text stringlengths 1 7.76k | source stringlengths 17 81 |
|---|---|
256 Chapter 8 • Applications of Integrals §8.1 Example 8.7 y = ex2 y 1 2 3 x 0 1 2 3 x2 + y2 = 9 y = 2cosx2 Use Monte Carlo integration to approximate the area of the region in the first quadrant above the curves y = ex2 and y = 2 cos x2, and inside the circle x2 + y2 = 9. Solution: The region is the shaded area shown i... | ElementaryCalculus_Page_266_Chunk3801 |
Area Between Curves • Section 8.1 257 10. Find the area between r = 1+cos θ and r = 2+2cos θ. 11. Find the area between r = 1+cos θ and r = 2+cos θ. B x2 −y2 = 1 y 2 x 1 2 x2 + y2 = 4 O P 12. Find the area of the region in the first quadrant between the unit hyperbola x2 −y2 = 1 and the circle x2 + y2 = 4 (i.e. the shad... | ElementaryCalculus_Page_267_Chunk3802 |
258 Chapter 8 • Applications of Integrals §8.2 8.2 Average Value of a Function planet d ✸ Sun Figure 8.2.1 According to Kepler’s laws of planetary motion, a planet revolv- ing around the Sun follows an elliptical orbit, with the Sun at one focus of the ellipse, as in Figure 8.2.1. The distance d between the planet and ... | ElementaryCalculus_Page_268_Chunk3803 |
Average Value of a Function • Section 8.2 259 〈f 〉≈ 1 b −a nX i=1 f (xi) · b −a n = 1 b −a n X i=1 f (xi)∆xi Note that the last summation on the right is just a Riemann sum for the definite integral Rb a f (x)dx, with the points x∗ i chosen to be the right endpoints of the intervals [xi−1, xi] for i = 1 to n. Thus, taki... | ElementaryCalculus_Page_269_Chunk3804 |
260 Chapter 8 • Applications of Integrals §8.2 Example 8.11 x y 0 5 −5 3 −3 (x, y) (4,0) d Figure 8.2.3 x2 25 + y2 9 = 1 Find the average distance from the ellipse x2 25+ y2 9 = 1 to the point (4,0). Solution: Let d represent the distance from any point (x, y) on the ellipse to the point (4,0), as in Figure 8.2.3. If (... | ElementaryCalculus_Page_270_Chunk3805 |
Average Value of a Function • Section 8.2 261 Example 8.12 The Monte Carlo method is easy to implement in Octave/MATLAB. Typically only a “one-liner” is needed, owing to Octave’s vectorization—i.e. the ability to perform mathematical operations on entire arrays of objects all at once. For example, recall from Example 8... | ElementaryCalculus_Page_271_Chunk3806 |
262 Chapter 8 • Applications of Integrals §8.2 E R s C 11. An electric circuit with a supplied voltage (electromotive force) E, a capacitor with capacitance C, and a resistor with resistance R, is shown in the picture on the right. When a switch s in the circuit is opened at time t = 0 the current I through the circuit... | ElementaryCalculus_Page_272_Chunk3807 |
Arc Length and Curvature • Section 8.3 263 8.3 Arc Length and Curvature Just like the area of a plane region can be found using calculus, so too can the length of a plane curve. Along the way the mystery mentioned in a footnote in Chapter 1 will finally be solved: what is the length of the hypotenuse of a right triangle... | ElementaryCalculus_Page_273_Chunk3808 |
264 Chapter 8 • Applications of Integrals §8.3 Example 8.13 y x 0 1 y = coshx s 1 Find the arc length of the curve y = cosh x over [0,1]. Solution: Since dy dx = sinh x, then the arc length s is: s = Z1 0 s 1+ µ dy dx ¶2 dx = Z1 0 p 1+sinh2 x dx = Z1 0 cosh x dx = sinh x ¯¯¯¯ 1 0 = sinh 1 −sinh 0 = sinh 1 ≈1.1752 Examp... | ElementaryCalculus_Page_274_Chunk3809 |
Arc Length and Curvature • Section 8.3 265 s = 4 a Zπ/2 0 s a4 −(a2 −b2)a2sin2 θ a2 −a2sin2 θ a cos θ dθ = 4 Zπ/2 0 s a2 −(a2 −b2) sin2 θ ✘✘✘✘✘ 1−sin2 θ ✘✘✘ cos θ dθ = 4a Zπ/2 0 s 1−a2 −b2 a2 sin2 θ dθ s = 4a Zπ/2 0 p 1−e2 sin2 θ dθ (since e2 = c2 a2 = a2 −b2 a2 ) The last integral is a special case of the elliptic int... | ElementaryCalculus_Page_275_Chunk3810 |
266 Chapter 8 • Applications of Integrals §8.3 The parametric formula for arc length can be derived by dividing all sides of the infinitesimal right triangle in Figure 8.3.1(b) by dt, then applying the Pythagorean Theorem to the resulting noninfinitesimal right triangle: ds dt = sµdx dt ¶2 + µ dy dt ¶2 ⇒ ds = sµdx dt ¶2 ... | ElementaryCalculus_Page_276_Chunk3811 |
Arc Length and Curvature • Section 8.3 267 Example 8.16 Prove that the circumference of a circle of radius R is 2πR. Solution: Use the polar curve r = R for 0 ≤θ ≤2π. Then dr dθ = 0, so: s = Z2π 0 s r2 + µ dr dθ ¶2 dθ = Z2π 0 p R2 +02 dθ = Z2π 0 R dθ = R θ ¯¯¯¯ 2π 0 = 2πR ✓ Curvature y x 1 0 1 y = x2 Figure 8.3.2 In Ch... | ElementaryCalculus_Page_277_Chunk3812 |
268 Chapter 8 • Applications of Integrals §8.3 Similar to how the instantaneous rate of change of a function at a point is the average rate of change over an infinitesimal interval, the curvature of a curve at a point can be defined as the average curvature over an infinitesimal length of the curve: The curvature κ of a c... | ElementaryCalculus_Page_278_Chunk3813 |
Arc Length and Curvature • Section 8.3 269 For a parametric curve x = x(t), y = y(t), the curvature κ will be a function of the parameter t. Since dy dx = y′(t) x′(t) by formula (7.14) in Section 7.6, then by formula (7.15): d2y dx2 = d dt µ dy dx ¶ dx dt = d dt µ y′(t) x′(t) ¶ x′(t) = x′(t) y′′(t) −y′(t) x′′(t) (x′(t)... | ElementaryCalculus_Page_279_Chunk3814 |
270 Chapter 8 • Applications of Integrals §8.3 Exercises A For Exercises 1-10, find the arc length of the given curve over the given interval. 1. y = x3/2 ; 1 ≤x ≤4 2. y = x2 ; 0 ≤x ≤1 3. y = x2/3 ; 1 ≤x ≤8 4. y = x2 4 −ln x 2 ; 1 ≤x ≤2 5. y = x4 4 + 1 8x2 ; 1 ≤x ≤2 6. y = ln ex +1 ex −1 ; 1 ≤x ≤2 7. x = et cos t, y = e... | ElementaryCalculus_Page_280_Chunk3815 |
Surfaces and Solids of Revolution • Section 8.4 271 8.4 Surfaces and Solids of Revolution Long before calculus was invented the ancient Greeks (e.g. Archimedes) discovered the formu- las for the volume and surface area of familiar three-dimensional objects such as the sphere.7 Volumes and surface areas of arbitrary sol... | ElementaryCalculus_Page_281_Chunk3816 |
272 Chapter 8 • Applications of Integrals §8.4 The surface area S of the surface of revolution obtained by revolving the curve y = f (x) ≥0 around the x-axis for a ≤x ≤b is: S = Zb a dS = Zb a 2π f (x) q 1+(f ′(x))2 dx (8.12) For a general curve y = f (x), possibly negative in [a,b], the surface area S is: S = Zb a dS ... | ElementaryCalculus_Page_282_Chunk3817 |
Surfaces and Solids of Revolution • Section 8.4 273 Now suppose you revolve the region between a curve y = f (x) ≥0 and the x-axis around the x-axis, for a ≤x ≤b (see Figure 8.4.3(a)). This produces a solid of revolution in three dimensions, as in Figure 8.4.3(b). Notice that this solid consists of the surface of revol... | ElementaryCalculus_Page_283_Chunk3818 |
274 Chapter 8 • Applications of Integrals §8.4 Example 8.20 Show that the volume of a sphere of radius r is 4 3πr3. y x −r r y = p r2 −x2 Solution: Use the circle x2 + y2 = r2. Revolve the region between the upper half of the circle y = f (x) = p r2 −x2 and the x-axis around the x-axis over the interval [−r,r], as in t... | ElementaryCalculus_Page_284_Chunk3819 |
Surfaces and Solids of Revolution • Section 8.4 275 To find the volume V of that solid, at a point x in [a,b) form an infinitesimal strip of width dx from the x-axis up to the curve y = f (x), as in Figure 8.4.5(a). y x 0 a x b y = f (x) (a) Revolve region around y-axis f (x) f (x+ dx) y x ds dx dy (b) Infinitesimal strip... | ElementaryCalculus_Page_285_Chunk3820 |
276 Chapter 8 • Applications of Integrals §8.4 The volume dV in formula (8.16) can be generalized to dV = 2πrhw, where r is the distance from the axis of revolution to a generic vertical strip of infinitesimal width w in the region, and h is the height of the strip. Example 8.23 Suppose the region bounded by the curves ... | ElementaryCalculus_Page_286_Chunk3821 |
Applications in Physics and Statistics • Section 8.5 277 8.5 Applications in Physics and Statistics This chapter concludes with a few applications showing how some familiar discrete sums can be replaced by integrals, which are essentially continuous sums. Center of Gravity Suppose a thin uniform rod has n > 1 masses m1... | ElementaryCalculus_Page_287_Chunk3822 |
278 Chapter 8 • Applications of Integrals §8.5 A region can be thought of as a lamina—a thin plate with uniform density. Take the area of the region as its mass, which makes sense given the uniform density. For the region between two curves y = f1(x) and y = f2(x) over [a,b], with f1(x) ≥f2(x), take a vertical slice of... | ElementaryCalculus_Page_288_Chunk3823 |
Applications in Physics and Statistics • Section 8.5 279 The center of gravity (¯x, ¯y) of the region between the curves y = f1(x) and y = f2(x) over [a,b], with f1(x) ≥f2(x), is given by: ¯x = My M = Zb a x(f1(x)−f2(x)) dx Zb a (f1(x)−f2(x)) dx and ¯y = Mx M = Zb a 1 2 ((f1(x))2 −(f2(x))2) dx Zb a (f1(x)−f2(x)) dx (8.... | ElementaryCalculus_Page_289_Chunk3824 |
280 Chapter 8 • Applications of Integrals §8.5 Work Suppose that a constant force displaces an object along a line in the same direction in which the force is applied. The work done by the force is defined as the force times the displacement. For example, if the constant force F moves an object from position x = a to x ... | ElementaryCalculus_Page_290_Chunk3825 |
Applications in Physics and Statistics • Section 8.5 281 Before continuing, some possible confusion needs to be cleared up. First, force is always a vector—it has both a magnitude and a direction. For the forces considered here, which act in a single dimension (e.g. along the x-axis), by convention the direction of the... | ElementaryCalculus_Page_291_Chunk3826 |
282 Chapter 8 • Applications of Integrals §8.5 Probability Suppose you flip two evenly balanced pennies and let X be the number of heads in the result. Then X is a discrete random variable—discrete because it can take only a discrete set of values (0, 1 and 2); random because its value is left to chance. The probability... | ElementaryCalculus_Page_292_Chunk3827 |
Applications in Physics and Statistics • Section 8.5 283 Example 8.27 Let X be the lifetime—i.e. the time to failure—of an electronic component. If the average lifetime of the component is 700 days, then the probability density function f (x) for the random variable X is f (x) = ( λ e−λx if x ≥0, 0 if x < 0 (8.20) wher... | ElementaryCalculus_Page_293_Chunk3828 |
284 Chapter 8 • Applications of Integrals §8.5 11. Recall that the ideal gas law states that PV = RT, where R is a constant, P is the pressure, V is the volume, and T is the temperature. It can be shown that the work W done by an ideal gas in expanding the volume from Va to Vb is W = ZVb Va P dV . Calculate W. 12. Veri... | ElementaryCalculus_Page_294_Chunk3829 |
Applications in Physics and Statistics • Section 8.5 285 21. This exercise is related to Einstein’s famous law E = mc2. The relativistic momentum p of a particle of mass m moving at a speed v along a straight line (say, the x-axis) is p = mv q 1−v2 c2 , where c is the speed of light. The relativistic force on the parti... | ElementaryCalculus_Page_295_Chunk3830 |
CHAPTER 9 Infinite Sequences and Series 9.1 Sequences and Series In the 5th century B.C. the ancient Greek philosopher Zeno of Elea devised several paradoxes, the most famous of which—The Dichotomy—asserts that if space is infinitely divisible then motion is impossible. The argument goes like this: imagine a line segment... | ElementaryCalculus_Page_296_Chunk3831 |
Sequences and Series • Section 9.1 287 A sequence is an ordered list of objects, which in this book will always be real numbers. Sequences can be finite or infinite: finite if there is a last number in the list, infinite if every number in the list is followed by another number (i.e. a “successor”). Sequences should not be... | ElementaryCalculus_Page_297_Chunk3832 |
288 Chapter 9 • Infinite Sequences and Series §9.1 Example 9.2 For integers n ≥0 define an = 2n+1 3n+2. Is { an } a convergent sequence? If so then find its limit. Solution: By L’Hôpital’s Rule, treating an integer n ≥0 as a real-valued variable x, lim n→∞an = lim n→∞ 2n+1 3n+2 = lim n→∞ 2 3 = 2 3 . Thus the sequence is c... | ElementaryCalculus_Page_298_Chunk3833 |
Sequences and Series • Section 9.1 289 An infinite series is the sum of an infinite sequence. If the infinite sequence is { an }∞ n=0 then the series can be written as ∞ X n=0 an = a0 + a1 + a2 +···+ an +··· or simply as Pan when the initial value of the index n is understood. There is a natural way to define the sum of su... | ElementaryCalculus_Page_299_Chunk3834 |
290 Chapter 9 • Infinite Sequences and Series §9.1 Example 9.5 Show that ∞ X n=1 1 2n = 1. Solution: This is a geometric progression with a = 1 2 and r = 1 2. So by formula (9.6) the sum is: ∞ X n=1 1 2n = ∞ X n=0 µ1 2 ¶ · µ1 2 ¶n = a 1−r = 1 2 1−1 2 = 1 ✓ Example 9.6 Write the repeating decimal 0.17 = 0.17171717... as ... | ElementaryCalculus_Page_300_Chunk3835 |
Sequences and Series • Section 9.1 291 The geometric progression also doesn’t help when considering only the distances and ignor- ing time. Even assuming each of those distances could be traveled, the partial sums approach 1 but never actually reach it. The limit of a sequence is defined in terms of an inequality—you ca... | ElementaryCalculus_Page_301_Chunk3836 |
292 Chapter 9 • Infinite Sequences and Series §9.1 19. In this exercise you will prove a formula for the general number Fn in the Fibonacci sequence. Denote the positive and negative solutions to the equation x2−x−1 = 0 by φ+ = 1+ p 5 2 and φ−= 1− p 5 2 , respectively. Note that φ+ is the golden ratio mentioned in Examp... | ElementaryCalculus_Page_302_Chunk3837 |
Tests for Convergence • Section 9.2 293 9.2 Tests for Convergence There are many ways to determine if a sequence converges—two are listed below. In all cases changing or removing a finite number of terms in a sequence does not affect its convergence or divergence: 1. Monotone Bounded Test: A sequence that is bounded and... | ElementaryCalculus_Page_303_Chunk3838 |
294 Chapter 9 • Infinite Sequences and Series §9.2 Some tests for convergence of a series are listed below: 1. n-th Term Test: If Pan converges then lim n→∞an = 0. 2. Ratio Test: For a series Pan of positive terms let R = lim n→∞ an+1 an . Then (a) if R < 1 then the series converges, (b) if R > 1 (including R = ∞) then ... | ElementaryCalculus_Page_304_Chunk3839 |
Tests for Convergence • Section 9.2 295 Most of the above tests have fairly short proofs or at least intuitive explanations. For exam- ple, the n-th Term Test follows from the definition of convergence of a series: if Pan converges to a number L then since each term an = sn −sn−1 is the difference of successive partial ... | ElementaryCalculus_Page_305_Chunk3840 |
296 Chapter 9 • Infinite Sequences and Series §9.2 Figure 9.2.2 shows why the Integral Test works. a1 a2 a3 a4 a5 y x 0 1 2 3 4 5 y = f (x) (a) Z∞ 1 f (x) dx > ∞ X n=2 an a1 a2 a3 a4 y x 0 1 2 3 4 5 y = f (x) (b) Z∞ 1 f (x) dx < ∞ X n=1 an Figure 9.2.2 Integral Test In Figure 9.2.2(a) the area R∞ 1 f (x)dx is greater th... | ElementaryCalculus_Page_306_Chunk3841 |
Tests for Convergence • Section 9.2 297 The divergence part of the Comparison Test is clear enough to understand, but for the con- vergence part with 0 ≤an ≤bn for all n larger than some N, ignore the (finite) number of terms before aN and bN. Since Pbn converges then its partial sums must be bounded. The partial sums f... | ElementaryCalculus_Page_307_Chunk3842 |
298 Chapter 9 • Infinite Sequences and Series §9.2 A series Pan is telescoping if an = bn −bn+1 for some sequence { bn }. Assume the series Pan and sequence { bn } both start at n = 1. Then the partial sum sn for Pan is sn = a1 + a2 + ··· + an = (b1 −b2) + (b2 −b3) + ··· + (bn −bn+1) = b1 −bn+1 for n ≥1. Thus, since b1 ... | ElementaryCalculus_Page_308_Chunk3843 |
Tests for Convergence • Section 9.2 299 10. ∞ X n=2 1 n p ln n 11. ∞ X n=1 n! (2n)! 12. ∞ X n=1 n en 13. ∞ X n=1 1 cosh2 n 14. ∞ X n=1 n! 2n 15. ∞ X n=1 1 pn 16. ∞ X n=1 1 n(2n−1) 17. ∞ X n=1 ln(n+1) n2 For Exercises 18-21 determine whether the given series is convergent. If convergent then find its sum. 18. ∞ X n=1 1 (... | ElementaryCalculus_Page_309_Chunk3844 |
300 Chapter 9 • Infinite Sequences and Series §9.3 9.3 Alternating Series In the last section the harmonic series ∞ X n=1 1 n = 1 + 1 2 + 1 3 + 1 4 + 1 5 + ··· was shown to diverge. If you were to alternate the signs of successive terms, as in ∞ X n=1 (−1)n−1 n = 1 −1 2 + 1 3 −1 4 + 1 5 −··· (9.7) then it turns out that... | ElementaryCalculus_Page_310_Chunk3845 |
Alternating Series • Section 9.3 301 Example 9.14 Determine if ∞ X n=2 (−1)n ln n is convergent. Solution: For the general term an = (−1)n ln n , since ln(n+1) > ln n for n ≥2 and ln n →∞as n →∞, then |an| decreases to 0 as n →∞. Thus, by the Alternating Series Test the series converges. The series P∞ n=1 (−1)n−1 n con... | ElementaryCalculus_Page_311_Chunk3846 |
302 Chapter 9 • Infinite Sequences and Series §9.3 One unusual feature of a conditionally convergent series is that its terms can be rearranged to converge to any number, a result known as Riemann’s Rearrangement Theorem. For example, the alternating harmonic series 1 −1 2 + 1 3 −1 4 + 1 5 −··· consists of one divergent... | ElementaryCalculus_Page_312_Chunk3847 |
Power Series • Section 9.4 303 9.4 Power Series A power series is an infinite series whose terms involve constants an and powers of x −c, where x is a variable and c is a constant: P an (x−c)n. In many cases c will be 0. For example, the geometric progression ∞ X n=0 rn = 1 + r + r2 + r3 + ··· = 1 1−r converges when |r|... | ElementaryCalculus_Page_313_Chunk3848 |
304 Chapter 9 • Infinite Sequences and Series §9.4 Example 9.16 Find the interval of convergence of the power series ∞ X n=0 xn n! . Solution: For fn(x) = xn n! , r(x) = lim n→∞ ¯¯¯¯ fn+1(x) fn(x) ¯¯¯¯ = lim n→∞ ¯¯¯¯ xn+1/(n+1)! xn/n! ¯¯¯¯ = |x| · lim n→∞ ¯¯¯¯ 1 n+1 ¯¯¯¯ = |x| · 0 = 0 for any fixed x. Thus, r(x) = 0 < 1 ... | ElementaryCalculus_Page_314_Chunk3849 |
Power Series • Section 9.4 305 Notice that the above statement says nothing about the convergence of f ′(x) or R f (x)dx at the endpoints of the interval |x −c| < R. In each case convergence at the endpoints can be checked individually. Example 9.19 Write the power series form of the derivative of f (x) = ∞ X n=0 xn an... | ElementaryCalculus_Page_315_Chunk3850 |
306 Chapter 9 • Infinite Sequences and Series §9.4 The general Bessel equation of order m, d2y dx2 + 1 x dy dx + µ 1−m2 x2 ¶ y = 0 (9.13) for m = 0,1,2,..., has a solution Jm(x), called Bessel’s function of order m: Jm(x) = ∞ X n=0 (−1)n 1 n! · (n+ m)! ³ x 2 ´2n+m (9.14) For example, the Bessel function J1(x) of order 1... | ElementaryCalculus_Page_316_Chunk3851 |
Power Series • Section 9.4 307 Exercises A For Exercises 1-8 find the interval of convergence of the given power series. 1. ∞ X n=1 nxn (n+1)2 2. ∞ X n=1 nxn 2n 3. ∞ X n=1 n2 (x−2)n 4. ∞ X n=0 (x+4)n 2n 5. ∞ X n=1 (x+1)n nn 6. ∞ X n=1 nn xn 7. ∞ X n=0 (−1)n xn 8. ∞ X n=1 nxn n+1 9. Note that power series of the form P∞ ... | ElementaryCalculus_Page_317_Chunk3852 |
308 Chapter 9 • Infinite Sequences and Series §9.5 9.5 Taylor’s Series In the previous section a few functions, e.g. f (x) = 1 1−x, turned out to be the sum of a power series. This section will discuss a general method for representing a function as a power series, called a Taylor’s series.11 Suppose that a function f (... | ElementaryCalculus_Page_318_Chunk3853 |
Taylor’s Series • Section 9.5 309 Example 9.20 Find the Taylor’s series for f (x) = ex about x = 0. Solution: Since d dx (ex) = ex, then for all n ≥0, f (n)(x) = ex ⇒ f (n)(0) = e0 = 1 . Thus, by Taylor’s formula with c = 0:12 ex = ∞ X n=0 f (n)(0) n! xn = ∞ X n=0 xn n! = 1 + x + x2 2! + x3 3! + x4 4! + x5 5! + ··· For... | ElementaryCalculus_Page_319_Chunk3854 |
310 Chapter 9 • Infinite Sequences and Series §9.5 Example 9.22 Find the Taylor’s series for f (x) = sin x about x = 0. Solution: The derivatives of f (x) = sin x repeat every four derivatives: f (x) = sin x , f ′(x) = cos x , f ′′(x) = −sin x , f ′′′(x) = −cos x , f (4)(x) = sin x So at x = 0: f (0) = 0 , f ′(x) = 1 , ... | ElementaryCalculus_Page_320_Chunk3855 |
Taylor’s Series • Section 9.5 311 Example 9.24 The function ln x is not defined at x = 0 and hence has no Taylor’s series about x = 0. Instead, find the Taylor’s series for f (x) = ln(1+ x) about x = 0. Solution: Take successive derivatives: f (x) = ln(1+ x) , f ′(x) = 1 1+ x , f ′′(x) = − 1 (1+ x)2 , f ′′′(x) = 1 · 2 (1... | ElementaryCalculus_Page_321_Chunk3856 |
312 Chapter 9 • Infinite Sequences and Series §9.5 Define the n-th degree Taylor polynomial Pn(x) for a function f (x) about x = c by Pn(x) = nX k=0 f (k)(c) k! (x−c)k = f (c) + f ′(c) 1! (x−c) + f ′′(c) 2! (x−c)2 + ··· + f (n)(c) n! (x−c)n for x in the interval of convergence for the full Taylor’s series. In other words... | ElementaryCalculus_Page_322_Chunk3857 |
Taylor’s Series • Section 9.5 313 Remainder Theorem:13If Pn(x) is the n-th degree Taylor polynomial about x = c for a function f (x) in some interval containing x = c, then for all x in that interval, f (x) = Pn(x) + Rn(x) , (9.17) where Rn(x) = f (n+1)(c +θ(x−c)) (n+1)! (x−c)n+1 (9.18) for some number θ between 0 and ... | ElementaryCalculus_Page_323_Chunk3858 |
314 Chapter 9 • Infinite Sequences and Series §9.5 For Exercises 12-15 replace the function f (x) by its Taylor’s series about x = 0 to evaluate the given indefinite integral R f (x) dx (up to the first three nonzero terms in the series). 12. Z sin x x dx 13. Z cos(x2) dx 14. Z e−x2 dx 15. Zp 1+ x6 dx 16. Use d dx (tan−1 ... | ElementaryCalculus_Page_324_Chunk3859 |
APPENDIX A Answers and Hints to Selected Exercises Chapter 1 Section 1.1 (p. 6) 1. 2t 2. 19.6t 3. −32t+2 4. 3t2 Section 1.2 (p. 14) 1. 0 3. 2x+2 5. − 1 (x+1)2 7. −2 x3 9. 1 2 p x+1 11. 2x+3 2 p x2+3x+4 Section 1.3 (p. 20) 5. Hint: Use the sine double-angle formula. 7. Hint: Use Exercise 5 and the sine addition formula.... | ElementaryCalculus_Page_325_Chunk3860 |
316 Appendix A: Answers and Hints to Selected Exercises 15. xx2(x+2xlnx) 17. xsinx ³ cos x lnx+ sin x x ´ 19. 15.5 hours 21. 12 hours Section 2.4 (p. 55) 1. ln3(3x−3−x) 2 3. (ln2)2 22x 2x 5. 2x (ln2)(x2+1) 7. cos(log2 πx) xln2 9. 3x2 Chapter 3 Section 3.1 (p. 61) 1. y = 4x−3 3. y = −6x+10 5. y = 4x 7. y = x+3 9. y = 24... | ElementaryCalculus_Page_326_Chunk3861 |
Appendix A: Answers and Hints to Selected Exercises 317 7. local maximum at x = ln2, inflection pt at x = ln4, increasing for x < ln2, decreasing for x > ln2, concave up for x > ln4, concave down for x < ln4, horizontal asymptote: y = 0 Section 4.3 (p. 117) 1. x = 0.450184 3. x = 0.567143 5. x = 1.414213 11. global maxi... | ElementaryCalculus_Page_327_Chunk3862 |
318 Appendix A: Answers and Hints to Selected Exercises 13. −9 p 9−x2 + 1 3(9−x2)3/2 +C 15. −1 9 p −9x2 +36x−32−2 3 sin−1¡3x−6 2 ¢ +C Section 6.4 (p. 184) 1. −ln|x|+ln|x−1|+C 3. 1 5 ln|2x−1|−1 5 ln|x+2|+C 5. 1 x + 1 2 ln|x−1|−1 2 ln|x+1|+C 7. 2ln|x|+ 1 x −2ln|x+1|+C 9. −3ln|x|+ 2 x +3ln|x−1|+ 1 x−1 +C 11. 1 3 tan−1 x−1... | ElementaryCalculus_Page_328_Chunk3863 |
Appendix A: Answers and Hints to Selected Exercises 319 5. Focus: (−3,−239 16 ), vertex: (−3,−15), direc- trix: y = −241 16 7. Focus: ¡1 2, 5 4 ¢ , vertex: ¡1 2, 3 2 ¢ , directrix: y = 7 4 9. Foci: (−1± p 13,−3), vertexes: (−4,−3) and (2,−3), directrices: x = −1 ± 9 p 13, asymptotes: y = ± 2 3(x+1)−3 11. Foci: ( p 2, p... | ElementaryCalculus_Page_329_Chunk3864 |
320 Appendix A: Answers and Hints to Selected Exercises Chapter 9 Section 9.1 (p. 291) 1. Converges to 0 2. Converges to 1 3 3. Converges to 0 5. Divergent 7. Divergent 9. 6 11. 32 13. 113 999 14. 1 15. 1 4 20. 132 7 ft 24. No Section 9.2 (p. 298) 6. Divergent 7. Convergent 8. Divergent 9. Convergent 10. Divergent 11. ... | ElementaryCalculus_Page_330_Chunk3865 |
GNU Free Documentation License Version 1.3, 3 November 2008 Copyright © 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc. <http://fsf.org/> Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. Preamble The purpose of this License is to make a m... | ElementaryCalculus_Page_331_Chunk3866 |
322 GNU Free Documentation License A “Modified Version” of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another lan- guage. A “Secondary Section” is a named appendix or a front-matter section of the Document that deals exclusi... | ElementaryCalculus_Page_332_Chunk3867 |
GNU Free Documentation License 323 A section “Entitled XYZ” means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language. (Here XYZ stands for a specific section name mentioned below, such as “Acknowledgments”, “Dedication... | ElementaryCalculus_Page_333_Chunk3868 |
324 GNU Free Documentation License you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated lo- cation until at least one year after the last time you distribute an Opaque copy (directly or through yo... | ElementaryCalculus_Page_334_Chunk3869 |
GNU Free Documentation License 325 J. Preserve the network location, if any, given in the Document for public access to a Trans- parent copy of the Document, and likewise the network locations given in the Document for previous versions it was based on. These may be placed in the “History” section. You may omit a netwo... | ElementaryCalculus_Page_335_Chunk3870 |
326 GNU Free Documentation License list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers. The combined work need only contain one copy of this License, and multiple identical Invari- ant Sections may be replaced with a single copy. If there... | ElementaryCalculus_Page_336_Chunk3871 |
GNU Free Documentation License 327 You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement b... | ElementaryCalculus_Page_337_Chunk3872 |
328 GNU Free Documentation License “Massive Multiauthor Collaboration Site” (or “MMC Site”) means any World Wide Web server that publishes copyrightable works and also provides prominent facilities for anybody to edit those works. A public wiki that anybody can edit is an example of such a server. A “Massive Multiautho... | ElementaryCalculus_Page_338_Chunk3873 |
History This section contains the revision history of the book. For persons making modifications to the book, please record the pertinent information here, following the format in the first item below. 1. VERSION: 0.1 Date: 2016-01-24 Author(s): Michael Corral Title: Elementary Calculus Modification(s): Initial version 2.... | ElementaryCalculus_Page_339_Chunk3874 |
Index Symbols ∆x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 〈f 〉. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .258 ˙f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 dx. . . . . . . . . . . . . . . . . . . . . . . . ... | ElementaryCalculus_Page_340_Chunk3875 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 13 accelerating.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .34 acceleration.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .33 algebraic curve . . . . . . . . . . . . . . . . . . . . . . . . . . 80 alternating series . . . . . . . . . . . . . .... | ElementaryCalculus_Page_340_Chunk3876 |
. . . . . 305 Bessel functions order m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 order zero . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Bessel’s equation .. . . . . . . . . . . . . . . . . . . . . . 305 order m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 Beta function.. . .... | ElementaryCalculus_Page_340_Chunk3877 |
Index 331 circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 definition . . . . . . . . . . . . . . . . . . . . . . . . . . 202 involute . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 circular functions.. . . . . . . . . . . . . . . . . . . . . .230 closed interval . . . . . . ... | ElementaryCalculus_Page_341_Chunk3878 |
. . . . . . . . 267 parametric .. . . . . . . . . . . . . . . . . . . . . . . . 269 polar .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 cycloid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 D Darboux’s Theorem. . . . . . . . . . . . . . . . . . . . .122 decelerating . . . . ... | ElementaryCalculus_Page_341_Chunk3879 |
. . . . . . . . . . . . . . . . 21 differentiable .. . . . . . . . . . . . . . . . . . . . . . . . . . . 14 differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 differential equation . . . . . . . . . . . . . . . . . 36, 47 differentiation .. . . . . . . . . . . . . . . . . . . . . . . . . . 14 i... | ElementaryCalculus_Page_341_Chunk3880 |
. . . . 203 construction.. . . . . . . . . . . . . . . . . . . . . . .203 definition . . . . . . . . . . . . . . . . . . . . . . . . . . 202 diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 directrix.. . . . . . . . . . . . . . . . . . . . . . . . . . .209 eccentric angle . . . . . . . . . . . . . . . ... | ElementaryCalculus_Page_341_Chunk3881 |
332 Index latus rectum .. . . . . . . . . . . . . . . . . . . . . . 209 major auxiliary circle .. . . . . . . . . . . . . 243 major axis. . . . . . . . . . . . . . . . . . . . . . . . . .203 minor auxiliary circle .. . . . . . . . . . . . . 243 minor axis . . . . . . . . . . . . . . . . . . . . . . . . . 203 principal ... | ElementaryCalculus_Page_342_Chunk3882 |
50 exponential distribution.. . . . . . . . . . . . . . .283 exponential functions.. . . . . . . . . . . . . . . . . . .45 exponential growth .. . . . . . . . . . . . . . . . . . . . . 51 Extended Mean Value Theorem . . . . . . . . 121 F factorial.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34 Ferm... | ElementaryCalculus_Page_342_Chunk3883 |
. .45 floor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 hyperbolic.. . . . . . . . . . . . . . . . . . . . . . . . .230 infinitesimal . . . . . . . . . . . . . . . . . . . . . . . . 72 inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 inverse hyperbolic . . . . . . . . . . . ... | ElementaryCalculus_Page_342_Chunk3884 |
. . . . . .281 horizontal asymptote . . . . . . . . . . . . . . . . . . . . 67 hyperbola. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .216 | ElementaryCalculus_Page_342_Chunk3885 |
Index 333 asymptotes. . . . . . . . . . . . . . . . . . . . . . . . .217 center .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 conjugate axis. . . . . . . . . . . . . . . . . . . . . .217 construction.. . . . . . . . . . . . . . . . . . . . . . .219 definition . . . . . . . . . . . . . . . . . . . . . . ... | ElementaryCalculus_Page_343_Chunk3886 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 infinite series . . . . . . . . . . . . . . . . . . . . . . . . 4, 289 convergent . . . . . . . . . . . . . . . . . . . . . . . . . 289 divergent. . . . . . . . . . . . . . . . . . . . . . . . . . .289 partial sum . . . . . . . . . . . . . . . . . . . . . . . . ... | ElementaryCalculus_Page_343_Chunk3887 |
. . . . . . . . . . . . . . . . . . . . . . . . 195 tabular method .. . . . . . . . . . . . . . . . . . . 163 interval of convergence. . . . . . . . . . . . . . . . .303 inverse function . . . . . . . . . . . . . . . . . . . . . . . . . 38 inverse hyperbolic functions . . . . . . . . . . . 234 inverse trigonometric fun... | ElementaryCalculus_Page_343_Chunk3888 |
. . . . . . . 64 right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 | ElementaryCalculus_Page_343_Chunk3889 |
334 Index Limit Comparison Test. . . . . . . . . . . . . . . . .294 limits of integration .. . . . . . . . . . . . . . . . . . . 139 line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 tangent. . . . . . . . . . . . . ... | ElementaryCalculus_Page_344_Chunk3890 |
. . . . . . . 277, 278 Monotone Bounded Test. . . . . . . . . . . . . . . .293 Monte Carlo integration.. . . . . . . . . . . . . . .255 Monte Carlo method . . . . . . . . . . . . . . . . . . . 260 N nappe.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .221 natural logarithm . . . . . . . . . . . . . ... | ElementaryCalculus_Page_344_Chunk3891 |
. . 210 directrix.. . . . . . . . . . . . . . . . . . . . . . . . . . .210 eccentricity . . . . . . . . . . . . . . . . . . . . . . . . 210 focal radius . . . . . . . . . . . . . . . . . . . . . . . . 212 focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 latus rectum .. . . . . . . . . . . . . . .... | ElementaryCalculus_Page_344_Chunk3892 |
. . . . . . . . . . . . . . . . . 4, 303 differentiation . . . . . . . . . . . . . . . . . . . . . 304 | ElementaryCalculus_Page_344_Chunk3893 |
Index 335 integration.. . . . . . . . . . . . . . . . . . . . . . . .304 interval of convergence . . . . . . . . . . . . 303 radius of convergence . . . . . . . . . . . . . . 303 principal axis. . . . . . . . . . . . . . . . . . . . . . . . . . .203 of an ellipse . . . . . . . . . . . . . . . . . . . . . . . . 203 prob... | ElementaryCalculus_Page_345_Chunk3894 |
. .4 recurrence relation . . . . . . . . . . . . . . . . . . . . . 288 reflection property.. . . . . . . . . . . . . . . . . . . . .206 ellipse.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .206 hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . 220 parabola . . . . . . . . . . . . . . . . . . . . .... | ElementaryCalculus_Page_345_Chunk3895 |
. . . .293 convergent . . . . . . . . . . . . . . . . . . . . . . . . . 287 divergent. . . . . . . . . . . . . . . . . . . . . . . . . . .287 Fibonacci.. . . . . . . . . . . . . . . . . . . . . . . . . .288 limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Monotone Bounded Test . . . . . . . . . .... | ElementaryCalculus_Page_345_Chunk3896 |
336 Index snap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Snell’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 solid of revolution .. . . . . . . . . . . . . . . . . . . . . 273 disc method . . . . . . . . . . . . . . . . . . . . . . . . 273 speed . . . . . . . . .... | ElementaryCalculus_Page_346_Chunk3897 |
. . . . . . . . . . . . . . . . . . . .72 torque .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 torus.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .276 trajectory envelope . . . . . . . . . . . . . . . . . . . . . 213 translation . . . . . . . . . . . . . . . . . . . . . .... | ElementaryCalculus_Page_346_Chunk3898 |
Michael Biehl The Shallow and the Deep A biased introduction to neural networks and old school machine learning | The+Shallow+and+the+Deep_Page_1_Chunk3899 |
The Shallow and the Deep | The+Shallow+and+the+Deep_Page_3_Chunk3900 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.