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Unformatted text preview: ion of the tangent plane. D EFINITION 8.6. Suppose is the plane given by is differentiable at . The tangent plane of at the point Notice this is the graph of the linear approximation. E XERCISE 14. Suppose is differentiable at . Is it possible to ﬁnd a plane such that the error in approximating the surface using is strictly less than the error in approximating the surface using the tangent plane? 9 Partial Derivatives, Continuous Functions, and Differentiable Functions (5.25.3) We now look at the various connections between partial derivatives, continuous functions, and differentiable functions. Recall in one variable: differentiable at implies is continuous at . In two dimensions, we will have the analogous result that if is differentiable at , then is continuous at . T HEOREM 9.1. Let . If is differentiable at , then is continuous at . 15 WATERLOO SOS E XAM AID: MATH237 E XERCISE 15. Show the converse to Theorem 9.1 is not true, i.e. ﬁnd a function which is continuous at , but is not differentiable at . Hint: Look at a previous exercise. What about the partial derivatives that we have developed? Is their existence sufﬁcient for continuity? E XAMPLE 18. Deﬁne by if if Prove , yet is not continuous at . S OLUTION. Using the deﬁnition of partial derivatives, Thus, both partial derivatives exist. However, recall we showed in example 6 that the limit as does not exist. Indeed, for we get while for , we get for for Thus, is not continuous at . It can be tedious to always go back to the deﬁnition of differentiability to prove a function is differentiable. Fortunately, we have a shortcut. T HEOREM 9.2. Let . If and are continuous at generalizes to dimensions in the natural way.) R EMARK 9.3. It is important to distinguish between (a) (b) , which is a function , and with respect to , evaluated at is continuous at . , you must prove , then is differentiable at . (This theorem , which is the partial derivative of For example, when proving the partial derivative with respect to the function is continuous at , i.e. 16 WATERLOO SOS E XAM AID: MATH237 E XAMPLE 19. Let be deﬁned by . Determine where is differentiable. S OLUTION. By differentiation, for . By the continuity theorems, is continuous for . By symmetry, is continuous for . Then by Theorem 9.1, is differentiable for all . However, we have not ﬁnished the question  we have not determined differentiability at . In cases like this, we will have to go back to the deﬁnition of differentiability. I leave the ﬁnal step of proving is not differentiable at as an exercise. E XERCISE 16 [C HALLENGE ]. Show the converse to Theorem 9.2 is not true, i.e. ﬁnd a function which is differentiable at , yet the partial derivatives are not continuous at . Hint: Try thinking about a function which oscillates rapidly as you approach the origin. E XERCISE 17. Find a function whose directional derivative exists in every direction at a point , yet is not differentiable at . Hint: Find such a function that is not even continuous at . If you’re interested, you can also ﬁnd such a function which is continuous, but not differentiable. 10 Linear Approximation Revisited (5.4) Recall by deﬁnition of differentiability, the linear approximation for is a good approximation if is differentiable. Now that we can apply Theorem 9.2, we can usually verify the validity of approximations much easier. E XAMPLE 20. Discuss the validity of the approximation S OLUTION. Write . Recall in example 15, by standard differentiation rules we get and at , so the linear approximation is By the continuity theorems, and are continuous at , so tha...
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.
 Spring '10
 WILKIE
 Scalar, Sets

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