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)-eigenspaces for WN , and the cusp forms decompose similarly. This WN is the Atkin-Lehner involution. This is the “substitute” for the the operator S = 0 −1 in Γ(1). 1 0 83 9 Hecke theory for Γ0(N ) III Modular Forms and L-functions 9 Hecke theory for Γ0(N ) Note that it is possible to do this for other congruence sub... |
he following properties: Proposition. – We have L∗ kf = 0 iff f is holomorphic. 89 10 Modular forms and rep theory III Modular Forms and L-functions – If f ∈ WK(Γ(1)), then g ≡ L∗ kf ∈ Wk−2(Γ(1)). Thus, L∗ k is a “lowering” operator. Proof. The first part is clear. For the second part, note that we have f (γ(z)) = (cz + ... |
e group (e.g. SLn(R), Sp2n(R)), and Γ by some arithmetic subgroup. This leads to the general theory of automorphic forms, and is one half of the Langlands’ program. 95 Index Index Bn, 15 G-module, 48 Gr,k, 75 L(χ, s), 20 L(f, s), 57 L∗ k, 89 M (f, s), 9 M∗, 39 Mk, 39 Mk(Γ(1)), 39 O(f (n)), 57 R(n), 51 R∗ k, 93 S, 28 S∗... |
ssible from a WYSIWYG technical document environment. CengageBrain.com To access additional course materials and companion resources, please visit www.cengagebrain.com. At the CengageBrain.com home page, search for the ISBN of your title (from the back cover of your book) using the search box at the top of the page. Th... |
a. Since 1 x 1 moves back and forth infinitely often x Module 10.1A gives an ani ma tion of the TEC relationship between motion along a parametric x curve x f t , f y tt t graphs of and as functions of . Clicking on t and motion along the TRIG gives you the family of parametric curves x a cos bt y c sin dt and and click... |
4:12 PM Page 666 666 CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES 19–22 Describe the motion of a particle with position t varies in the given interval. x, y as 19. x 3 2 cos t , y 1 2 sin t 2 t 32 , 20. x 2 sin t , y 4 cos t , 0 t 32 21. x 5 sin t , y 2 cos t t 5 , 22. x sin t , y cos2t 2 t 2 , 23. Suppose a c... |
f the graph. Then let n vary while keeping d constant. What happens when n d 1 ? 4. What happens if or . Take larger and larger values for and speculate on what would happen if we were to is irrational? Experiment with an irrational number like and a e 2 graph the hypocycloid for all real values of . b 1 s2 5. If the c... |
, then the length of t to , C is and t , is traversed dx dt 2 dy dt 2 dt L y Notice that the formula in Theorem 5 is consistent with the general formulas ds2 dx2 dy2 of Section 8.1. and L x ds EXAMPLE 4 tion 10.1, If we use the representation of the unit circle given in Example 2 in Sec- x cos t y sin t 0 t 2 then dxd... |
rea correct to four decimal places. x 57. 58. 59. , x t sin t x sin t , x 1 te t y t cos t y sin 2t 1e t , , 0 t 1 60 61–63 Find the exact area of the surface obtained by rotating the given curve about the -axis. x 0 t 1 61. 62. 63. y t 2 , x t 3 x 3t t 3 x a cos3 , , , y 3t 2 y a sin3 , 0 t 1 , 0 2 ; 64. Graph the cur... |
ht to remove additional content at any time if subsequent rights restrictions require it.www.EngineeringEBooksPdf.com 97817_10_ch10_p680-689.qk_97817_10_ch10_p680-689 11/3/10 4:13 PM Page 680 680 CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES 2 r 2 x 2 y 2 tan y x which can be deduced from Equations 1 or simply ... |
horizontal or vertical. r 1 sin of Example 7, find the slope of the tangent line SOLUTION Using Equation 3 with , we have dy dx dr d sin r cos dr cos r sin d cos sin 1 sin cos cos cos 1 sin sin cos 1 2 sin 1 2 sin2 sin cos 1 2 sin 1 sin 1 2 sin (a) The slope of the tangent at the point where 3 is dy dx 3 cos31 2 sin3 1 ... |
R COORDINATES 55–60 Find the slope of the tangent line to the given polar curve at the point specified by the value of . 55. r 2 sin , 6 56. r 2 sin , 3 57. r 1 , 58. r cos3 , 59. r cos 2 4 , 60. r 1 2 cos 3 , 61–64 Find the points on the given curve where the tangent line is horizontal or vertical. 61. 63. r 3 cos r 1 ... |
On one curve the origin is reached at 32 0, or 0, 32 , the origin satisfies r 1 sin r 3 sin r 3 sin , it satisfies 2, 6) ( 3 0 and and Thus, to find all points of intersection of two polar curves, it is recommended that you draw the graphs of both curves. It is especially convenient to use a graphing calculator or comput... |
because by , then the standard equation of a parabola p 0 and downward if y ax 2 . [see Figure 4, parts (a) and (b)]. The is replaced is unchanged when becomes x 1 1 x y . y y=_p 0 (0, p) x y 0 x=_p y ( p, 0) ( p, 0) x 0 x x=_p (a) ≈=4py, p>0 (b) ≈=4py, p<0 (c) ¥=4px, p>0 (d) ¥=4px, p<0 FIGURE 4 y ¥+10x=0 5 ”_ , 0’ 2 ... |
that b a2 1 the equation of the hyperbola is c 2 a 2 b 2 5 and 2 4 a 1 and . The foci are ab 2 (0, s52) . Thus and y 2 4x 2 1 Shifted Conics As discussed in Appendix C, we shift conics by taking the standard equations 5 and replacing and by x h y k , and and , EXAMPLE 6 1, 2 5, 2 , . Find an equation of the ellipse wi... |
a fixed positive number (called the eccentricity). The set of all points in the plane such that P F e l PF Pl e (that is, the ratio of the distance from is a conic section. The conic is F to the distance from l is the constant ) e (a) an ellipse if e 1 (b) a parabola if e 1 (c) a hyperbola if e 1 PROOF Notice that if th... |
helion distance and aphelion distance, respectively. In Figure 1 the sun is at the focus , so at perihelion we have and, from Equation 7, 0 r a1 e2 1 e cos 0 a1 e1 e 1 e a1 e Similarly, at aphelion and r a1 e . 8 The perihelion distance from a planet to the sun is distance is a1 e . a1 e and the aphelion EXAMPLE 5 (a) ... |
rdinates to sketch the cochleoid by hand. Then graph it with a machine to check your sketch. r 24 3 cos Also graph the ellipse obtained by rotation about the origin through an angle and its directrix. 23 . ; 20. Graph the ellipse 21–24 Find the slope of the tangent line to the given curve at the point corresponding to ... |
ly affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.www.EngineeringEBooksPdf.com 97817_11_ch11_p713-721.qk_97817_11_ch11_p713-721 11/3/10 5:28 PM Page 714 714 CHAPTER 11 INFINITE SEQUENCES AND SERIES 11.1 Se... |
s EXAMPLE 6 Calculate lim n l ln n n . SOLUTION Notice that both numerator and denominator approach infinity as can’t apply l’Hospital’s Rule directly because it applies not to sequences but to functions of a real variable. However, we can apply l’Hospital’s Rule to the related function f x ln xx and obtain . We n l lim... |
ms suggest that the sequence is increasing and the terms are approaching 6. To confirm that the sequence is increasing, we use mathematical induction to show an1 an that that it is true for n 1 , then we have for all n k . This is true for a2 4 a1 . If we assume because n 1 so and Thus ak1 ak ak1 6 ak 6 1 2 ak1 6 1 2 ak... |
population after years and and are positive constants that depend on the species and its environment. Suppose that the population in year 0 is (a) Show that if pn values for its limit are 0 and is convergent, then the only possible . p0 0 . b a a b pn1 ba pn (b) Show that . (c) Use part (b) to show that if , then in o... |
sum is the limit of the sequence of partial sums. So, by taking the sum of sufficiently many terms, we can get as . The table close as we like to the number 3 shows the first ten partial sums and the sn graph in Figure 2 shows how the sequence of partial sums approaches .3 5 10 3 20 9 40 27 5 3 10 sn 5.000000 1.666667 3... |
s convergent, find its sum. 17. 19. 20. 3 64 9 3 4 16 10 2 0.4 0.08 2 0.5 0.125 0.03125 18. 4 3 9 4 27 16 SECTION 11.2 SERIES 735 21. 23. 25. n1 n1 n0 60.9n1 3n1 4 n n 3 n1 22. 24. 26. n1 n0 n1 10 n 9n1 1 (s2 )n e n 3n1 27– 42 Determine whether the series is convergent or divergent. If it is convergent, find its sum. 27.... |
to prove your guess. (c) Show that the given infinite series is convergent, and find its sum. 90. In the figure there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length 1, find the total area occup... |
uire it.www.EngineeringEBooksPdf.com 97817_11_ch11_p742-751.qk_97817_11_ch11_p742-751 11/3/10 5:29 PM Page 742 742 CHAPTER 11 INFINITE SEQUENCES AND SERIES Estimating the Sum of a Series Suppose we have been able to use the Integral Test to show that a series is convergent s and we now want to find an approximation to t... |
monic series, then sn 1 ln n (b) The harmonic series diverges, but very slowly. Use part (a) to show that the sum of the first million terms is less than 15 and the sum of the first billion terms is less than 22. tn tn1 lnn 1 ln n 1 n 1 as a difference of areas to show that fore is a decreasing sequence. tn tn tn1 0 . Th... |
bsequent rights restrictions require it.www.EngineeringEBooksPdf.com 97817_11_ch11_p742-751.qk_97817_11_ch11_p742-751 11/3/10 5:30 PM Page 750 750 CHAPTER 11 INFINITE SEQUENCES AND SERIES Therefore the remainder Rn for the given series satisfies With n 100 we have Rn Tn 1 2n2 R100 1 21002 0.00005 Using a programmable ca... |
is , which is the absolute value of the first neglected term. smaller than is the remainder Rn s sn s sn bn1 You can see geometrically why the Alternating Series Estimation Theorem is true by looking at Figure 1 (on page 752). Notice that s s5 b6, lies between any two consecutive partial sums. and so on. Notice also tha... |
n N Putting successively equal to n , , 4 , . . . in N 2 N 1 N aN1 aN r aN2 aN1 r aN r 2 aN3 aN2 r aN r 3 , we obtain and, in general, 5 Now the series aNk aN r k for all k 1 k1 aN r k aN r aN r 2 aN r 3 is convergent because it is a geometric series with together with the Comparison Test, shows that the series 0 r 1 ... |
and use the fact that there is an integer .] whenever n N L r 1 sn an r that that N 42. Around 1910, the Indian mathematician Srinivasa Ramanujan discovered the formula 1 2 s2 9801 n0 4n!1103 26390n n!4 3964n . William Gosper used this series in 1985 to compute the first 17 million digits of (a) Verify that the series ... |
4 and divergent when . Now so the series converges when 2 x 4 The Ratio Test gives no information when x 4 separately. If we put x 4 and x 2 series, which is divergent. If Alternating Series Test. Thus the given power series converges for , the harmonic , which converges by the 2 x 4 . , the series is and diverges whe... |
1ncn 9n 31. If k is a positive integer, find the radius of convergence of the series n0 n!k kn! x n q p 32. Let and be real numbers with whose interval of convergence is (a) (c) p, q p, q (b) (d) p, q p, q p q . Find a power series 33. Is it possible to find a power series whose interval of conver- gence is 0, ? Explain.... |
quation and obtain ln1 0 C . ln1 n1 1n1 x n n x 1 The radius of convergence is the same as for the original series: R 1 . v EXAMPLE 7 Find a power series representation for f x tan1x . SOLUTION We observe that the power series for 11 x 2 found in Example 1. f x 11 x 2 and find the required series by integrating tan1x ob... |
fficients are given by the formula cn f na n! Substituting this formula for cn a , then it must be of the following form. expansion at back into the series, we see that if f has a power series f x 6 x an n0 f na n! f a f a 1! x a f a 2! x a2 f a 3! x a3 Taylor and Maclaurin The Taylor series is named after the English m... |
a Notice that, as better approximation to n y 1 T¡ y=sin x 0 1 T∞ x T£ FIGURE 2 by the Squeeze Theorem. It follows that of its Maclaurin series by Theorem 8. We have two power series expansions for e x , the Maclaurin series in Equation 11 and the Taylor series in Equation 13. The first is better if we are interested i... |
en in in the series for e x ex 2 n0 x 2n n! n0 1n x 2n n! 1 x 2 1! x 4 2! x 6 3! Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review ... |
rially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.www.EngineeringEBooksPdf.com 97817_11_ch11_p782-791.qk_97817_11_ch11_p782-791 11/3/10 5:32 PM Page 791 WRITING PROJECT HOW NEWTON DISCOVERED THE BINOMI... |
oximation to find sin 12 correct to six decimal x is this approximation accurate to within 0.00005 ? SOLUTION (a) Notice that the Maclaurin series sin x x x 3 3! x 5 5! x 7 7! is alternating for all nonzero values of , and the successive terms decrease in size x 1 because in approximating , so we can use the Alternating... |
hese x 0.9 a 1 f f and 1.3. to f x . T3x ; 3–10 Find the Taylor polynomial f . Graph and a centered at the number a 2 f x 1x , 3. for the function T3 on the same screen. f 11–12 Use a computer algebra system to find the Taylor polynomials polynomials and on the same screen. centered at f . Then graph these n 2, 3, 4, 5 ... |
Rule to show that l 0 f 0 lim and lim l f 0 for Planck’s Law. So this law models blackbody radiation better than the Rayleigh-Jeans Law for short wavelengths. 2. Use a Taylor polynomial to show that, for large wavelengths, Planck’s Law gives approxi- mately the same values as the Rayleigh-Jeans Law. ; 3. Graph as give... |
f f is an even function, show that c1 c3 c5 0 (c) Use Taylor’s Inequality to estimate the accuracy of the approximation f x Tnx when x lies in the given interval. 62. If f x ex 2 , show that f 2n0 2n! n! . Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in pa... |
l sum of the harmonic series is not an integer. n 1 be the largest power of 2 that is less than or equal to and let 26 Prove that if 2k Hint: Let of all odd integers that are less than or equal to M2ksn M2km showing that each of its terms is an even integer, except for the last one. , an integer. Then . The right side ... |
la for the distance between two points in a plane is easily extended to the following three-dimensional formula. Distance Formula in Three Dimensions The distance P1x1, y1, z1 and is P2x2, y2, z2 P1P2 sx2 x12 y2 y12 z2 z12 P1P2 between the points z 0 FIGURE 10 The plane y=x y x Copyright 2010 Cengage Learning. All Righ... |
e third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.www.Eng... |
dimensional vector p p1, p2, p3, p4, p5, p6 might represent the prices of six dif ferent ingredients required to make a particular product. Four-dimensional vectors in relativity theory, where the first three compo nents specify a position in space and the fourth represents time. x, y, z, t are used a a1, a2, . . . , an... |
e direction as the given vector. 23. 25 24. 4, 2, 4 26. Find a vector that has the same direction as 2, 4, 2 but has length 6. 37. A clothesline is tied between two poles, 8 m apart. The line is quite taut and has negligible sag. When a wet shirt with a mass of 0.8 kg is hung at the middle of the line, the mid point is... |
2j k 5i 4j 2 k 25 24 12 0 these vectors are perpendicular by 7 . cos 0 2 0 2 if and negative for is Because as measuring the positive for extent to which a and b point in the same direction. The dot product is positive if a and b point in the same general direction, 0 if they are perpendicular, and negative if they po... |
diagonal of one of its faces. 57. A molecule of methane, CH4 , is structured with the four hydro- gen atoms at the vertices of a regular tetrahedron and the carbon atom at the centroid. The bond angle is the angle formed by the H— C—H combination; it is the angle between the lines that join the carbon atom to two of t... |
d. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the rig... |
1 of Theorem 11. 24. Prove Property 2 of Theorem 11. 25. Prove Property 3 of Theorem 11. 26. Prove Property 4 of Theorem 11. 27. Find the area of the parallelogram with vertices A2, 1 , B0, 4 , C4, 2 , and D2, 1 . 4 ft 30° 36 lb 28. Find the area of the parallelogram with vertices , and L1, 3, 6 , N3, 7, 3 . M3, 8, 6 ... |
c , we could write the equations of as that appear in the denominators of Equations 3 are direction numbers of . If one of a b , , and L , that is, com, we can still eliminate . For . Notice that the numbers a b , is L L 0 t , or L z z0 c y y0 b x x0 Figure 4 shows the line point where it intersects the in Example 2 an... |
in both planes, it is perpendicular to both of the is given by the cross product L v n1 n2 i 1 1 j 1 2 3 5i 2 j 3 k k 1 and so the symmetric equations of can be written as , y, z that satisfy both NOTE Since a linear equation in represents a plane and two nonparallel , and planes intersect in a line, it follows that tw... |
. The plane that passes through the point 2x y 2z 2 dicular to the planes 1, 5, 1 and x 3z 4 and is perpen- 40. The plane that passes through the line of intersection of the y 2z 3 and is perpendicular to the planes plane x z 1 and x y 2z 1 41– 44 Use intercepts to help sketch the plane. 41. 2x 5y z 10 43. 6x 3y 4z 6 4... |
nsions of the conic sections in the plane. (See Section 10.5 for a review of conic sections.) EXAMPLE 3 Use traces to sketch the quadric surface with equation x 2 y 2 9 z2 4 1 SOLUTION By substituting which we recognize as an equation of an ellipse. In general, the horizontal trace in the plane , we find that the trace ... |
escribe and sketch the surface. 3. x 2 z 2 1 4. 4x 2 y 2 4 . 9. (a) Find and identify the traces of the quadric surface x 2 y2 z2 1 the graph of the hyperboloid of one sheet in Table 1. and explain why the graph looks like (b) If we change the equation in part (a) to x 2 y2 z2 1 , how is the graph affected? (c) What if... |
a) Find a vector perpendicular to the plane through the points , and C1, 4, 3 . A1, 0, 0 B2, 0, 1 , (b) Find the area of triangle ABC . 3. If u and v are the vectors shown in the figure, find u v and u v . Is u v directed into the page or out of it? | v |=3 45° | u |=2 4. Calculate the given quantity if comp a b (a) (c) ... |
describe the motion of objects through space. In particular, we will use them to derive Kepler’s laws of planetary motion. © Christos Georghiou / Shutterstock 863 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third pa... |
e. (This is especially true in Figure 8. See Exercise 50.) The next example shows how to cope with this problem. EXAMPLE 7 curve is called a twisted cubic. Use a computer to draw the curve with vector equation rt t, t 2, t 3. This SOLUTION We start by using the computer to plot the curve with parametric equations x t y... |
r this curve and use these equations and a computer to graph the curve. 47. If two objects travel through space along two different curves, it’s often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) The curves might intersect, but we need to know whether t... |
erivative of , that is, r where nite integrals (antiderivatives). Rt rt . We use the notation x rt dt for indefi- EXAMPLE 5 If rt 2 cos t i sin t j 2t k , then y rt dt y 2 cos t dt i y sin t dt j y 2t dt k 2 sin t i cos t j t 2 k C where C is a vector constant of integration, and y2 0 rt dt [2 sin t i cos t j t 2 k]0 2 ... |
y Equation 6, then we may be able to solve for as a function of : . Thus, Then the curve can be reparametrized in terms of by substituting for : rt3 if is the position vector of the point 3 units of length along the for instance, curve from its starting point. is already given in terms of a parameter and t r rts s 3 st... |
N , so its normal vector is B 1 2 s2 , 0, 1 s2 A simpler normal vector is 1, 0, 1 , so an equation of the osculating plane is 1x 0 0y 1 1z 0 2 or z x 2 EXAMPLE 8 Find and graph the osculating circle of the parabola y x 2 SOLUTION From Example 5, the curvature of the parabola at the origin is the radius of the osculatin... |
he negative -axis is to be joined x 1 y 1 . smoothly to a track along the line of degree 5 such that the func(a) Find a polynomial tion defined by P Px for F Fx 0 if x 0 if 0 x 1 if x 1 Px 1 is continuous and has continuous slope and continuous curvature. ; (b) Use a graphing calculator or computer to draw the graph of ... |
E 7 This resolution is illustrated in Figure 7. Let’s look at what Formula 7 says. The first thing to notice is that the binormal vector B is absent. No matter how an object moves through space, its acceleration always lies in the plane of T and N (the osculating plane). (Recall that T gives the direction of motion and ... |
. at t i e t j et k , v0 k , r0 j k 19. The position function of a particle is given by rt t 2, 5t, t 2 16t . When is the speed a minimum? 20. What force is required so that a particle of mass m has the posi- tion function rt t 3 i t 2 j t 3 k ? 21. A force with magnitude 20 N acts directly upward from the -plane on an... |
s orbit is called the Clarke Geosynchronous Orbit after Arthur C. Clarke, who first proposed the idea in 1945. The first such satellite, Syncom II, was launched in July 1963.) 5.98 1024 kg ; its radius is 13 Review Concept Check 1. What is a vector function? How do you find its derivative and 6. (a) What is the definition ... |
there is no friction between the road and the tires. The loss of friction could occur, for example, if the road is covered with a film of water or ice. The rated speed the maximum speed that a car can attain without skidding. Suppose a car of mass vR. to the weight of the car, and a force to, the road (see the figure). i... |
k and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.www.EngineeringEBooksPdf.com 97817_14_ch14_p901-909.qk_97817... |
and vertical traces are parabolas (see Figure 9). z FIGURE 9 Graph of h(x, y)=4≈+¥ x y Computer programs are readily available for graphing functions of two variables. In most x k are drawn for equally spaced such programs, traces in the vertical planes values of and parts of the graph are eliminated using hidden line ... |
remains fixed. f x, y, z x, y, z f z ≈+¥+z@=9 EXAMPLE 15 Find the level surfaces of the function ≈+¥+z@=4 f x, y, z x 2 y 2 z2 SOLUTION The level surfaces are of concentric spheres with radius , the value of sphere with center O x 2 y 2 z2 k sk f x, y, z k 0 . (See Figure 20.) Thus, as remains fixed. , where . These for... |
46. f x, y lnx 2 4y 2 47. f x, y ye x 48. f x, y y sec x 49. f x, y sy 2 x 2 50. f x, y yx 2 y2 51–52 Sketch both a contour map and a graph of the function and compare them. 51. f x, y x 2 9y 2 52. f x, y s36 9x 2 4y 2 at the point 53. A thin metal plate, located in the xy . The level curves of -plane, has temperature ... |
oes not refer to the direction of must approach the same limit no mat. Thus, if we can find two different paths of approach along does a, b has different limits, then it follows that approaches f x, y limx, y l a, b f x, y a, b f x, y to x, y a, b and f stays within the domain of . f x, y f x, y l L1 If x, y l a, b not ... |
x=0. is continuous except where h y above the -axis. x 0 . The graph in Figure 9 shows the break in the graph of Functions of Three or More Variables Everything that we have done in this section can be extended to functions of three or more variables. The notation lim x, y, z l a, b, c f x, y, z L f x, y, z means that ... |
PM Page 926 926 CHAPTER 14 PARTIAL DERIVATIVES By averaging these values we get the estimate perature is percent that the relative humidity rises. 96F and the relative humidity is 70%, the heat index rises about G70 0.9 . This says that, when the temfor every 0.9F f In general, if y b b y , where keeping fixed, say tx f... |
y 2 4y Therefore fxx fyx x x 3x 2 2xy 3 6x 2y 3 fxy 3x 2 y 2 4y 6xy 2 fyy y y 3x 2 2xy 3 6xy 2 3x 2 y 2 4y 6x 2 y 4 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook a... |
hen PL, K bLK 1 This is the Cobb-Douglas production function that we discussed in Section 14.1. 14.3 Exercises C T 1. The temperature (in at a location in the Northern Hemiy sphere depends on the longitude , latitude , and time , so we can write beginning of January. (a) What are the meanings of the partial derivatives... |
as. Calculate TP PV and a b . 88. The gas law for a fixed mass m T perature the gas constant. Show that , pressure P , and volume of an ideal gas at absolute temis PV mRT , where is R V P V V T T P 1 89. For the ideal gas of Exercise 88, show that T P T V T mR 90. The wind-chill index is modeled by the function W 13.12 ... |
x, y)= xy ≈+¥ f(0, 0)=0 if (x, y)≠(0, 0), is called the linear approximation or the tangent plane approximation of a, b. f , where has continuous first are not continuous? Figure 4 pictures such a z f x, y at f We have defined tangent planes for surfaces partial derivatives. What happens if function; its equation is fx a... |
be 75 cm, 60 cm, EXAMPLE 6 and 40 cm, and each measurement is correct to within mate the largest possible error when the volume of the box is calculated from these measurements. cm. Use differentials to esti- 0.2 SOLUTION If the dimensions of the box are , , and , its volume is x y z V xyz and so dV V x dx V y dy V z ... |
n turn, a funcdeals with the case where and each of the variables and t z f tt, ht t z tion of a variable . This means that is indirectly a function of , , and the f t Chain Rule gives a formula for differentiating as a function of . We assume that is difare continuous ferentiable (Definition 14.4.7). Recall that this i... |
f subsequent rights restrictions require it.www.EngineeringEBooksPdf.com 97817_14_ch14_p950-959.qk_97817_14_ch14_p950-959 11/8/10 1:30 PM Page 953 SECTION 14.5 THE CHAIN RULE 953 , where Fx, f x 0 y f x for all apply Case 1 of the Chain Rule to differentiate both sides of the equation y respect to . Since both and are ... |
332 ms c . (This is the is the speed of sound, about where Doppler effect.) Suppose that, at a particular moment, you 34 ms 1.2 ms2 are in a train traveling at . A train is approaching you from the opposite direction on the other track at , and sounds its whistle, which has a frequency of 460 Hz. At that instant, what ... |
ts the rate of change of the direction of gent line to the curve of intersection of the surface plane through shown in Figure 5. z x 3 3xy 4y 2 1, 2, 0 in the direction of and the vertical u SOLUTION Formula 6 gives Du f x, y fxx, y cos 6 fyx, y sin 6 3x 2 3y s3 2 3x 8y 1 2 1 2[3 s3 x 2 3x (8 3s3 )y] Therefore Du f 1, ... |
18 Fx0, y0, z0 rt0 0 P r ª(t¸ ) 0 S x FIGURE 9 C y rt0 to any curve Equation 18 says that the gradient vector at , is perpendicular to the tangent vector . (See Figure 9.) If Fx0, y0, z0 0 , it is therefore natural to define the tangent plane to the level surface Fx, y, z k Px0, y0, z0 at and has normal vector Fx0, y0, ... |
l, which we take to be the origin. The temperature at the point at (a) Find the rate of change of 2, 1, 3 . 120 . is in the direction 1, 2, 2 1, 2, 2 toward the point T (b) Show that at any point in the ball the direction of greatest increase in temperature is given by a vector that points toward the origin. 32. The te... |
mum or minimum at fxa, b 0 partial derivatives of exist there, then f f a, b and and the first-order . fya, b 0 tx f x, b PROOF Let local maximum (or minimum) at , so But mat’s Theorem to the function ta fxa, b a f . If has a local maximum (or minimum) at ta 0 t , then has a by Fermat’s Theorem (see Theorem 3.1.4). . Si... |
om 97817_14_ch14_p970-979.qk_97817_14_ch14_p970-979 11/8/10 1:32 PM Page 974 974 CHAPTER 14 PARTIAL DERIVATIVES in The five critical points of the function Example 4 are shown in red in the contour map of in Figure 9. f f _3 FIGURE 9 y 2 1 _1.48 _1 7 3 _0.8 _ 3 _10 _ 20 _ 30 3 x v EXAMPLE 5 Find the shortest distance fr... |
r a local maximum or minimum at each critical point. Explain your f f reasoning. Then use the Second Derivatives Test to confirm your predictions. 3. f x, y 4 x 3 y 3 3xy y 1 _1 2 1 0 3.7 3.2 _1 3.7 4 4.2 3.2 1 6 5 x ; Graphing calculator or computer required 1. Homework Hints available at stewartcalculus.com Copyright ... |
ot. ■ Lids cost approximately $50.00 each, regardless of dimensions. ■ Welding costs approximately $0.18 per foot for material and labor combined. Give justification of any further assumptions or simplifications made of the details of construction. 3. Describe how any of your assumptions or simplifications may affect the ... |
ngenuity is required. In the present example you might notice that if we multiply and y by , by , then the left sides of these equations will be identical. Doing this, we have by x, 2 3 4 z Another method for solving the system of equations (2 –5) is to solve each of Equations 2, 3, and 4 for and then to equate the res... |
(b) Does give a larger value than the one in part (a)? subject to the constraint sx sy 5 . f 25, 0 ; (c) Solve the problem by graphing the constraint equation f and several level curves of . (d) Explain why the method of Lagrange multipliers fails to solve the problem. (e) What is the significance of f 9, 4 ? . 23. Cons... |
ncoming water can be apportioned in different volumes to each turbine, so the goal is to determine how to distribute water among the turbines to give the maximum total energy production for any rate of flow. Using experimental evidence and Bernoulli’s equation, the following quadratic models were determined for the powe... |
_ch14_p990-996 11/8/10 1:33 PM Page 993 CHAPTER 14 REVIEW 993 f 19–22 Find all second partial derivatives of . z xe2y v r coss 2t f x, y 4x 3 xy 2 f x, y, z x k y lz m 22. 20. 21. 19. 23. If z xy xe yx , show that x z x y z y xy z . 24. If z sinx sin t , show that z x 2z x t z t 2z x 2 25–29 Find equations of (a) the t... |
_ch14_p990-996 11/8/10 1:33 PM Page 996 (b) It was Thomas Simpson (1710–1761) who formulated Newton’s method as we know it today and who extended it to functions of two variables as in part (a). (See the biography of Simpson on page 537.) The example that he gave to illustrate the method was to solve the system of equa... |
97-1005 11/8/10 3:33 PM Page 1001 SECTION 15.1 DOUBLE INTEGRALS OVER RECTANGLES 1001 The sum in Definition 5, m n i1 j1 f xij*, yij* A is called a double Riemann sum and is used as an approximation to the value of the double integral. [Notice how similar it is to the Riemann sum in for a function of a happens to be a po... |
of the solid , and take the m 3 n 2 , in part (a). 2. If to estimate the value of R 0, 4 1, 2 , use a Riemann sum with m 2 , n 3 . Take the sample points to be (a) the lower right corners and (b) the upper left corners of the rectangles. 1 x y 2 dA xx R 3. (a) Use a Riemann sum with xx R xexy dA of points to be upper r... |
ion 15.1.) x 3y 2 dA y2 0 y2 1 yy R x 3y 2 dy dx y2 [xy y 3] y1 y2 dx y2 0 x 7 dx x 2 2 0 7x 2 0 12 Notice the negative answer in Example 2; is nothing is wrong with that. The function f not a positive function, so its integral doesn’t represent a volume. From Figure 3 we see that , so the value of the f R integral is ... |
viously defined in Section 15.1. The procedure that we have used is reasonable because the and so they contribute nothing to the intevalues of D R gral. This means that it doesn’t matter what rectangle we use as long as it contains . xx D f x, y dA as the volume of the , we can still interpret z f x, y f (the graph of )... |
0 y1 x siny 2 dy dx yy siny 2 dA D D x, y 0 x 1, x y 1 D in Figure 15. Then from Figure 16 we see that an alternative D x, y 0 y 1, 0 x y This enables us to use reverse order: 5 to express the double integral as an iterated integral in the y1 0 y1 x siny 2 dy dx yy siny 2 dA D y1 0 yy 0 siny 2 dx dy y1 0 [x siny 2 ]x0... |
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