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5.24Table 5.4Interval]0, t1 []t1 , 1[]1, t2 [Sign of ẏ(t)+−−]t2 , +∞[+Behavior of y(t)0 ) y(t1 )y(t1 ) ( −∞+∞ ( y(t2 )y(t2 ) ) +∞we find its zeros: t1 ≈ 0.5 and t2 ≈ 1.5 in the region t ≥ 0.Then, by compiling Table 5.4 we find the regions of monotonicity and the localextreme values y(t1 ) ≈ 13 (a maximum) and y(t2 ) ≈ 4 (a minimum).Now, by studying both graphs x = x(t) and y = y(t) simultaneously, we make asketch of the trajectory of the point in the plane (Fig. 5.24c).This sketch can be made more precise. For example, one can determine theasymptotics of the trajectory.y(t)Since limt→1 x(t)= −1 and limt→1 (y(t) + x(t)) = 2, the line y = −x + 2 is anasymptote for both ends of the trajectory, corresponding to t approaching 1.
It is5.4 Differential Calculus Used to Study Functions259also clear that the line x = 0 is a vertical asymptote for the portion of the trajectorycorresponding to t → +∞.We find nextyx =ẏt1 − 5t 2 + 2t 4=.ẋt1 + t22As one can easily see, the function 1−5u+2udecreases monotonically from 1 to −11+uas u increases from 0 to 1 and increases from −1 to +∞ as u increases from 1 to+∞.From the monotonic nature of yx , one can draw conclusions about the convexityof the trajectory on the corresponding regions. Taking account of what has just beensaid, one can construct the following, more precise sketch of the trajectory of thepoint (Fig. 5.24d).If we had considered the trajectory for t < 0 as well, the fact that x(t) and y(t)are odd functions would have added to the curves already drawn in the xy-plane thecurves obtained from them by reflection in the origin.We now summarize some of these results as very general recommendations forthe order in which to proceed when constructing the graph of a function given analytically.
Here they are:10 Give the domain of definition of the function.20 Note the specific properties of the function if they are obvious (for example,evenness or oddness, periodicity, identity to some well-known functions up to simple coordinate changes).30 Determine the asymptotic behavior of the function under approach to boundarypoints of the domain of definition and, in particular, find asymptotes if they exist.40 Find the intervals of monotonicity of the function and exhibit its local extremevalues.50 Determine the convexity properties of the graph and indicate the points of inflection.60 Note any characteristic points of the graph, in particular points of intersectionwith the coordinate axes, provided there are such and they are amenable to computation.5.4.6 Problems and Exercises1. Letx = (x1 , .
. . , xn ) and α = (α1 , . . . , αn ), where xi ≥ 0, αi > 0 for i = 1, . . . , nand ni=1 αi = 1. For any number t = 0 we consider the mean of order t of thenumbers x1 , . . . , xn with weights αi :Mt (x, α) = ni=1 1/tαx xit.2605Differential CalculusIn particular, when α1 = · · · = αn = n1 , we obtain the harmonic, arithmetic, andquadratic means for t = −1, 1, 2 respectively.Show thata) limt→0 Mt (x, α) = x1α1 · · · xnαn , that is, in the limit one can obtain the geometric mean;b) limt→+∞ Mt (x, α) = max1≤i≤n xi ;c) limt→−∞ Mt (x, α) = min1≤i≤n xi ;d) Mt (x, α) is a nondecreasing function of t on R and is strictly increasing ifn > 1 and the numbers xi are all nonzero.2.
Show that |1 + x|p ≥ 1 + px + cp ϕp (x), where cp is a constant depending onlyon p,!|x|2 for |x| ≤ 1,ϕp (x) =if 1 < p ≤ 2,|x|p for |x| > 1,and ϕp (x) = |x|ρ on R if 2 < p.3. Verify that cos x < ( sinx x )3 for 0 < |x| < π2 .4. Study the function f (x) and construct its graph ifa) f (x) = arctan log2 cos(πx + π4 );b) f (x) = arccos( 32 − sin x);#c) f (x) = 3 x(x + 3)2 ;d) Construct the curve defined in polar coordinates by the equation ϕ = ρ 2ρ+1 ,ρ ≥ 0, and exhibit its asymptotics;e) Show how, knowing the graph of the function y = f (x), one can obtain thegraph of the following functions f (x) + B, Af (x), f (x + b), f (ax), and, in particular −f (x) and f (−x).5.
Show that if f ∈ C(]a, b[) and the inequalityf (x1 ) + f (x2 )x 1 + x2≤f22holds for any points x1 , x2 ∈ ]a, b[, then the function f is convex on ]a, b[.6. Show thata) if a convex function f : R → R is bounded, it is constant;b) iff (x)f (x)= lim= 0,x→+∞ xxfor a convex function f : R → R, then f is constant.c) for any convex function f defined on an open interval a < x < +∞ (or−∞ < x < a), the ratio f (x)x tends to a finite limit or to infinity as x tends to infinityin the domain of definition of the function.limx→−∞7.
Show that if f : ]a, b[ → R is a convex function, then5.4 Differential Calculus Used to Study Functions261a) at any point x ∈ ]a, b[ it has a left-hand derivative f− and a right-hand derivative f+ , defined asf (x + h) − f (x),h→−0hf− (x) = limf+ (x) = limh→+0f (x + h) − f (x),hand f− (x) ≤ f+ (x);b) the inequality f+ (x1 ) ≤ f− (x2 ) holds for x1 , x2 ∈ ]a, b[ and x1 < x2 ;c) the set of cusps of the graph of f (x) (for which f− (x) = f+ (x)) is at mostcountable.8. The Legendre transform21 of a function f : I → R defined on an interval I ⊂ Ris the functionf ∗ (t) = sup tx − f (x) .x∈IShow thata) The set I ∗ of values of t ∈ R for which f ∗ (t) ∈ R (that is, f ∗ (t) = ∞) iseither empty or consists of a single point, or is an interval of the line, and in this lastcase the function f ∗ (t) is convex on I ∗ .b) If f is a convex function, then I ∗ = ∅, and for f ∗ ∈ C(I ∗ ) ∗ ∗= sup xt − f ∗ (t) = f (x)ft∈I ∗for any x ∈ I .
Thus the Legendre transform of a convex function is involutive, (itssquare is the identity transform).c) The following inequality holds:xt ≤ f (x) + f ∗ (t)for x ∈ I and t ∈ I ∗ .d) When f is a convex differentiable function, f ∗ (t) = txt − f (xt ), where xtis determined from the equation t = f (x). Use this relation to obtain a geometricinterpretation of the Legendre transform f ∗ and its argument t, showing that theLegendre transform is a function defined on the set of tangents to the graph of f .e) The Legendre transform of the function f (x) = α1 x α for α > 1 and x ≥ 0 isthe function f ∗ (t) = β1 t β , where t ≥ 0 and α1 + β1 = 1.
Taking account of c), usethis fact to obtain Young’s inequality, which we already know:xt ≤21 A.M.1 α 1 βx + t .αβLegendre (1752–1833) – famous French mathematician.2625Differential Calculusf) The Legendre transform of the function f (x) = ex is the function f ∗ (t) =t ln et , t > 0, and the inequalityxt ≤ ex + t lnteholds for x ∈ R and t > 0.9. Curvature and the radius and center of curvature of a curve at a point.
Suppose apoint is moving in the plane according to a law given by a pair of twice-differentiablecoordinate functions of time: x = x(t), y = y(t). In doing so, it describes a certaincurve, which is said to be given in the parametric form x = x(t), y = y(t). A specialcase of such a definition is that of the graph of a function y = f (x), where one maytake x = t, y = f (t). We wish to find a number that characterizes the curvature ofthe curve at a point, as the reciprocal of the radius of a circle serves as an indicationof the amount of bending of the circle. We shall make use of this comparison.a) Find the tangential and normal components at and an respectively of the acceleration a = (ẍ(t), ÿ(t)) of the point, that is, write a as the sum at + an , where atis collinear with the velocity vector v(t) = (ẋ(t), ẏ(t)), so that at points along thetangent to the trajectory and an is directed along the normal to the trajectory.b) Show that the relationr=|v(t)||an (t)|holds for motion along a circle of radius r.c) For motion along any curve, taking account of b), it is natural to call thequantityr(t) =|v(t)||an (t)|the radius of curvature of the curve at the point (x(t), y(t)).Show that the radius of curvature can be computed from the formular(t) =(ẋ 2 + ẏ 2 )3/2.|ẋ ÿ − ẍ ẏ|d) The reciprocal of the radius of curvature is called the absolute curvature of aplane curve at the point (x(t), y(t)).
Along with the absolute curvature we considerthe quantityk(t) =ẋ ÿ − ẍ ẏ,(ẋ 2 + ẏ 2 )3/2called the curvature.Show that the sign of the curvature characterizes the direction of turning of thecurve relative to its tangent. Determine the physical dimension of the curvature.5.5 Complex Numbers and Elementary Functions263e) Show that the curvature of the graph of a function y = f (x) at a point(x, f (x)) can be computed from the formulak(x) =y (x).[1 + (y )2 ]3/2Compare the signs of k(x) and y (x) with the direction of convexity of the graph.f) Choose the constants a, b, and R so that the circle (x − a)2 + (y − b)2 = R 2has the highest possible order of contact with the given parametrically defined curvex = x(t), y = y(t). It is assumed that x(t) and y(t) are twice differentiable and that(ẋ(t0 ), ẏ(t0 )) = (0, 0).This circle is called the osculating circle of the curve at the point (x0 , y0 ).
Itscenter is called the center of curvature of the curve at the point (x0 , y0 ). Verify thatits radius equals the radius of curvature of the curve at that point, as defined in b).g) Under the influence of gravity a particle begins to slide without any preliminary impetus from the tip of an iceberg of parabolic cross-section. The equationof the cross-section is x + y 2 = 1, where x ≥ 0, y ≥ 0. Compute the trajectory ofmotion of the particle until it reaches the ground.5.5 Complex Numbers and the Connections Amongthe Elementary Functions5.5.1 Complex NumbersJust as the equation x 2 = 2 has no solutions in the domain Q of rational numbers,no solutions in the domain R of real numbers.
And, justthe equation x 2 = −1 has√as we adjoin the symbol 2 as a solution of√x 2 = 2 and connect it with rationalnumbers to get new numbers of the form r1 + 2r2 , where r1 , r2 ∈ Q, we introducethe symbol i as a solution of x 2 = −1 and attach this number, which lies outside thereal numbers, to real numbers and arithmetic operations in R.One remarkable feature of this enlargement of the field R of real numbers, amongmany others, is that in the resulting field C of complex numbers, every algebraicequation with real or complex coefficients now has a solution.Let us now carry out this program.a.
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