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Unformatted text preview: rst, but our strategy is to try to ﬁnd points at which we know the value of and without a calculator. For example, we should know or be able to work out , , and so on. We know how to handle odd multiples of (it is , , , we look ), and we are given , whose denominator happens to be a multiple of . Writing for an odd multiple of that will be approximately 9.5. Even making the estimate that , we have . Deﬁne given by . Recall we showed in example 15 the linear approximation at is The calculator gives us a value of about , for an error of about . E XERCISE 9. Calculate the following approximately (compare with the calculator value): (a) (b) . at . Notation: For and a point , deﬁne By rearranging in the linear approximation formula, 12 WATERLOO SOS E XAM -AID: MATH237 which in our new notation becomes That is, we have a formula for the approximate change in increment form of the linear approximation formula. due to a change . We call this the E XERCISE 10. Suppose a triangle has two sides (say and ) of length 1 m, and the angle between them is . If is increased by 1 cm and decreases by 1 cm, estimate the change in area. The linear approximation formula becomes unwieldy when generalizing to higher dimensions. To help us out, we introduce a new deﬁnition. D EFINITION 7.3. Let . The gradient of at is deﬁned by D EFINITION 7.4. For we deﬁne the linear approximation of at by We write for sufﬁciently close to . In the increment form, we have for sufﬁciently close to zero. ( .) , and E XERCISE 11. Deﬁne linear approximation for by at . . Find the gradient of and the E XERCISE 12. Use linear approximation to estimate . Compare with the calculator value. 8 Differentiability (5.1)
and had for sufﬁciently close In the last section, we deﬁned the linear approximation of to , where It is a natural question to ask: how good is the approximation? D EFINITION 8.1. The error in the linear approximation is deﬁned by 13 We will consider how large error is relatively small. T HEOREM 8.2. If and WATERLOO SOS E XAM -AID: MATH237 is compared to the displacement . In one variable, we see the exists, then . We use this idea to deﬁne an analogous notion of differentiability in higher dimensions. D EFINITION 8.3. A function is differentiable at such that if and only if there is a linear function where Fortunately, all our work with linear approximation has not been in vain. T HEOREM 8.4. If is differentiable at then is the linear approximation of at , i.e. R EMARK 8.5. (a) Thus to prove with linear function and . is differentiable, we simply need to prove , where (b) To prove it is not differentiable, either this limit is not , or the partial derivatives of even exist (and so the linear approximation does not exist). don’t is (c) This theorem also tells us that the linear approximation for is a good approximation if differentiable, that is, the error goes to faster than the magnitude of the displacement. E XAMPLE 17. Let be deﬁned by if if Determine whether is differentiable at . exist. Recall from S OLUTION. We ﬁrst check if exists, that is, if the partial derivatives at example 13 that they do (or work it out again), and 14 Since , the linear approximation is WATERLOO SOS E XAM -AID: MATH237 , and the error is if if and the magnitude of displacement is Now along the line , we have so by deﬁnition, is not differentiable at . E XERCISE 13. Deﬁne . Prove that is differentiable at Contrast this to the case in single variable, when is differentiable at , yet not at yet not at . . Now that we have the deﬁnition of differentiability, we can give a formal deﬁnit...
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