ob rown, CCC Dundalk Math 253 Calculus 3, Chapter 13 Section 6 Completed 1 Eercise 1: Let z f, be a function, graphed as a contour map below. Let A and correspond to points in the domain of f. 2 A i) Estimate the value of and eplain the meaning of. Moving past the point 15 1 f b about per unit moved Note that is an instantaneous rate of change, which we estimated here with an average rate of change. 2 A ii) Estimate the value of and eplain the meaning of. 15 1 Moving past the point f b about per unit moved Note that is an instantaneous rate of change, which we estimated here with an average rate of change. iii) Estimate the average rate of change of f from A to, and eplain its meaning. 2 A 15 Moving past the point A in the direction of, 1 f b about per unit moved
ob rown, CCC Dundalk Math 253 Calculus 3, Chapter 13 Section 6 Completed 2 Directional Derivative Let P, and Q be points in the domain of a function f. Suppose that we want to compute the rate of change of f at P in the direction of Q. P, Let u u1i j be the unit vector in the direction from P to Q. Q If Q is h units awa from P, what are the coordinates of Q? Q Therefore, what is the average rate of change of f from P to Q? change in f distance from P toq In order to determine the instantaneous rate of change of f at P in the direction from P to Q, Def.: Let z f, be a function, and let u u1i j be a unit vector. Provided that the limit eists, the directional derivative of f in the direction of u, denoted is given b Eercise 2: If u i, show that D u f, f,. u i u 1 and u 2 Then, D u f, D i f, Note: If u j, then D u f, D j f, f,. Theorem: Let z f, be a function, and let u differentiable function of and, then D u f, u1i j be a unit vector. If f is a
ob rown, CCC Dundalk Math 253 Calculus 3, Chapter 13 Section 6 Completed 3 Note: The directional derivative is defined in the direction of a unit vector. If v is not a unit vector, then define D v f as D u f, where u 2 2 Eercise 3: Draw a contour diagram for f,, and use the limit definition in order to calculate D u f 1,2 ) in the direction of v 3i 4 j. Check our answer b using the Theorem at the bottom of page 2. First, let u D u f 1,2) Now, re-calculate D u f 1,2 ), this time b using the Theorem at the bottom of page 2. f, f 1,2 ) f, f 1,2 ) Thus, b the Theorem, D v f 1,2 ) D u f 1,2 )
ob rown, CCC Dundalk Math 253 Calculus 3, Chapter 13 Section 6 Completed 4 2 Eercise 4: Draw a contour diagram for g,, and use the limit definition in order to calculate D v g 2,3) in the direction of v i j. Check our answer b using the Theorem at the bottom of page 2. First, let u g, g 2,3) g, g 2,3) Thus, b the Theorem, D v g 2,3) D u g 2,3) Gradient The gradient of a function of two variables is a vector-valued function of two variables with man important uses. Def.: Let z f, be a function of and such that f, and f, eist. f a, is called the f a, f a, is a f a, is read as del f or grad f. At a general point, in the domain of f, we write f,
ob rown, CCC Dundalk Math 253 Calculus 3, Chapter 13 Section 6 Completed 5 Eercise 5: Determine and sketch 2 2 f 1,2), where f,. From Eercise 3, f 1,2) and f 1,2 ) Thus, f 1,2 ) Eercise 6: Determine and sketch 2 g 2,3), where g,. From Eercise 4, g 2,3) and g 2,3) Thus, g 2,3) Alternate Form of the Directional Derivative Theorem: Let z f, be a function, and let u differentiable function of and, then D u f, u1i j be a unit vector. If f is a Eercise 7: Use this Theorem to redo Eercise 3. v 3i 4 j u f, 2 2 f, f 1,2 ) Thus, b the Theorem, D u f 1,2 )
ob rown, CCC Dundalk Math 253 Calculus 3, Chapter 13 Section 6 Completed 6 Properties of the Gradient Let z f, be a function that is differentiable at the point,. 1. If f, ), then D f, u ) for all u that is, in an direction.) 2. The direction of maimum increase of f at, is given b The maimum value of D f, u ) is 3. The direction of maimum decrease of f at, is given b The minimum value of D f, u ) is 4. If f, ), then f, ) is Eercise 8: Prove Gradient Propert #2. D f, u ) f, ) u We see, then, that the maimum value of D f, u ) occurs for, which is when Rewording of Properties 2 and 4: The gradient vector at a point points in the direction of the greatest rate of change of f at that point. This implies that f, ) is normal to the contour of f that passes through,. This fact also implies that f, ) points in the direction of increasing f.
ob rown, CCC Dundalk Math 253 Calculus 3, Chapter 13 Section 6 Completed 7 Note: The definitions and properties of the directional derivative and of the gradient are similar for functions of three or more variables. See page 923 in the tet. Eercise 9: Given a contour diagram of f, which vector that is most likel the gradient vector of f at P? Circle our answer here: green vector, blue vector, red vector. 1 9 P 8 Eercise 1: Given a contour diagram of f, which illustration most likel contains correctl drawn gradient vectors of f at P and at Q? Left Illustration or Right Illustration? Left Illustration Right Illustration 1 1 9 P 9 P Q Q 8 8 Although the gradient, like the directional derivative, gives information about instantaneous rates of change, these diagrams onl hint at average rates of change. Nevertheless, f most likel