SECTION LINEAR APPROXIMATIONS AND DIFFERENTIALS A lighthouse is located on a small island 3 km away from the

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1 SECTION LINEAR APPROXIMATIONS AND DIFFERENTIALS as a function of body length L (measured in centimeters) is W 0.12L If, over 10 million years, the average length of a certain species of fish evolved from 15 cm to 20 cm at a constant rate, how fast was this species brain growing when the average length was 18 cm? 35. Two sides of a triangle have lengths 12 m and 15 m. The angle between them is increasing at a rate of 2min. How fast is the length of the third side increasing when the angle between the sides of fixed length is 60? 36. Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley P (see the figure). The point Q is on the floor 12 ft directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 2 fts. How fast is cart B moving toward Q at the instant when cart A is 5 ft from Q? 37. A P 12 f t Q A television camera is positioned 4000 ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Let s assume the rocket rises vertically and its speed is 600 fts when it has risen 3000 ft. (a) How fast is the distance from the television camera to the rocket changing at that moment? B (b) If the television camera is always kept aimed at the rocket, how fast is the camera s angle of elevation changing at that same moment? 38. A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P? 39. A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is 3, this angle is decreasing at a rate of 6 radmin. How fast is the plane traveling at that time? 40. A Ferris wheel with a radius of 10 m is rotating at a rate of one revolution every 2 minutes. How fast is a rider rising when his seat is 16 m above ground level? 41. A plane flying with a constant speed of 300 kmh passes over a ground radar station at an altitude of 1 km and climbs at an angle of 30. At what rate is the distance from the plane to the radar station increasing a minute later? 42. Two people start from the same point. One walks east at 3 mih and the other walks northeast at 2 mih. How fast is the distance between the people changing after 15 minutes? 43. A runner sprints around a circular track of radius 100 m at a constant speed of 7 ms. The runner s friend is standing at a distance 200 m from the center of the track. How fast is the distance between the friends changing when the distance between them is 200 m? 44. The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one o clock? 3.10 LINEAR APPROXIMATIONS AND DIFFERENTIALS y {a, f(a)} y=ƒ y=l(x) We have seen that a curve lies very close to its tangent line near the point of tangency. In fact, by zooming in toward a point on the graph of a differentiable function, we noticed that the graph looks more and more like its tangent line. (See Figure 2 in Section 2.7.) This observation is the basis for a method of finding approximate values of functions. The idea is that it might be easy to calculate a value f a of a function, but difficult (or even impossible) to compute nearby values of f. So we settle for the easily computed values of the linear function L whose graph is the tangent line of f at a, f a. (See Figure 1.) In other words, we use the tangent line at a, f a as an approximation to the curve y f x when x is near a. An equation of this tangent line is y f a f ax a and the approximation 0 x 1 f x f a f ax a FIGURE 1 is called the linear approximation or tangent line approximation of f at a. The linear

2 2 4 8 CHAPTER 3 DIFFERENTIATION RULES function whose graph is this tangent line, that is, 2 Lx f a f ax a is called the linearization of f at a. V EXAMPLE 1 Find the linearization of the function f x sx 3 at a 1 and use it to approximate the numbers s3.98 and s4.05. Are these approximations overestimates or underestimates? SOLUTION The derivative of f x x 3 12 is f x 1 2 x sx 3 and so we have f 1 2 and f Putting these values into Equation 2, we see that the linearization is Lx f 1 f 1x x x 4 The corresponding linear approximation (1) is sx x 4 (when x is near 1) In particular, we have 7 y= + x 4 4 y (1, 2) y= œ x+3 _3 0 1 x FIGURE 2 s and s The linear approximation is illustrated in Figure 2. We see that, indeed, the tangent line approximation is a good approximation to the given function when x is near l. We also see that our approximations are overestimates because the tangent line lies above the curve. Of course, a calculator could give us approximations for s3.98 and s4.05, but the linear approximation gives an approximation over an entire interval. M In the following table we compare the estimates from the linear approximation in Example 1 with the true values. Notice from this table, and also from Figure 2, that the tangent line approximation gives good estimates when x is close to 1 but the accuracy of the approximation deteriorates when x is farther away from 1. x From Lx Actual value s3.9 s3.98 s4 s4.05 s4.1 s5 s

3 SECTION LINEAR APPROXIMATIONS AND DIFFERENTIALS How good is the approximation that we obtained in Example 1? The next example shows that by using a graphing calculator or computer we can determine an interval throughout which a linear approximation provides a specified accuracy. EXAMPLE 2 For what values of x is the linear approximation sx x 4 L(x) P 4.3 _4 10 FIGURE 3 y= œ x _1 3 y= œ x Q y= œ x Q accurate to within 0.5? What about accuracy to within 0.1? SOLUTION Accuracy to within 0.5 means that the functions should differ by less than 0.5: Equivalently, we could write sx x4 0.5 sx x 4 sx This says that the linear approximation should lie between the curves obtained by shifting the curve y sx 3 upward and downward by an amount 0.5. Figure 3 shows the tangent line y 7 x4 intersecting the upper curve y sx at P and Q. Zooming in and using the cursor, we estimate that the x-coordinate of P is about 2.66 and the x-coordinate of Q is about Thus we see from the graph that the approximation P y= œ x sx x 4 _2 FIGURE is accurate to within 0.5 when 2.6 x 8.6. (We have rounded to be safe.) Similarly, from Figure 4 we see that the approximation is accurate to within 0.1 when 1.1 x 3.9. M APPLICATIONS TO PHYSICS Linear approximations are often used in physics. In analyzing the consequences of an equation, a physicist sometimes needs to simplify a function by replacing it with its linear approximation. For instance, in deriving a formula for the period of a pendulum, physics textbooks obtain the expression a T t sin for tangential acceleration and then replace sin by with the remark that sin is very close to if is not too large. [See, for example, Physics: Calculus, 2d ed., by Eugene Hecht (Pacific Grove, CA: Brooks/Cole, 2000), p. 431.] You can verify that the linearization of the function f x sin x at a 0 is Lx x and so the linear approximation at 0 is (see Exercise 42). So, in effect, the derivation of the formula for the period of a pendulum uses the tangent line approximation for the sine function. Another example occurs in the theory of optics, where light rays that arrive at shallow angles relative to the optical axis are called paraxial rays. In paraxial (or Gaussian) optics, both sin and cos are replaced by their linearizations. In other words, the linear approximations sin and cos 1 sin x x

4 2 5 0 CHAPTER 3 DIFFERENTIATION RULES are used because is close to 0. The results of calculations made with these approximations became the basic theoretical tool used to design lenses. [See Optics, 4th ed., by Eugene Hecht (San Francisco: Addison-Wesley, 2002), p. 154.] In Section we will present several other applications of the idea of linear approximations to physics. N If dx 0, we can divide both sides of Equation 3 by dx to obtain We have seen similar equations before, but now the left side can genuinely be interpreted as a ratio of differentials. y dy f x dx 0 x x+îx x FIGURE 5 P y=ƒ Q dx=îx R S Îy dy DIFFERENTIALS The ideas behind linear approximations are sometimes formulated in the terminology and notation of differentials. If y f x, where f is a differentiable function, then the differential dx is an independent variable; that is, dx can be given the value of any real number. The differential dy is then defined in terms of dx by the equation 3 So dy is a dependent variable; it depends on the values of x and dx. If dx is given a specific value and x is taken to be some specific number in the domain of f, then the numerical value of dy is determined. The geometric meaning of differentials is shown in Figure 5. Let Px, f x and Qx x, f x x be points on the graph of f and let dx x. The corresponding change in y is The slope of the tangent line PR is the derivative f x. Thus the directed distance from S to R is f x dx dy. Therefore dy represents the amount that the tangent line rises or falls (the change in the linearization), whereas y represents the amount that the curve y f x rises or falls when x changes by an amount dx. EXAMPLE 3 Compare the values of y and dy if y f x x 3 x 2 2x 1 and x changes (a) from 2 to 2.05 and (b) from 2 to SOLUTION (a) We have dy f x dx y f x x f x f f y f 2.05 f N Figure 6 shows the function in Example 3 and a comparison of dy and y when a 2. The viewing rectangle is 1.8, 2.5 by 6, 18. y= + -2x+1 In general, dy f x dx 3x 2 2x 2 dx When x 2 and dx x 0.05, this becomes dy (2, 9) dy Îy (b) f y f 2.01 f When dx x 0.01, FIGURE 6 dy M

5 SECTION LINEAR APPROXIMATIONS AND DIFFERENTIALS Notice that the approximation y dy becomes better as x becomes smaller in Example 3. Notice also that dy was easier to compute than y. For more complicated functions it may be impossible to compute y exactly. In such cases the approximation by differentials is especially useful. In the notation of differentials, the linear approximation (1) can be written as f a dx f a dy For instance, for the function f x sx 3 in Example 1, we have If a 1 and dx x 0.05, then dy f x dx dx 2sx 3 dy s and s4.05 f 1.05 f 1 dy just as we found in Example 1. Our final example illustrates the use of differentials in estimating the errors that occur because of approximate measurements. V EXAMPLE 4 The radius of a sphere was measured and found to be 21 cm with a possible error in measurement of at most 0.05 cm. What is the maximum error in using this value of the radius to compute the volume of the sphere? SOLUTION If the radius of the sphere is r, then its volume is V 4 3r 3. If the error in the measured value of r is denoted by dr r, then the corresponding error in the calculated value of V is V, which can be approximated by the differential When r 21 and dr 0.05, this becomes dv 4r 2 dr dv The maximum error in the calculated volume is about 277 cm. M NOTE Although the possible error in Example 4 may appear to be rather large, a better picture of the error is given by the relative error, which is computed by dividing the error by the total volume: V V dv V 4r 2 dr 4 3 dr 3r 3 r Thus the relative error in the volume is about three times the relative error in the radius. In Example 4 the relative error in the radius is approximately drr and it produces a relative error of about in the volume. The errors could also be expressed as percentage errors of 0.24% in the radius and 0.7% in the volume.

6 2 5 2 CHAPTER 3 DIFFERENTIATION RULES 3.10 EXERCISES 1 4 Find the linearization Lx of the function at a. 1. f x x 4 3x 2, a 1 2. f x ln x, 3. f x cos x, a 2 4. f x x 34, ; 5. Find the linear approximation of the function f x s1 x at a 0 and use it to approximate the numbers s0.9 and s0.99. Illustrate by graphing f and the tangent line. ; 6. Find the linear approximation of the function tx s 3 1 x at a 0 and use it to approximate the numbers s and s Illustrate by graphing t and the tangent line. ; 7 10 Verify the given linear approximation at a 0. Then determine the values of x for which the linear approximation is accurate to within s 3 1 x x 8. tan x x Find the differential of each function. 11. (a) y x 2 sin 2x (b) y lns1 t (a) 11 2x 4 1 8x 12. (a) y s1 2s (b) y e u cos u y u 1 u 1 (b) y 1 r (a) y e tan t (b) y s1 ln z (a) Find the differential dy and (b) evaluate dy for the given values of x and dx. 15. y e x 10, x 0, 16. y 1x 1, x 1, 17. y tan x, x 4, 18. y cos x, x 3, dx Compute y and dy for the given values of x and dx x. Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and y. 19. y 2x x 2, x 2, x y sx, x 1, x y 2x, x 4, x y e x, x 0, x 0.5 dx 0.01 dx 0.1 dx 0.05 e x 1 x Use a linear approximation (or differentials) to estimate the given number e a 1 a tan s Explain, in terms of linear approximations or differentials, why the approximation is reasonable ln Let and (a) Find the linearizations of f, t, and h at a 0. What do you notice? How do you explain what happened? ; (b) Graph f, t, and h and their linear approximations. For which function is the linear approximation best? For which is it worst? Explain. 33. sec f x x 1 2 The edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum possible error, relative error, and percentage error in computing (a) the volume of the cube and (b) the surface area of the cube. 34. The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm. (a) Use differentials to estimate the maximum error in the calculated area of the disk. (b) What is the relative error? What is the percentage error? 35. The circumference of a sphere was measured to be 84 cm with a possible error of 0.5 cm. (a) Use differentials to estimate the maximum error in the calculated surface area. What is the relative error? (b) Use differentials to estimate the maximum error in the calculated volume. What is the relative error? 36. Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispherical dome with diameter 50 m. 37. (a) Use differentials to find a formula for the approximate volume of a thin cylindrical shell with height h, inner radius r, and thickness r. (b) What is the error involved in using the formula from part (a)? 38. One side of a right triangle is known to be 20 cm long and the opposite angle is measured as 30, with a possible error of 1. (a) Use differentials to estimate the error in computing the length of the hypotenuse. (b) What is the percentage error? 30. hx 1 ln1 2x x tx e

7 LABORATORY PROJECT TAYLOR POLYNOMIALS If a current I passes through a resistor with resistance R, Ohm s Law states that the voltage drop is V RI. If V is constant and R is measured with a certain error, use differentials to show that the relative error in calculating I is approximately the same (in magnitude) as the relative error in R. 40. When blood flows along a blood vessel, the flux F (the volume of blood per unit time that flows past a given point) is proportional to the fourth power of the radius R of the blood vessel: F kr 4 (This is known as Poiseuille s Law; we will show why it is true in Section 8.4.) A partially clogged artery can be expanded by an operation called angioplasty, in which a balloon-tipped catheter is inflated inside the artery in order to widen it and restore the normal blood flow. Show that the relative change in F is about four times the relative change in R. How will a 5% increase in the radius affect the flow of blood? 41. Establish the following rules for working with differentials (where c denotes a constant and u and v are functions of x). (a) dc 0 (b) dcu c du (c) du v du dv (d) duv u dv v du (e) d u v v du u dv v 2 (f) dx n nx n 1 dx 42. On page 431 of Physics: Calculus, 2d ed., by Eugene Hecht (Pacific Grove, CA: Brooks/Cole, 2000), in the course of deriving the formula T 2sLt for the period of a pendulum of length L, the author obtains the equation a T t sin for the tangential acceleration of the bob of the pendulum. He then says, for small angles, the value of in radians is very nearly the value of sin ; they differ by less than 2% out to about 20. (a) Verify the linear approximation at 0 for the sine function: ; (b) Use a graphing device to determine the values of x for which sin x and x differ by less than 2%. Then verify Hecht s statement by converting from radians to degrees. 43. Suppose that the only information we have about a function f is that f 1 5 and the graph of its derivative is as shown. (a) Use a linear approximation to estimate f 0.9 and f 1.1. (b) Are your estimates in part (a) too large or too small? Explain. y 1 sin x x 0 1 y=fª(x) 44. Suppose that we don t have a formula for tx but we know that t2 4 and tx sx 2 5 for all x. (a) Use a linear approximation to estimate t1.95 and t2.05. (b) Are your estimates in part (a) too large or too small? Explain. x L A B O R AT O R Y P R O J E C T ; TAYLOR POLYNOMIALS The tangent line approximation Lx is the best first-degree (linear) approximation to f x near x a because f x and Lx have the same rate of change (derivative) at a. For a better approximation than a linear one, let s try a second-degree (quadratic) approximation Px. In other words, we approximate a curve by a parabola instead of by a straight line. To make sure that the approximation is a good one, we stipulate the following: (i) Pa f a ( P and f should have the same value at a.) (ii) Pa f a ( P and f should have the same rate of change at a.) (iii) Pa f a (The slopes of P and f should change at the same rate at a.) 1. Find the quadratic approximation Px A Bx Cx 2 to the function f x cos x that satisfies conditions (i), (ii), and (iii) with a 0. Graph P, f, and the linear approximation Lx 1 on a common screen. Comment on how well the functions P and L approximate f. 2. Determine the values of x for which the quadratic approximation f x Px in Problem 1 is accurate to within 0.1. [Hint: Graph y Px, y cos x 0.1, and y cos x 0.1 on a common screen.]

8 2 5 4 CHAPTER 3 DIFFERENTIATION RULES 3. To approximate a function f by a quadratic function P near a number a, it is best to write P in the form Px A Bx a Cx a 2 Show that the quadratic function that satisfies conditions (i), (ii), and (iii) is Px f a f ax a 1 2 f ax a 2 4. Find the quadratic approximation to f x sx 3 near a 1. Graph f, the quadratic approximation, and the linear approximation from Example 2 in Section 3.10 on a common screen. What do you conclude? 5. Instead of being satisfied with a linear or quadratic approximation to f x near x a, let s try to find better approximations with higher-degree polynomials. We look for an nth-degree polynomial T nx c 0 c 1x a c 2x a 2 c 3x a 3 c nx a n such that T n and its first n derivatives have the same values at x a as f and its first n derivatives. By differentiating repeatedly and setting x a, show that these conditions are satisfied if c 0 f a, c 1 f a, c f a, and in general where k! k. The resulting polynomial T nx f a f ax a c k f k a k! f a x a 2 2! is called the nth-degree Taylor polynomial of f centered at a. f n a x a n n! 6. Find the 8th-degree Taylor polynomial centered at a 0 for the function f x cos x. Graph f together with the Taylor polynomials T 2, T 4, T 6, T 8 in the viewing rectangle [ 5, 5] by [ 1.4, 1.4] and comment on how well they approximate f HYPERBOLIC FUNCTIONS Certain even and odd combinations of the exponential functions e x and e x arise so frequently in mathematics and its applications that they deserve to be given special names. In many ways they are analogous to the trigonometric functions, and they have the same relationship to the hyperbola that the trigonometric functions have to the circle. For this reason they are collectively called hyperbolic functions and individually called hyperbolic sine, hyperbolic cosine, and so on. DEFINITION OF THE HYPERBOLIC FUNCTIONS sinh x e x e x 2 cosh x e x e x 2 tanh x sinh x cosh x csch x 1 sinh x sech x 1 cosh x coth x cosh x sinh x