p, are < f (p). the values of f to the right of p, but close to p, are < f (p), and the values of f to the left of p, but close to p, are > f (p). Negative Derivative and Decreasing behavior Similarly, Theorem If f 0(p) < 0 then : and the values of f to the left of p, but close to p, are > f (p). Negative Derivative and Decreasing behavior Similarly, Theorem If f 0(p) < 0 then : the values of f to the right of p, but close to p, are < f (p), the values of f to the left of p, but close to p, are > f (p). Negative Derivative and Decreasing behavior Similarly, Theorem If f 0(p) < 0 then : the values of f to the right of p, but close to p, are < f (p), and Negative Derivative and Decreasing behavior Similarly, Theorem If f 0(p) < 0 then : the values of f to the right of p, but close to p, are < f (p), and the values of f to the left of p, but close to p, are > f (p). Local Maxima and Minima A function f is said to have a local maximum at a point p if there is a neighborhood U of p such that that for all x 2 U in the domain of f , the value f (x) is ≥ f (p). A function f is said to have a local minimum at a point p if there is a neighborhood U of p such that that for all x 2 U in the domain of f , the value f (x) is ≤ f (p). Local Maxima and Minima local maximum local minimum Figure: Local Maxima and Minima f (p) ≥ f (x) for all x in a neighborhood of p. Suppose also that f 0(p) exists. Then f 0(p) must be 0. If f (p) ≤ f (x) for all x in a neighborhood of p, and f 0(p) exists, then f 0(p) is 0. Note that we are requiring that f be defined in a neighborhood of p, and so on both sides of p. The local Maxima/Minima theorem Theorem Suppose f is defined in a neighborhood of a point p 2 R, and Suppose also that f 0(p) exists. Then f 0(p) must be 0. If f (p) ≤ f (x) for all x in a neighborhood of p, and f 0(p) exists, then f 0(p) is 0. Note that we are requiring that f be defined in a neighborhood of p, and so on both sides of p. The local Maxima/Minima theorem Theorem Suppose f is defined in a neighborhood of a point p 2 R, and f (p) ≥ f (x) for all x in a neighborhood of p. Then f 0(p) must be 0. If f (p) ≤ f (x) for all x in a neighborhood of p, and f 0(p) exists, then f 0(p) is 0. Note that we are requiring that f be defined in a neighborhood of p, and so on both sides of p. The local Maxima/Minima theorem Theorem Suppose f is defined in a neighborhood of a point p 2 R, and f (p) ≥ f (x) for all x in a neighborhood of p. Suppose also that f 0(p) exists. If f (p) ≤ f (x) for all x in a neighborhood of p, and f 0(p) exists, then f 0(p) is 0. Note that we are requiring that f be defined in a neighborhood of p, and so on both sides of p. The local Maxima/Minima theorem Theorem Suppose f is defined in a neighborhood of a point p 2 R, and f (p) ≥ f (x) for all x in a neighborhood of p. Suppose also that f 0(p) exists. Then f 0(p) must be 0. Note that we are requiring that f be defined in a neighborhood of p, and so on both sides of p. The local Maxima/Minima theorem Theorem Suppose f is defined in a neighborhood of a point p 2 R, and f (p) ≥ f (x) for all x in a neighborhood of p. Suppose also that f 0(p) exists. Then f 0(p) must be 0. If f (p) ≤ f (x) for all x in a neighborhood of p, and f 0(p) exists, then f 0(p) is 0. The local Maxima/Minima theorem Theorem Suppose f is defined in a neighborhood of a point p 2 R, and f (p) ≥ f (x) for all x in a neighborhood of p. Suppose also that f 0(p) exists. Then f 0(p) must be 0. If f (p) ≤ f (x) for all x in a neighborhood of p, and f 0(p) exists, then f 0(p) is 0. Note that we are requiring that f be defined in a neighborhood of p, and so on both sides of p.
Load more

Recommended publications
  • Section 8.8: Improper Integrals
    Section 8.8: Improper Integrals One of the main applications of integrals is to compute the areas under curves, as you know. A geometric question. But there are some geometric questions which we do not yet know how to do by calculus, even though they appear to have the same form. Consider the curve y = 1=x2. We can ask, what is the area of the region under the curve and right of the line x = 1? We have no reason to believe this area is finite, but let's ask. Now no integral will compute this{we have to integrate over a bounded interval. Nonetheless, we don't want to throw up our hands. So note that b 2 b Z (1=x )dx = ( 1=x) 1 = 1 1=b: 1 − j − In other words, as b gets larger and larger, the area under the curve and above [1; b] gets larger and larger; but note that it gets closer and closer to 1. Thus, our intuition tells us that the area of the region we're interested in is exactly 1. More formally: lim 1 1=b = 1: b − !1 We can rewrite that as b 2 lim Z (1=x )dx: b !1 1 Indeed, in general, if we want to compute the area under y = f(x) and right of the line x = a, we are computing b lim Z f(x)dx: b !1 a ASK: Does this limit always exist? Give some situations where it does not exist. They'll give something that blows up.
    [Show full text]
  • Notes on Calculus II Integral Calculus Miguel A. Lerma
    Notes on Calculus II Integral Calculus Miguel A. Lerma November 22, 2002 Contents Introduction 5 Chapter 1. Integrals 6 1.1. Areas and Distances. The Definite Integral 6 1.2. The Evaluation Theorem 11 1.3. The Fundamental Theorem of Calculus 14 1.4. The Substitution Rule 16 1.5. Integration by Parts 21 1.6. Trigonometric Integrals and Trigonometric Substitutions 26 1.7. Partial Fractions 32 1.8. Integration using Tables and CAS 39 1.9. Numerical Integration 41 1.10. Improper Integrals 46 Chapter 2. Applications of Integration 50 2.1. More about Areas 50 2.2. Volumes 52 2.3. Arc Length, Parametric Curves 57 2.4. Average Value of a Function (Mean Value Theorem) 61 2.5. Applications to Physics and Engineering 63 2.6. Probability 69 Chapter 3. Differential Equations 74 3.1. Differential Equations and Separable Equations 74 3.2. Directional Fields and Euler’s Method 78 3.3. Exponential Growth and Decay 80 Chapter 4. Infinite Sequences and Series 83 4.1. Sequences 83 4.2. Series 88 4.3. The Integral and Comparison Tests 92 4.4. Other Convergence Tests 96 4.5. Power Series 98 4.6. Representation of Functions as Power Series 100 4.7. Taylor and MacLaurin Series 103 3 CONTENTS 4 4.8. Applications of Taylor Polynomials 109 Appendix A. Hyperbolic Functions 113 A.1. Hyperbolic Functions 113 Appendix B. Various Formulas 118 B.1. Summation Formulas 118 Appendix C. Table of Integrals 119 Introduction These notes are intended to be a summary of the main ideas in course MATH 214-2: Integral Calculus.
    [Show full text]
  • Optimization and Gradient Descent INFO-4604, Applied Machine Learning University of Colorado Boulder
    Optimization and Gradient Descent INFO-4604, Applied Machine Learning University of Colorado Boulder September 11, 2018 Prof. Michael Paul Prediction Functions Remember: a prediction function is the function that predicts what the output should be, given the input. Prediction Functions Linear regression: f(x) = wTx + b Linear classification (perceptron): f(x) = 1, wTx + b ≥ 0 -1, wTx + b < 0 Need to learn what w should be! Learning Parameters Goal is to learn to minimize error • Ideally: true error • Instead: training error The loss function gives the training error when using parameters w, denoted L(w). • Also called cost function • More general: objective function (in general objective could be to minimize or maximize; with loss/cost functions, we want to minimize) Learning Parameters Goal is to minimize loss function. How do we minimize a function? Let’s review some math. Rate of Change The slope of a line is also called the rate of change of the line. y = ½x + 1 “rise” “run” Rate of Change For nonlinear functions, the “rise over run” formula gives you the average rate of change between two points Average slope from x=-1 to x=0 is: 2 f(x) = x -1 Rate of Change There is also a concept of rate of change at individual points (rather than two points) Slope at x=-1 is: f(x) = x2 -2 Rate of Change The slope at a point is called the derivative at that point Intuition: f(x) = x2 Measure the slope between two points that are really close together Rate of Change The slope at a point is called the derivative at that point Intuition: Measure the
    [Show full text]
  • Two Fundamental Theorems About the Definite Integral
    Two Fundamental Theorems about the Definite Integral These lecture notes develop the theorem Stewart calls The Fundamental Theorem of Calculus in section 5.3. The approach I use is slightly different than that used by Stewart, but is based on the same fundamental ideas. 1 The definite integral Recall that the expression b f(x) dx ∫a is called the definite integral of f(x) over the interval [a,b] and stands for the area underneath the curve y = f(x) over the interval [a,b] (with the understanding that areas above the x-axis are considered positive and the areas beneath the axis are considered negative). In today's lecture I am going to prove an important connection between the definite integral and the derivative and use that connection to compute the definite integral. The result that I am eventually going to prove sits at the end of a chain of earlier definitions and intermediate results. 2 Some important facts about continuous functions The first intermediate result we are going to have to prove along the way depends on some definitions and theorems concerning continuous functions. Here are those definitions and theorems. The definition of continuity A function f(x) is continuous at a point x = a if the following hold 1. f(a) exists 2. lim f(x) exists xœa 3. lim f(x) = f(a) xœa 1 A function f(x) is continuous in an interval [a,b] if it is continuous at every point in that interval. The extreme value theorem Let f(x) be a continuous function in an interval [a,b].
    [Show full text]
  • Lecture 13 Gradient Methods for Constrained Optimization
    Lecture 13 Gradient Methods for Constrained Optimization October 16, 2008 Lecture 13 Outline • Gradient Projection Algorithm • Convergence Rate Convex Optimization 1 Lecture 13 Constrained Minimization minimize f(x) subject x ∈ X • Assumption 1: • The function f is convex and continuously differentiable over Rn • The set X is closed and convex ∗ • The optimal value f = infx∈Rn f(x) is finite • Gradient projection algorithm xk+1 = PX[xk − αk∇f(xk)] starting with x0 ∈ X. Convex Optimization 2 Lecture 13 Bounded Gradients Theorem 1 Let Assumption 1 hold, and suppose that the gradients are uniformly bounded over the set X. Then, the projection gradient method generates the sequence {xk} ⊂ X such that • When the constant stepsize αk ≡ α is used, we have 2 ∗ αL lim inf f(xk) ≤ f + k→∞ 2 P • When diminishing stepsize is used with k αk = +∞, we have ∗ lim inf f(xk) = f . k→∞ Proof: We use projection properties and the line of analysis similar to that of unconstrained method. HWK 6 Convex Optimization 3 Lecture 13 Lipschitz Gradients • Lipschitz Gradient Lemma For a differentiable convex function f with Lipschitz gradients, we have for all x, y ∈ Rn, 1 k∇f(x) − ∇f(y)k2 ≤ (∇f(x) − ∇f(y))T (x − y), L where L is a Lipschitz constant. • Theorem 2 Let Assumption 1 hold, and assume that the gradients of f are Lipschitz continuous over X. Suppose that the optimal solution ∗ set X is not empty. Then, for a constant stepsize αk ≡ α with 0 2 < α < L converges to an optimal point, i.e., ∗ ∗ ∗ lim kxk − x k = 0 for some x ∈ X .
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • The Infinite and Contradiction: a History of Mathematical Physics By
    The infinite and contradiction: A history of mathematical physics by dialectical approach Ichiro Ueki January 18, 2021 Abstract The following hypothesis is proposed: \In mathematics, the contradiction involved in the de- velopment of human knowledge is included in the form of the infinite.” To prove this hypothesis, the author tries to find what sorts of the infinite in mathematics were used to represent the con- tradictions involved in some revolutions in mathematical physics, and concludes \the contradiction involved in mathematical description of motion was represented with the infinite within recursive (computable) set level by early Newtonian mechanics; and then the contradiction to describe discon- tinuous phenomena with continuous functions and contradictions about \ether" were represented with the infinite higher than the recursive set level, namely of arithmetical set level in second or- der arithmetic (ordinary mathematics), by mechanics of continuous bodies and field theory; and subsequently the contradiction appeared in macroscopic physics applied to microscopic phenomena were represented with the further higher infinite in third or higher order arithmetic (set-theoretic mathematics), by quantum mechanics". 1 Introduction Contradictions found in set theory from the end of the 19th century to the beginning of the 20th, gave a shock called \a crisis of mathematics" to the world of mathematicians. One of the contradictions was reported by B. Russel: \Let w be the class [set]1 of all classes which are not members of themselves. Then whatever class x may be, 'x is a w' is equivalent to 'x is not an x'. Hence, giving to x the value w, 'w is a w' is equivalent to 'w is not a w'."[52] Russel described the crisis in 1959: I was led to this contradiction by Cantor's proof that there is no greatest cardinal number.
    [Show full text]
  • Maxima and Minima
    Basic Mathematics Maxima and Minima R Horan & M Lavelle The aim of this document is to provide a short, self assessment programme for students who wish to be able to use differentiation to find maxima and mininima of functions. Copyright c 2004 [email protected] , [email protected] Last Revision Date: May 5, 2005 Version 1.0 Table of Contents 1. Rules of Differentiation 2. Derivatives of order 2 (and higher) 3. Maxima and Minima 4. Quiz on Max and Min Solutions to Exercises Solutions to Quizzes Section 1: Rules of Differentiation 3 1. Rules of Differentiation Throughout this package the following rules of differentiation will be assumed. (In the table of derivatives below, a is an arbitrary, non-zero constant.) y axn sin(ax) cos(ax) eax ln(ax) dy 1 naxn−1 a cos(ax) −a sin(ax) aeax dx x If a is any constant and u, v are two functions of x, then d du dv (u + v) = + dx dx dx d du (au) = a dx dx Section 1: Rules of Differentiation 4 The stationary points of a function are those points where the gradient of the tangent (the derivative of the function) is zero. Example 1 Find the stationary points of the functions (a) f(x) = 3x2 + 2x − 9 , (b) f(x) = x3 − 6x2 + 9x − 2 . Solution dy (a) If y = 3x2 + 2x − 9 then = 3 × 2x2−1 + 2 = 6x + 2. dx The stationary points are found by solving the equation dy = 6x + 2 = 0 . dx In this case there is only one solution, x = −1/3.
    [Show full text]
  • Calculus 120, Section 7.3 Maxima & Minima of Multivariable Functions
    Calculus 120, section 7.3 Maxima & Minima of Multivariable Functions notes by Tim Pilachowski A quick note to start: If you’re at all unsure about the material from 7.1 and 7.2, now is the time to go back to review it, and get some help if necessary. You’ll need all of it for the next three sections. Just like we had a first-derivative test and a second-derivative test to maxima and minima of functions of one variable, we’ll use versions of first partial and second partial derivatives to determine maxima and minima of functions of more than one variable. The first derivative test for functions of more than one variable looks very much like the first derivative test we have already used: If f(x, y, z) has a relative maximum or minimum at a point ( a, b, c) then all partial derivatives will equal 0 at that point. That is ∂f ∂f ∂f ()a b,, c = 0 ()a b,, c = 0 ()a b,, c = 0 ∂x ∂y ∂z Example A: Find the possible values of x, y, and z at which (xf y,, z) = x2 + 2y3 + 3z 2 + 4x − 6y + 9 has a relative maximum or minimum. Answers : (–2, –1, 0); (–2, 1, 0) Example B: Find the possible values of x and y at which fxy(),= x2 + 4 xyy + 2 − 12 y has a relative maximum or minimum. Answer : (4, –2); z = 12 Notice that the example above asked for possible values. The first derivative test by itself is inconclusive. The second derivative test for functions of more than one variable is a good bit more complicated than the one used for functions of one variable.
    [Show full text]
  • Convergence Rates for Deterministic and Stochastic Subgradient
    Convergence Rates for Deterministic and Stochastic Subgradient Methods Without Lipschitz Continuity Benjamin Grimmer∗ Abstract We extend the classic convergence rate theory for subgradient methods to apply to non-Lipschitz functions. For the deterministic projected subgradient method, we present a global O(1/√T ) convergence rate for any convex function which is locally Lipschitz around its minimizers. This approach is based on Shor’s classic subgradient analysis and implies generalizations of the standard convergence rates for gradient descent on functions with Lipschitz or H¨older continuous gradients. Further, we show a O(1/√T ) convergence rate for the stochastic projected subgradient method on convex functions with at most quadratic growth, which improves to O(1/T ) under either strong convexity or a weaker quadratic lower bound condition. 1 Introduction We consider the nonsmooth, convex optimization problem given by min f(x) x∈Q for some lower semicontinuous convex function f : Rd R and closed convex feasible → ∪{∞} region Q. We assume Q lies in the domain of f and that this problem has a nonempty set of minimizers X∗ (with minimum value denoted by f ∗). Further, we assume orthogonal projection onto Q is computationally tractable (which we denote by PQ( )). arXiv:1712.04104v3 [math.OC] 26 Feb 2018 Since f may be nondifferentiable, we weaken the notion of gradients to· subgradients. The set of all subgradients at some x Q (referred to as the subdifferential) is denoted by ∈ ∂f(x)= g Rd y Rd f(y) f(x)+ gT (y x) . { ∈ | ∀ ∈ ≥ − } We consider solving this problem via a (potentially stochastic) projected subgradient method.
    [Show full text]
  • Math 105: Multivariable Calculus Seventeenth Lecture (3/17/10)
    Daily Summary Fast Taylor Series Critical Points and Extrema Lagrange Multipliers Math 105: Multivariable Calculus Seventeenth Lecture (3/17/10) Steven J Miller Williams College [email protected] http://www.williams.edu/go/math/sjmiller/ public html/341/ Bronfman Science Center Williams College, March 17, 2010 1 Daily Summary Fast Taylor Series Critical Points and Extrema Lagrange Multipliers Summary for the Day 2 Daily Summary Fast Taylor Series Critical Points and Extrema Lagrange Multipliers Summary for the day Fast Taylor Series. Critical Points and Extrema. Constrained Maxima and Minima. 3 Daily Summary Fast Taylor Series Critical Points and Extrema Lagrange Multipliers Fast Taylor Series 4 Daily Summary Fast Taylor Series Critical Points and Extrema Lagrange Multipliers Formula for Taylor Series Notation: f twice differentiable function ∂f ∂f Gradient: f =( ∂ ,..., ∂ ). ∇ x1 xn Hessian: ∂2f ∂2f ∂x1∂x1 ∂x1∂xn . ⋅⋅⋅. Hf = ⎛ . .. ⎞ . ∂2f ∂2f ⎜ ∂ ∂ ∂ ∂ ⎟ ⎝ xn x1 ⋅⋅⋅ xn xn ⎠ Second Order Taylor Expansion at −→x 0 1 T f (−→x 0) + ( f )(−→x 0) (−→x −→x 0)+ (−→x −→x 0) (Hf )(−→x 0)(−→x −→x 0) ∇ ⋅ − 2 − − T where (−→x −→x 0) is the row vector which is the transpose of −→x −→x 0. − − 5 Daily Summary Fast Taylor Series Critical Points and Extrema Lagrange Multipliers Example 3 Let f (x, y)= sin(x + y) + (x + 1) y and (x0, y0) = (0, 0). Then ( f )(x, y) = cos(x + y)+ 3(x + 1)2y, cos(x + y) + (x + 1)3 ∇ and sin(x + y)+ 6(x + 1)2y sin(x + y)+ 3(x + 1)2 (Hf )(x, y) = − − , sin(x + y)+ 3(x + 1)2 sin(x + y) − − so 0 3 f (0, 0)= 0, ( f )(0, 0) = (1, 2), (Hf )(0, 0)= ∇ 3 0 which implies the second order Taylor expansion is 1 0 3 x 0 + (1, 2) (x, y)+ (x, y) ⋅ 2 3 0 y 1 3y = x + 2y + (x, y) = x + 2y + 3xy.
    [Show full text]
  • Lecture 13: Extrema
    Math S21a: Multivariable calculus Oliver Knill, Summer 2013 Lecture 13: Extrema An important problem in multi-variable calculus is to extremize a function f(x, y) of two vari- ables. As in one dimensions, in order to look for maxima or minima, we consider points, where the ”derivative” is zero. A point (a, b) in the plane is called a critical point of a function f(x, y) if ∇f(a, b) = h0, 0i. Critical points are candidates for extrema because at critical points, all directional derivatives D~vf = ∇f · ~v are zero. We can not increase the value of f by moving into any direction. This definition does not include points, where f or its derivative is not defined. We usually assume that a function is arbitrarily often differentiable. Points where the function has no derivatives are not considered part of the domain and need to be studied separately. For the function f(x, y) = |xy| for example, we would have to look at the points on the coordinate axes separately. 1 Find the critical points of f(x, y) = x4 + y4 − 4xy + 2. The gradient is ∇f(x, y) = h4(x3 − y), 4(y3 − x)i with critical points (0, 0), (1, 1), (−1, −1). 2 f(x, y) = sin(x2 + y) + y. The gradient is ∇f(x, y) = h2x cos(x2 + y), cos(x2 + y) + 1i. For a critical points, we must have x = 0 and cos(y) + 1 = 0 which means π + k2π. The critical points are at ... (0, −π), (0, π), (0, 3π),.... 2 2 3 The graph of f(x, y) = (x2 + y2)e−x −y looks like a volcano.
    [Show full text]