SI Calculus II
Weber State University
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This 3 page Class Notes was uploaded by Rosa Farrell on Wednesday October 28, 2015. The Class Notes belongs to MATH 1220 at Weber State University taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/230807/math-1220-weber-state-university in Math at Weber State University.
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Date Created: 10/28/15
ADDITIONAL NOTES AND EXERCISES FOR 71 These notes probably belong towards the end of chapter 6 rather than at the start of chapter 7 Unfortunately I was unable to write them in time for that In this discussion we will concern ourselves with linear approximations Some of this we have already discussed in class De nition 1 Let f z I 7gt R be a function on the open interval I Let a E I If f is differentiable at a then the linear approximation of f at a is given by 1 1 m 111 WWI a 1 Notice that L is simply the equation of the tangent line to f at z a The point is that if f is differentiable at a then L is a good approximation of f for 1 near a Put another way7 if f is differentiable at a then under a microscope f will look very much like a straight line Exercise 2 Sketch a graph of a nonlinear function f that illustrates this point Example 3 Let 13 At 1 2 the linear approximation of f is given by 2 fx mLz 322z722312z728 since 312 Example 4 Let xz 4 Then 2 The linear approximation to f at z 5 is z 7 5 z 7 5 lt3 fltzgtLltzgt M 6 3 As an immediate application we can approximate square roots of numbers near 10 by hand Computing m is difficult to compute directly since m is irrational The linear approximation gives us a simple way of estimating Notice that 6 7 5 19 7 4 W6 f6T3g316 It turns out that m w 316227766 correct to 8 decimal places so our linear approximation is only accurate to one decimal p ace Exercise 5 In the preceding example our approximation of was an overes timate Show that we always get an overestimate that is7 show that Lz 2 for z gt 74 Hint What does the second derivative tell you about With advanced calculators and computing software it may not appear necessary to use linear approximations Keep in mind7 however that when computing with irrational numbers both calculators and software use approximations As such7 it can be helpful to have some idea of how the approximations happen Although these approximations may not be linear7 an advantage of studying linear approximations initially is that they are relatively simp e 1 2 ADDITIONAL NOTES AND EXERCISES FOR 71 It should also be pointed out that one can estimate how good our linear approxi mation is without actually knowing the precise value ofwhat we are estimatingi We discuss this more when we consider approximate integration methods like Simpson s ruler As a practical example7 consider the trigonometric functions sinz and cos zi These functions come up frequently in any science where angles play a roleli Exercise 6 Show that the linear approximation of sinz at z 0 is z and the linear approximation of cos z at z 0 is 1 Do not use the subsequent discussion We will see in math 1220 that 00 7 71n12n1 7 Is 15 5 s1nzigmizigmiiu and 00 707L127 12 I4 6 717 i 7 H CO 7 2n 2 24 Notice that if z is close to zero then sinz m z and cosz m 1 because the higher order terms are much smaller than z This linearization simpli es many calculations without serious loss of accuracy and indeed makes otherwise intractable calculations feasible De nition 7 Let y be a differentiable function We de ne a new indepen dent variable dzi The domain of dz is any real number We de ne dy f zdz and note that dy is a dependent variable We say that dz and dy are differentials Notice that dy is a function both of z since is a function of z and of dzi Here is the reason for introducing differentials Fix a point Pa7 Let 7 Az z 7 d Note that if z is near a then Az is small Set dz Az and let Ay 7 fai If Az is small then dy m Ay and dy is the linear approximation of for z near We illustrate the point in gure 1 Exercise 8 Let z4i If a 1 and dz Az what are Ay and dy Exercise 9 Let If a l and dz Az i what are Ay and dy Exercise 10 Let sin2zi If a 7r and dz Az W72 what are Ay and dy Exercise 11 Using differentials estimate the amount of paint needed to apply a coat of paint 002 cm think to a sphere with diameter 40 meters Recall that the volume of a sphere of radius 7 is given by the formula V 47 Notice that you are given that d7quot 0 02 1For example optics and mechanics ADDITIONAL NOTES AND EXERCISES FOR 71 y fx FIGURE 1 Differentials
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