CHAPTER 11 Vector-Valued Functions
|
|
- Francis Moody
- 6 years ago
- Views:
Transcription
1 CHAPTER Vector-Valued Functions Section. Vector-Valued Functions Section. Differentiation and Integration of Vector-Valued Functions.... Section. Velocit and Acceleration Section. Tangent Vectors and Normal Vectors Section.5 Arc Length and Curvature Review Eercises Problem Solving
2 CHAPTER Vector-Valued Functions Section. Solutions to Odd-Numbered Eercises Vector-Valued Functions. rt 5ti tj t k Component functions: ft 5t gt t ht t Domain:,,. rt ln ti e t j tk Component functions: ft ln t gt e t ht t Domain:, rt Ft Gt cos ti sin tj tk cos ti sin tj cos ti tk Domain:, i rt Ft Gt sin t Domain:, j cos t sin t k cos ti sin t cos tj sin tk cos t 9. rt t i t j (a) r i (b) r j (c) rs s i s j s i sj (d) r t r t i tj i j t t i tj i j t t i tj. rt ln ti j tk t (a) r ln i j k (b) r is not defined. ln does not eist. (c) (d) rt lnt i j t k t r t r ln ti j tk i j k t ln ti t j tk 9
3 Chapter Vector-Valued Functions. rt sin ti cos tj tk 5. rt sin t cos t t t rt ut t t t 8 t t t t t 5t t, a scalar. The dot product is a scalar-valued function. 7. rt ti tj t k, t 9. t, t, t Thus,. Matches (b) rt ti t j e.75t k, t t, t, e.75t Thus,. Matches (d). (a) View from the negative -ais:,, (c) View from the -ais:,, (b) View from above the first octant:,, (d) View from the positive -ais:,,. t 5. t, t 7. cos, sin t Ellipse 9. sec, tan. t. cos t, sin t, t Hperbola 9 t t Line passing through the points: (,, ),, 5, 5,, (,, ) (,, 5) 5 t Circular heli 7 5. sin t, cos t, e t 7. t, t, t e t, t (,, ) 5 (,, )
4 Section. Vector-Valued Functions 9. rt t i tj t k. Parabola Heli 5 rt sin ti cos t t j cos t k. (a) (b) (c) π π 8π π π π π π The heli is translated units back on the -ais. The height of the heli increases at a faster rate. The orientation of the heli is reversed. (d) (e) π π π The ais of the heli is the -ais. The radius of the heli is increased from to Let t, then t. rt ti tj Let t, then t. rt ti t j 9. 5 Let 5 cos t, then 5 sin t. rt 5 cos ti 5 sin tj 5. Let sec t, tan t. rt sec ti tan tj 5. The parametric equations for the line are t, 5t, 8t. One possible answer is rt ti 5tj 8tk (, 8, 8) 5 (,, ) r t ti, t r, r i r t ti tj, t r i, r j r t tj, t r j, r (Other answers possible)
5 Chapter Vector-Valued Functions 57. r t ti t j, t 59. r t ti, t r t tj, t (Other answers possible), Let t, then tand t. Therefore, t, t, t. rt ti tj t k ( ) ( ),, 5,,., sin t, cos t sin t t rt sin ti cos tj sin tk., Let sin t, then sin t and. sin t sin t sin t cos t, sin t, sin t ± cos t ± cos t t ± ± ± rt sin ti cos tj sin tk and rt sin ti cos tj sin tk 5., Subtracting, we have or ±. (,, ) Therefore, in the first octant, if we let t, then t, t, t. rt ti tj t k (,, )
6 Section. Vector-Valued Functions 7. t cos t t sin t t 9. 8 lim t ti t t t j t k i j k since t lim t t t lim t t. t (L Hôpital s Rule) lim t t i tj since cos t k t 7. cos t sin t lim lim. t t t (L Hôpital s Rule) lim t t i cos tj sin tk does not eist since lim t t does not eist. 75. rt ti t j 77. rt ti arcsin tj t k Continuous on, Continuous on,,, rt e t, t, tan t 8. See the definition on page 78. Discontinuous at t n Continuous on n, n rt t i t j tk (a) st rt k t i t j t k (b) st rt i t i t j tk (c) st rt 5j t i t j tk 85. Let rt t tj tk and ut ti tj tk. Then: lim t c rt ut lim t c t t t ti t t t tj t t t tk lim t c t lim t c t lim t c t lim t c t i lim t c t lim t c t lim t c t lim t c t j lim t c lim ti lim tj lim t c t c lim rt lim ut t c t c t lim t c t lim t c t lim t c t k t c tk lim ti lim tj lim t c t c t c tk 87. Let rt ti tj tk. Since r is continuous at t c, then lim rt rc. t c rc ci cj ck c, c, c are defined at c. r t t t lim r c c c rc t c Therefore, r is continuous at c. 89. True
7 Chapter Vector-Valued Functions Section. Differentiation and Integration of Vector-Valued Functions. rt t i tj, t. t t, t t r i j rt ti j r i j rt is tangent to the curve. r (, ) r 8 rt cos ti sin tj, t t cos t, t sin t r j rt sin ti cos tj r i r (, ) r rt is tangent to the curve. 5. rt ti t j (a) (b) (c) 8 r r r r i j r i j r r i j r r r This vector approimates r. 8 rt i tj r i j i j i j 7. rt cos ti sin tj tk,, t π (,, ) r r j k r i k rt sin ti cos tj k π r π t 9. rt ti 7t j t k. rt i tj t k rt a cos ti a sin tj k rt a cos t sin ti a sin t cos tj. rt e t i j 5. rt e t i rt t sin t, t cos t, t rt sin t t cos t, cos t t sin t, 7. rt t i t j (a) rt t i tj rt t i j (b) rt rt t t t 8t t
8 Section. Differentiation and Integration of Vector-Valued Functions 5 9. rt cos ti sin tj. (a) rt sin ti cos tj rt cos t i sin tj (b) rt rt sin t cos t cos t sin t rt t i tj t k (a) rt ti j t k rt i tk (b) rt rt t t t t t. rt cos t t sin t, sin t t cos t, t (a) rt sin t sin t t cos t, cos t cos t t sin t, t cos t, t sin t, rt cos t t sin t, sin t t cos t, (b) rt rt t cos tcos t t sin t t sin tsin t t cos t t 5. rt costi sintj t k, t rt sinti costj tk r r r i j k r r r r r i j k rt costi sintj k r i j k r r r i j k 7. rt t i t j 9. rt ti t j r Smooth on,,, r cos i sin j r cos sin i 9 sin cos j r n Smooth on n n, n an integer.,. r sin i cos j. r cos i sin j r for an value of Smooth on, rt t i t j t k rt i t j tk r is smooth for all t :,,,
9 Chapter Vector-Valued Functions 5. rt ti tj tan tk rt i j sec tk r is smooth for all t n Smooth on intervals of form n. n, n 7. rt ti tj t k, ut ti t j t k (a) rt i j tk (c) rt ut t t t 5 D t rt ut 8t 9t 5t (e) rt ut t i t t j t t k D t rt ut 8t i t t j t tk (b) (d) rt k rt ut ti 9t t j t t k D t rt ut i 9 tj t t k (f) rt t t t t D t rt t t rt sin ti cos tj rt cos ti sin tj rt rt 9 sin t cos t cos t sin t 7 sin t cos t cos maimum at minimum at rt rt rt rt 7 sin t cos t 9 sin t cos t9 cos t sin t arccos 7 sin t cos t t.97 5 t.5 9 sin t cos t9 cos t sin t and and for t n n,,,,....57, t.785. t π 7. rt lim t rt t rt t lim t t t i t t j t i t j t lim t ti tt t j t lim t i t tj i tj. ti j k dt t i tj tk C 5. t i j t k dt ln ti tj 5 t5 k C 7. t i t j tk dt t ti t j t k C 9. sec ti t j dt tan ti arctan tj C
10 t 5. 8ti tj k dt t i j tk i j k 5. a cos ti a sin tj k dt a sin ti Section. Differentiation and Integration of Vector-Valued Functions 7 a cos tj tk ai aj k 55. rt e t i e t j dt e t i e t j C 57. r i j C i C j rt e t i e t j rt j dt tj C r C i j rt i tj rt i tj dt ti t t j C r C rt ti t t j 59. rt te t i e t j k dt et i e t j tk C r i j C i j k C i j k rt i et et j t k e t i et j t k. See Definition of the Derivative of a Vector-Valued Function and Figure.8 on page 79.. At t t, the graph of ut is increasing in the,, and directions simultaneousl. 5. Let rt ti tj tk. Then crt cti ctj ctk and D t crt cti ctj ctk cti tj tk crt. 7. Let rt ti tj tk, then f trt f tti f ttj fttk. D t f trt f tt ftti f tt fttj ftt fttk f tti tj tk ftti tj tk f trt ftrt 9. Let rt ti tj tk. Then r f t f ti f tj f tk and D t r f t f t fti f t ftj f t ftk (Chain Rule) ft f ti f tj f tk ftr ft.
11 8 Chapter Vector-Valued Functions 7. Let rt ti tj tk, ut ti tj tk, and vt ti tj tk. Then: rt ut vt t t t t t t t t t t t t t t t D t rt ut vt t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t rt ut vt rt ut vt rt ut vt 7. False. Let rt cos ti sin tj k. rt d rt dt rt sin ti cos tj rt Section. Velocit and Acceleration. rt ti t j. rt t i tj vt rt i j at rt t, t, At,, t. v i j, a (, ) v vt rt ti j at rt i t, t, At,, t. v i j a i (, ) a 8 v 5. rt cos ti sin tj 7. vt rt sin ti cos tj at rt cos ti sin tj cos t, sin t, At,, t. v i j a i j a v (, ) rt t sin t, cos t vt rt cos t, sin t at rt sin t, cos t t sin t, cos t (ccloid) At,, t. v, i a, j ( π, ) v a π π
12 Section. Velocit and Acceleration 9 9. rt ti t 5j tk. vt i j k st vt 9 at rt ti t j t k vt i tj tk st t t 5t at j k. rt ti tj 9 t k 5. vt i j t 9 t k st t 9 t 8 t 9 t 9 at 9 t k rt t, cos t, sin t vt, sin t, cos t i sin tj cos tk st 9 sin t 9 cos t 5 at, cos t, sin t cos tj sin tk 7. (a) rt t, t, t, t t rt, t, r,, (b) r..,.,..,.,.5 t, t, t 9. at i j k, v, r. vt i j k dt ti tj tk C v C, vt ti tj tk, vt ti j k rt ti tj tk dt t i j k C r C, rt t i j k, r i j k i j k at tj tk, v 5j, r vt tj tk dt t j t k C v j k C 5j C 9 j k vt t 9 j t k rt t 9 j t k dt t 9 t j t t k C r j k C C j k rt t 9 t j t t k r 7 j k. The velocit of an object involves both magnitude and direction of motion, whereas speed involves onl magnitude. 5. rt 88 cos ti 88 sin t t j ti t t j 5
13 5 Chapter Vector-Valued Functions 7. rt v cos ti h v sin t gt j v ti v t t j v t when t v, v, v 9, v ftsec The maimum height is reached when the derivative of the vertical component is ero. t tv t t t t t t t t v t t. v v 5 v, v Maimum height: feet 9. t tv cos or t v cos t tv sin t h v v cos sin v cos h tan v sec h. rt ti.t.7t j, or (a)..7 (b) (c) feet. and (d) From Eercise 9, tan.7 sec v. v sec. cos v 7. ftsec... mph miles feet hr 58 sechour mile ftsec (a) rt cos ti sin t t j (b) Graphing these curves together with shows that. 5 CONTINUED
14 Section. Velocit and Acceleration 5. CONTINUED (c) We want t cos t and From t, the minimum angle occurs when t cos. Substituting this for t in t ields: sin cos cos, tan sec 7, tan tan 7, tan 8, tan 5,7 t sin t t. tan 8, ± 8,,5,7, tan 8,,,,8 8, rt v cos ti v sin t t j (a) We want to find the minimum initial speed v as a function of the angle. Since the bale must be thrown to the position, 8, we have v cos t 8 v sin t t. t v cos from the first equation. Substituting into the second equation and solving for v, we obtain: 5 sin v cos cos 5 v sin cos cos 5 We minimie f. sin cos cos f 5 cos sin sin cos sin cos cos f cos sin Substituting into the equation for v, v 8.78 feet per second. (b) If 5, v cos t v t 8 v sin t t v t t From part (a), v 8 v sin sin 5 cos v cos v sin cos v cos cos 5 tan v cos sin cos cos v ftsec.
15 5 Chapter Vector-Valued Functions 7. rt v cos ti v sin t t j v sin t t when The range is v cos t v cos v sin Hence, sin t and v t v sin. sin. sin (a),(b) rt cos ti sin t t j rt 5ti.t t j Maimum height:.5 feet Range:.557 feet 5 v ftsec, rt cos ti sin t t j rt.78ti 5.5t t j Maimum height:. feet Range: 7.88 feet 5 v ftsec 5 (c) 5, rt cos 5ti sin 5t t j rt.7ti.7t t j Maimum height:. feet Range:.5 feet v ftsec (d) 5, rt cos 5ti sin 5t t j rt.ti.t t j Maimum height:.5 feet Range:.5 feet v ftsec 8 (e), rt cos ti sin t t j rt ti 57.t t j Maimum height: 5.7 feet Range: feet v ftsec (f), rt cos ti sin t t j rt 7ti.t t j Maimum height: feet Range: feet v ftsec
16 Section. Velocit and Acceleration 5. rt v cos ti h v sin t.9t j cos ti.5 sin t.9t j The projectile hits the ground when.9t t.5 t. seconds. The range is therefore cos. 88. meters. The maimum height occurs when ddt. sin 9.8t t 5. sec The maimum height is.5 sin meters.. rt bt sinti b cos tj vt b cos ti b sin t j b cos ti b sin tj at b sin ti b cos tj b sinti costj vt b cost at b (a) vt when t,,,.... (b) vt is maimum when then vt b. t,,..., 5. vt b sinti b costj rt vt b sint cost b sint cost Therefore, rt and vt are orthogonal. 7. at b costi b sintj b costi sintj rt at is a negative multiple of a unit vector from, to cos t, sin t and thus at is directed toward the origin. 9. at b m F m b radsec vt b 8 ftsec 5. To find the range, set t h v then gt sin t gt v sin t h. B the Quadratic Formula, (discount the negative value) t v sin v sin gh g At this time, v sin v sin gh. g t v cos v sin v sin gh g v cos g v sin v sin gh v v cos g sin sin gh v.
17 5 Chapter Vector-Valued Functions 5. rt ti tj tk Position vector vt ti tj tk Velocit vector at ti tj tk Acceleration vector Speed vt t t t Orthogonal C, C is a constant. d dt t t t tt tt tt tt tt tt vt at 55. rt cos ti sin tj (a) vt rt sin ti cos tj vt sin t 9 cos t sin t cos t sin t at vt cos ti sin tj (c) 9 9 (b) t Speed (d) The speed is increasing when the angle between v and a is in the interval, The speed is decreasing when the angle is in the interval.,. Section. Tangent Vectors and Normal Vectors. rt t i tj. rt ti j, rt t t Tt rt ti j rt t ti j t T i j i j rt cos ti sin tj rt sin ti cos tj rt sin t cos t Tt rt sin ti cos tj rt T i j 5. rt ti t j tk 7. rt i tj k When t, r i k, t at,,. T r r i k Direction numbers: a, b, c Parametric equations: t,, t rt cos ti sin tj tk rt sin ti cos tj k When t, r j k, t at,,. T r r 5 j k 5 Direction numbers: a, b, c Parametric equations:, t, t
18 Section. Tangent Vectors and Normal Vectors rt cos t, sin t, rt sin t, cos t, When t, r,,, t r T r,, Direction numbers: a, b, c Parametric equations: t, t, at,,.. rt t, t, t rt, t, t When t, r,, 8, t at, 9, 8. T r r,, Direction numbers: a, b, c 8 Parametric equations: t, t 9, 8t 8. rt ti ln tj tk, t 5. rt i t j t k r i j k T rt i j k rt i j k r,, u8,, Hence the curves intersect. rt, t, r,, 8, Tangent line: t, t, t rt. r..i.j.5k.,.,.5 us,, s, u8,, cos r u8.97 r u rt ti t j, t 9. rt i tj Tt rt i tj rt t Tt t t i t j T i 5 5 j N T T 5 i j i j rt sin ti cos tj Tt rt sin ti cos tj rt Tt cos ti sin tj, Tt N rt cos ti sin tj k, t i j
19 5 Chapter Vector-Valued Functions. rt ti vt i at O Tt vt i vt i Tt O Nt Tt is undefined. Tt The path is a line and the speed is constant.. rt t i vt 8ti at 8i Tt vt 8ti vt 8t i Tt O Nt Tt is undefined. Tt The path is a line and the speed is variable. 5. rt ti j, vt i v i j, t t j, 7. at a j t j, Tt vt vt t t i t j t t i j T Nt Tt Tt N i j i j i j i j a T a T a N a N t t i t t j t t t i t j rt e t cos ti e t sin tj vt e t cos t sin ti e t cos t sin tj at e t sin ti e t cos tj At t T v v i j i j., Motion along r is counterclockwise. Therefore, N i j i j. a T a T e a N a N e 9. rt cos t t sin t i sin t t cos t j vt t cos t i t sin t j at cos t t sin t i t cos t sin t j Tt v v cos t i sin t j Motion along r is counterclockwise. Therefore Nt sin t i cos t j. a T a T a N a N t t
20 Section. Tangent Vectors and Normal Vectors 57. rt a costi a sintj. Speed: vt a vt a sinti a costj The speed is constant since a T. at a costi a sintj Tt vt vt sinti costj Nt Tt Tt costi sintj a T a T a N a N a 5. rt ti t j, t 7. rt ti tj tk t, t rt i t j Tt t i j t Nt i t j t r i j T 7 i j 7 N 7 i j 7, N T vt i j k at Tt v v i j k i j k Nt T is undefined. T are not defined. a T, a N 9. rt ti t j t k. rt ti cos tj sin tk vt i tj tk v i j k at j k Tt v v 5ti tj tk T i j k 5ti j k Nt T T 5t 5ti j k 5 5 5t 5t N 5i j k a T a T 5 a N a N vt i sin tj cos tk v i j at cos tj sin tk a k Tt v v i sin tj cos tk 5 T i j 5 Nt T cos tj sin tk T N k a T a T a N a N T N π π
21 58 Chapter Vector-Valued Functions. Tt rt rt Nt Tt Tt If at a T Tt a N Nt, then a T is the tangential component of acceleration and is the normal component of acceleration. a N 5. If a N, then the motion is in a straight line. 7. rt t sin t, cos t The graph is a ccloid. (a) rt t sin t, cos t vt cos t, sin t at sin t, cos t t = t = t = Tt vt vt cos t cos t, sin t Nt Tt Tt cos t sin t, cos t a T a T a N a N cos t sin t cos t cos t sin t sin t cos t sin cos t t cos t cos t cos t cos t cos t When t : a T, a N When t : a T, a N When t : a T, a N (b) Speed: s vt cos t ds dt sin t cos t a T When t : a T > the speed in increasing. When t : a T the height is maimum. When t : a T < the speed is decreasing.
22 Section. Tangent Vectors and Normal Vectors rt cos ti sin tj t k, t rt sin ti cos tj k Tt 7 7 sin ti cos tj k Nt cos ti sin tj r j 7 T 7 i k N j B k T N 7 i k 7 i 7 7 j k i k i k 7 N B T (,, π ) 5. From Theorem. we have: rt v t cos i h v t sin t j vt v cos i v sin tj at j Tt v cos i v sin tj v cos v sin t Nt v sin ti v cos j v cos v sin t a T a T v sin t v cos v sin t v a N a N cos v cos v sin t (Motion is clockwise.) Maimum height when v sin t ; (vertical component of velocit) At maimum height, a T and a N. 5. rt cos t, sin t, t, t (a) rt sint, cost, rt sin t cos t 5 mihr (b) a T and a N a T because the speed is constant. 55. rt a cos ti a sin tj From Eercise, we know a T and a N a. (a) Let. Then a N a a a or the centripetal acceleration is increased b a factor of when the velocit is doubled. (b) Let a a. Then a N a a a or the centripetal acceleration is halved when the radius is halved.
23 Chapter Vector-Valued Functions 57. v misec 59. v misec. Let Tt cos i sin j be the unit tangent vector. Then M sin i cos j cos i sin j and is rotated counterclockwise through an angle of from T. If ddt >, then the curve bends to the left and M has the same direction as. Thus, M has the same direction as which is toward the concave side of the curve. Tt dt dt N T T, dt dt sin i cos j d dt T d d M d dt. If ddt <, then the curve bends to the right and M has the opposite direction as T. Thus, again points to the concave side of the curve. N T T M T φ M T φ N. Using a a T T a N N, T T O, and T N, we have: v a va N T N Thus, a N v a vt a T T a N N va T T T va N T N va N T N va N v a. v Section.5 Arc Length and Curvature. rt ti tj. d, dt d, dt s 9 dt dt t d dt rt a cos ti a sin tj d dt a cos t sin t, a sin t cos t dt a d dt a sin t cos t s a cos tsin t a sin t cos t dt sin t dt a cos t a (, ) a 8 a a (, ) 8 a
24 Section.5 Arc Length and Curvature 5. (a) rt v cos ti h v sin t gt j cos 5ti sin 5t t j 5ti 5t t j (b) vt 5i 5 tj 5 t t 5 Maimum height: ft (c) 5t t t. Range: feet. (d) s 5 5 t dt.9 feet 7. rt ti tj tk d dt s dt d dt, d dt dt t (,, ) (,, ) 9. rt a cos ti a sin tj btk. d a sin t, dt s ( a,, πb) πb a sin t a cos t b dt a b dt a b t d a cos t, dt d dt b a b rt t i tj ln tk d t, dt s t t dt t t dt t d, dt d dt t t t t dt 8.7 πb ( a,, ). rt ti t j t k, t (a) r,,, r,, 8 distance CONTINUED
25 Chapter Vector-Valued Functions. CONTINUED (b) r,, r.5.5,.75,.5 r,, r.5.5,.75,.75 r,, 8 distance (c) Increase the number of line segments. (d) Using a graphing utilit, ou obtain rt cos t, sin t, t t (a) s u u u du t sin u cos u du t t 5 du 5u 5t (c) When s 5: When s : cos.8 sin.8.8,.8,. cos. 5 sin.95 5 (b) s 5 t cos s 5, sin s 5, rs cos s 5 i sin s j s 5 s 5 5 k ,.95,.789 (d) rs 5 5 sin s 5 5 cos s rs s i s j 9. rs i j Ts rs rs rs and Ts K Ts rs (The curve is a line.) rs cos s 5 i sin s j s 5 Ts rs 5 sin s i 5 Ts 5 cos s 5 i 5 sin s 5 j K Ts 5 5 k 5 cos s j 5 5 k
26 Section.5 Arc Length and Curvature. rt ti tj. rt ti t j vt i j Tt Tt i j 5 K Tt rt (The curve is a line.) vt i t j v i j at t j a j Tt t i j t Nt t i t j N i j K a N v 5. rt costi sintj7. rt 8 sinti 8 costj Tt sinti costj Tt costi sintj K Tt rt 8 rt a costi a sintj rt a sinti a costj Tt sinti costj Tt costi sintj K Tt rt a a 9. rt e t cos ti e t sin tj. rt e t sin t e t cos ti e t cos t e t sin tj Tt sin t cos ti cos t sin tj Tt cos t sin ti sin t cos tj rt cos t t sin t, sin t t cos t From Eercise, Section., we have: a N t K at Nt v t t t K Tt rt et et. rt ti t j t rt i tj tk Tt Tt K Tt rt i tj tk 5t 5ti j k 5t k t 5t 5 5t rt ti cos tj sin tk rt i sin tj cos tk Tt i sin tj cos tk 5 Tt cos tj sin tk 5 K Tt 5 rt 5 5
27 Chapter Vector-Valued Functions 7., Since is undefined. K, and the radius of curvature 9. K K (radius of curvature). a a a At : a a K K a a a (radius of curvature). (a) Point on circle: Center:, Equation:, (b) The circles have different radii since the curvature is different and r K. 5.,, 7. K Radius of curvature. Since the tangent line is horiontal at,, the normal line is vertical. The center of the circle is unit above the point, at, 5. Circle: 5 (, ) e, e, e, K r, K The slope of the tangent line at, is. The slope of the normal line is. Equation of normal line: or The center of the circle is on the normal line units awa from the point,. Since the circle is above the curve, and. Center of circle:, Equation of circle: 8 8 ± (, )
28 Section.5 Arc Length and Curvature 5 9. π π B A 5., K, (a) K is maimum when or at the verte,. π (b) lim K , 9, 55. K (a) K as. No maimum (b) lim K Curvature is at,. b 59. s K The curvature is ero when. a K 9 rt dt. The curve is a line. at.. Endpoints of the major ais: Endpoints of the minor ais: 8 ±,, ± K Therefore, since, K is largest when ± and smallest when. 5. f (a) K (b) For, K. f. At,, the circle of curvature has radius. Using the smmetr of the graph of f, ou obtain. For, K 5 5. f. At,, the circle of curvature has radius 5 K. Using the graph of f, ou see that the center of curvature is,. Thus, f 5. To graph these circles, use ± and ± 5. CONTINUED
29 Chapter Vector-Valued Functions 5. CONTINUED (c) The curvature tends to be greatest near the etrema of f, and K decreases as However, f and K do not have the same critical numbers. ±. 5 Critical numbers of f:, ± ±.77 Critical numbers of K:, ±.77, ±.8 7. (a) Imagine dropping the circle k into the parabola. The circle will drop to the point where the tangents to the circle and parabola are equal. and k k 5. Taking derivatives, k and Hence, Thus, k k k k k k Thus, k Finall, k.5, and the center of the circle is.5 units from the verte of the parabola. Since the radius of the circle is, the circle is.5 units from the verte. (b) In -space, the parabola or has a curvature of K at,. The radius of the largest sphere that will touch the verte has radius K. 9. Given f : K R K The center of the circle is on the normal line at a distance of R from,. Equation of normal line: Thus,,,. For e, When : Center of curvature:, (See Eercise 7) e, e, e e e e.
30 Section.5 Arc Length and Curvature r sin r cos r sin K r rr r r r cos sin sin sin cos sin sin 8 sin sin r e a, a > r ae a r a e a K r rr r a e a a e a e a r r a e a ea e aa (a) As, K. (b) As a, K r a sin r a cos r a sin K r rr r r r a cos a sin a sin a cos a sin a a a, a > r sin r 8 cos At the pole: K r f t 8. gt d d d dt gt ft d dt ftgt gtft ft ftgt gtft ft K d dt gt d ft dt ftgt gtft ft ft gt ft ftgt gtft ft ft gt ft ftgt gtft ft gt a sin a cos a sin K a cos cos a sin a cos a sin cos cos a cos a sin a cos a cos cos a cos a cos a csc Minimum: a Maimum: none K as 8. a N mk ds dt 55 lb 58 ft ftsec ft sec 7.5 lb
31 8 Chapter Vector-Valued Functions 85. Let r ti tj tk. Then r r t t t and Then, r dr dt t t t t t t tt tt tt r r. r ti tj tk. tt tt tt 87. Let r i j k where,, and are functions of t, and r r. d dt r r rr rdrdt rr rr rr r r r rr r r r (using Eercise 77) i j k i j k r r i j k i j k r r r r r 89. From Eercise 8, we have concluded that planetar motion is planar. Assume that the planet moves in the -plane with the sun at the origin. From Eercise 88, we have Since and r are both perpendicular to L, so is e. Thus, e lies in the -plane. Situate the coordinate sstem so that e lies along the positive -ais and is the angle between e and r. Let e e. Then r e r e cos re cos. Also, Thus, r L GM r r L L L L r r L r r L r GM e r r L GM r e cos r e. GM r e r r r GMre cos r and the planetar motion is a conic section. Since the planet returns to its initial position periodicall, the conic is an ellipse. Sun θ e r Planet 9. A r d Thus, d d da da dt dt r d dt L and r sweeps out area at a constant rate. Review Eercises for Chapter. rt ti csc tk. (a) Domain: t n, n an integer (b) Continuous ecept at t n, n an integer rt ln ti tj tk (a) Domain:, (b) Continuous for all t >
32 Review Eercises for Chapter 9 5. (a) r i (b) r i j 8 k (c) rc c i c j c k c i c j c k (d) r t r t i t j t k i j k ti tt j t t tk 7. rt cos ti sin tj 9. rt i tj t k. rt i sin tj k t cos t, t sin t, sin t, t t t. rt ti ln tj t k 5. One possible answer is: r t ti tj, t r t i tj, t r t ti, t 7. The vector joining the points is 7,,. One path is 9. rt 7t, t, 8 t.,, t t, t, t rt ti tj t k 5. lim t t i t j k i k
33 7 Chapter Vector-Valued Functions. rt ti t j, ut ti t j t k (a) rt i j (b) rt (c) rt ut t t t t t (d) ut rt 5ti t t j t k D t rt ut t t D t ut rt 5i t j t k (e) rt t t (f ) rt ut t t i t j t t tk D t rt t t t D t rt ut 8 t t i 8t j 9t t k 5. t and t are increasing functions at t t, and t is a decreasing function at t t. 7. cos ti t cos tj dt sin ti t sin t cos tj C 9. cos ti sin tj tk dt t dt t t ln t t C. rt ti e t j e t k dt t i e t j e t k C r j k C i j 5k C i j k rt t i e t j e t k. ti t j t k dt t t i j t k j e t i t j k dt e t i t j tk 5. e i 8j k 7. rt cos t, sin t, t vt rt cos t sin t, sin t cos t, vt 9 cos t sin t 9 sin t cos t 9 cos t sin tcos t sin t cos t sin t at vt cos tsin t cos t cos t, sin t cos t sin tsin t, cos t sin t cos t, sin t cos t sin t, 9. rt lnt, t, t rt t, t, r, 8,, t. Range v feet sin 75 sin 5 direction numbers Since r,,, the parametric equations are t, 8t, t. rt. r..,.8,.5. v 89.8 Range v.9 msec 9.8 sin 8 sin
34 Review Eercises for Chapter 7 5. rt 5ti 7. vt 5i vt 5 at Tt i Nt does not eist a T a N does not eist (The curve is a line.) rt ti tj vt i t j t vt t at tt j Tt Nt i tj ti j t t t i tj t a T ttt a N tt 9. rt e t i e t j 5. vt e t i e t j vt e t e t at e t i e t j Tt et i e t j e t e t Nt et i e t j e t e t a T et e t e t e t a N e t e t rt ti t j t k vt i tj tk v 5t at j k Tt Nt 5t a T 5t a N i tj tk 5t 5ti j k 5 5t 5 5 5t 5 5t 5. rt cos ti sin tj tk, cos t, sin t, t When t,,, rt sin ti cos tj k. Direction numbers when t a, b, c, t, t, t 55. v misec 57. rt ti tj, t 5 rt i j s b rt dt 5 9 dt a 5 t 5 8 (, ) 8 (, 5)
35 7 Chapter Vector-Valued Functions 59. rt cos ti sin tj rt cos t sin ti sin t cos tj rt cos t sin t sin t cos t cos t sin t s cos t sin t dt sin t. rt ti tj tk, t rt i j k b s a rt dt 9 dt 9 dt 9 8 ( 9,, ) (,, ) 8. rt 8 cos t, 8 sin t, t, t 5. rt ti sin tj cos tk, t rt < 8 sin t, 8 cos t,, rt 5 rt dt 5 5 dt b s a 8 (8,, ) π (, 8, ) 8 π rt i cos tj sin tk s rt dt cos t sin t dt 5 dt 5 t 5 7. rt ti tj 9. Line k rt ti t j t k rt i tj tk, r 5t rt j k r r i j t K r r r k t j k, r r 5 5t 5t ln K At, K 7 and r 7 77., K ] At, K and r. 75. The curvature changes abruptl from ero to a nonero constant at the points B and C.
CHAPTER 3 Applications of Differentiation
CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. 0 Section. Rolle s Theorem and the Mean Value Theorem. 07 Section. Increasing and Decreasing Functions and the First
More informationCHAPTER 3 Applications of Differentiation
CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. Section. Rolle s Theorem and the Mean Value Theorem. 7 Section. Increasing and Decreasing Functions and the First Derivative
More informationReview Exercises for Chapter 2
Review Eercises for Chapter 367 Review Eercises for Chapter. f 1 1 f f f lim lim 1 1 1 1 lim 1 1 1 1 lim 1 1 lim lim 1 1 1 1 1 1 1 1 1 4. 8. f f f f lim lim lim lim lim f 4, 1 4, if < if (a) Nonremovable
More informationMA 351 Fall 2007 Exam #1 Review Solutions 1
MA 35 Fall 27 Exam # Review Solutions THERE MAY BE TYPOS in these solutions. Please let me know if you find any.. Consider the two surfaces ρ 3 csc θ in spherical coordinates and r 3 in cylindrical coordinates.
More informationCHAPTER 3 Applications of Differentiation
CHAPTER Applications of Differentiation Section. Etrema on an Interval................... 0 Section. Rolle s Theorem and the Mean Value Theorem...... 0 Section. Increasing and Decreasing Functions and
More information13.1. For further details concerning the physics involved and animations of the trajectories of the particles, see the following websites:
8 CHAPTER VECTOR FUNCTIONS N Some computer algebra sstems provide us with a clearer picture of a space curve b enclosing it in a tube. Such a plot enables us to see whether one part of a curve passes in
More informationC H A P T E R 9 Topics in Analytic Geometry
C H A P T E R Topics in Analtic Geometr Section. Circles and Parabolas.................... 77 Section. Ellipses........................... 7 Section. Hperbolas......................... 7 Section. Rotation
More informationReview Exercises for Chapter 3. Review Exercises for Chapter r v 0 2. v ft sec. x 1 2 x dx f x x 99.4.
Review Eercises for Chapter 6. r v 0 sin. Let f, 00, d 0.6. v 0 00 ftsec changes from 0 to dr 00 cos d 6 0 d 0 r dr 80 00 6 96 feet 80 cos 0 96 feet 8080 f f fd d f 99. 00 0.6 9.97 00 Using a calculator:
More informationCHAPTER 3 Applications of Differentiation
CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. 78 Section. Rolle s Theorem and the Mean Value Theorem. 8 Section. Increasing and Decreasing Functions and the First
More informationMath 323 Exam 2 - Practice Problem Solutions. 2. Given the vectors a = 1,2,0, b = 1,0,2, and c = 0,1,1, compute the following:
Math 323 Eam 2 - Practice Problem Solutions 1. Given the vectors a = 2,, 1, b = 3, 2,4, and c = 1, 4,, compute the following: (a) A unit vector in the direction of c. u = c c = 1, 4, 1 4 =,, 1+16+ 17 17
More informationVector-Valued Functions
Vector-Valued Functions 1 Parametric curves 8 ' 1 6 1 4 8 1 6 4 1 ' 4 6 8 Figure 1: Which curve is a graph of a function? 1 4 6 8 1 8 1 6 4 1 ' 4 6 8 Figure : A graph of a function: = f() 8 ' 1 6 4 1 1
More information11 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS
PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS. Parametric Equations Preliminar Questions. Describe the shape of the curve = cos t, = sin t. For all t, + = cos t + sin t = 9cos t + sin t =
More informationCHAPTER 5 Logarithmic, Exponential, and Other Transcendental Functions
CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions Section 5. The Natural Logarithmic Function: Differentiation.... 9 Section 5. The Natural Logarithmic Function: Integration...... 98
More information1969 AP Calculus BC: Section I
969 AP Calculus BC: Section I 9 Minutes No Calculator Note: In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e).. t The asymptotes of the graph of the parametric
More informationSection Vector Functions and Space Curves
Section 13.1 Section 13.1 Goals: Graph certain plane curves. Compute limits and verify the continuity of vector functions. Multivariable Calculus 1 / 32 Section 13.1 Equation of a Line The equation of
More informationMath 101 chapter six practice exam MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Math 1 chapter si practice eam MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Which equation matches the given calculator-generated graph and description?
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. D) D: (-, 0) (0, )
Midterm Practice Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the domain and graph the function. ) G(t) = t - 3 ) 3 - -3 - - 3 - - -3
More informationExercises for Multivariable Differential Calculus XM521
This document lists all the exercises for XM521. The Type I (True/False) exercises will be given, and should be answered, online immediately following each lecture. The Type III exercises are to be done
More informationChapter Four. Derivatives. in the interior of a set S of real numbers means there is an interval centered at t 0
Chapter Four Derivatives 4 Derivatives Suppose f is a vector function and t 0 is a point in the interior of the domain of f ( t 0 in the interior of a set S of real numbers means there is an interval centered
More informationTangent and Normal Vectors
Tangent and Normal Vectors MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Navigation When an observer is traveling along with a moving point, for example the passengers in
More informationCHAPTER 6 Applications of Integration
PART II CHAPTER Applications of Integration Section. Area of a Region Between Two Curves.......... Section. Volume: The Disk Method................. 7 Section. Volume: The Shell Method................
More informationC) 2 D) 4 E) 6. ? A) 0 B) 1 C) 1 D) The limit does not exist.
. The asymptotes of the graph of the parametric equations = t, y = t t + are A) =, y = B) = only C) =, y = D) = only E) =, y =. What are the coordinates of the inflection point on the graph of y = ( +
More informationVECTOR FUNCTIONS. which a space curve proceeds at any point.
3 VECTOR FUNCTIONS Tangent vectors show the direction in which a space curve proceeds at an point. The functions that we have been using so far have been real-valued functions. We now stud functions whose
More informationAnswer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26.
Answer Key 969 BC 97 BC. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E.
More information13.1. For further details concerning the physics involved and animations of the trajectories of the particles, see the following websites:
8 CHAPTER 3 VECTOR FUNCTIONS N Some computer algebra sstems provie us with a clearer picture of a space curve b enclosing it in a tube. Such a plot enables us to see whether one part of a curve passes
More informationMATH 223 FINAL EXAM STUDY GUIDE ( )
MATH 3 FINAL EXAM STUDY GUIDE (017-018) The following questions can be used as a review for Math 3 These questions are not actual samples of questions that will appear on the final eam, but the will provide
More informationMAT 272 Test 1 Review. 1. Let P = (1,1) and Q = (2,3). Find the unit vector u that has the same
11.1 Vectors in the Plane 1. Let P = (1,1) and Q = (2,3). Find the unit vector u that has the same direction as. QP a. u =< 1, 2 > b. u =< 1 5, 2 5 > c. u =< 1, 2 > d. u =< 1 5, 2 5 > 2. If u has magnitude
More informationCHAPTER 2 Differentiation
CHAPTER Differentiation Section. The Derivative and the Slope of a Graph............. 9 Section. Some Rules for Differentiation.................. 56 Section. Rates of Change: Velocit and Marginals.............
More information2. Find the value of y for which the line through A and B has the given slope m: A(-2, 3), B(4, y), 2 3
. Find an equation for the line that contains the points (, -) and (6, 9).. Find the value of y for which the line through A and B has the given slope m: A(-, ), B(4, y), m.. Find an equation for the line
More informationAnalytic Geometry in Three Dimensions
Analtic Geometr in Three Dimensions. The Three-Dimensional Coordinate Sstem. Vectors in Space. The Cross Product of Two Vectors. Lines and Planes in Space The three-dimensional coordinate sstem is used
More informationEXAM 2 ANSWERS AND SOLUTIONS, MATH 233 WEDNESDAY, OCTOBER 18, 2000
EXAM 2 ANSWERS AND SOLUTIONS, MATH 233 WEDNESDAY, OCTOBER 18, 2000 This examination has 30 multiple choice questions. Problems are worth one point apiece, for a total of 30 points for the whole examination.
More information698 Chapter 11 Parametric Equations and Polar Coordinates
698 Chapter Parametric Equations and Polar Coordinates 67. 68. 69. 70. 7. 7. 7. 7. Chapter Practice Eercises 699 75. (a Perihelion a ae a( e, Aphelion ea a a( e ( Planet Perihelion Aphelion Mercur 0.075
More informationMTH 277 Test 4 review sheet Chapter , 14.7, 14.8 Chalmeta
MTH 77 Test 4 review sheet Chapter 13.1-13.4, 14.7, 14.8 Chalmeta Multiple Choice 1. Let r(t) = 3 sin t i + 3 cos t j + αt k. What value of α gives an arc length of 5 from t = 0 to t = 1? (a) 6 (b) 5 (c)
More informationDIFFERENTIATION. 3.1 Approximate Value and Error (page 151)
CHAPTER APPLICATIONS OF DIFFERENTIATION.1 Approimate Value and Error (page 151) f '( lim 0 f ( f ( f ( f ( f '( or f ( f ( f '( f ( f ( f '( (.) f ( f '( (.) where f ( f ( f ( Eample.1 (page 15): Find
More informationLesson 9.1 Using the Distance Formula
Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)
More informationCLEP Calculus. Time 60 Minutes 45 Questions. For each question below, choose the best answer from the choices given. 2. If f(x) = 3x, then f (x) =
CLEP Calculus Time 60 Minutes 5 Questions For each question below, choose the best answer from the choices given. 7. lim 5 + 5 is (A) 7 0 (C) 7 0 (D) 7 (E) Noneistent. If f(), then f () (A) (C) (D) (E)
More informationCALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS. Second Fundamental Theorem of Calculus (Chain Rule Version): f t dt
CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS d d d d t dt 6 cos t dt Second Fundamental Theorem of Calculus: d f tdt d a d d 4 t dt d d a f t dt d d 6 cos t dt Second Fundamental
More informationMotion in Space Parametric Equations of a Curve
Motion in Space Parametric Equations of a Curve A curve, C, inr 3 can be described by parametric equations of the form x x t y y t z z t. Any curve can be parameterized in many different ways. For example,
More informationAPJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, FEBRUARY 2017 MA101: CALCULUS PART A
A B1A003 Pages:3 (016 ADMISSIONS) Reg. No:... Name:... APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, FEBRUARY 017 MA101: CALCULUS Ma. Marks: 100 Duration: 3 Hours PART
More informationVector Functions & Space Curves MATH 2110Q
Vector Functions & Space Curves Vector Functions & Space Curves Vector Functions Definition A vector function or vector-valued function is a function that takes real numbers as inputs and gives vectors
More informationOHSx XM521 Multivariable Differential Calculus: Homework Solutions 14.1
OHSx XM5 Multivariable Differential Calculus: Homework Solutions 4. (8) Describe the graph of the equation. r = i + tj + (t )k. Solution: Let y(t) = t, so that z(t) = t = y. In the yz-plane, this is just
More informationSection Arclength and Curvature. (1) Arclength, (2) Parameterizing Curves by Arclength, (3) Curvature, (4) Osculating and Normal Planes.
Section 10.3 Arclength and Curvature (1) Arclength, (2) Parameterizing Curves by Arclength, (3) Curvature, (4) Osculating and Normal Planes. MATH 127 (Section 10.3) Arclength and Curvature The University
More informationlim 2 x lim lim sin 3 (9) l)
MAC FINAL EXAM REVIEW. Find each of the following its if it eists, a) ( 5). (7) b). c). ( 5 ) d). () (/) e) (/) f) (-) sin g) () h) 5 5 5. DNE i) (/) j) (-/) 7 8 k) m) ( ) (9) l) n) sin sin( ) 7 o) DNE
More informationAP Calculus AB/BC ilearnmath.net
CALCULUS AB AP CHAPTER 1 TEST Don t write on the test materials. Put all answers on a separate sheet of paper. Numbers 1-8: Calculator, 5 minutes. Choose the letter that best completes the statement or
More informationMultiple Choice. Circle the best answer. No work needed. No partial credit available. is continuous.
Multiple Choice. Circle the best answer. No work needed. No partial credit available. + +. Evaluate lim + (a (b (c (d 0 (e None of the above.. Evaluate lim (a (b (c (d 0 (e + + None of the above.. Find
More informationLecture D2 - Curvilinear Motion. Cartesian Coordinates
J. Peraire 6.07 Dynamics Fall 2004 Version. Lecture D2 - Curvilinear Motion. Cartesian Coordinates We will start by studying the motion of a particle. We think of a particle as a body which has mass, but
More informationIncreasing and Decreasing Functions and the First Derivative Test
Section 3.3 Increasing and Decreasing Functions and the First Derivative Test 3 Section 3.3 Increasing and Decreasing Functions and the First Derivative Test. f 8 3. 3, Decreasing on:, 3 3 3,,, Decreasing
More informationCHAPTER 6 Differential Equations
CHAPTER 6 Differential Equations Section 6. Slope Fields and Euler s Method.............. 55 Section 6. Differential Equations: Growth and Deca........ 557 Section 6. Separation of Variables and the Logistic
More informationChapter 14: Vector Calculus
Chapter 14: Vector Calculus Introduction to Vector Functions Section 14.1 Limits, Continuity, Vector Derivatives a. Limit of a Vector Function b. Limit Rules c. Component By Component Limits d. Continuity
More informationParticles in Motion; Kepler s Laws
CHAPTER 4 Particles in Motion; Kepler s Laws 4.. Vector Functions Vector notation is well suited to the representation of the motion of a particle. Fix a coordinate system with center O, and let the position
More informationChapter 5 Review. 1. [No Calculator] Evaluate using the FTOC (the evaluation part) 2. [No Calculator] Evaluate using geometry
AP Calculus Chapter Review Name: Block:. [No Calculator] Evaluate using the FTOC (the evaluation part) a) 7 8 4 7 d b) 9 4 7 d. [No Calculator] Evaluate using geometry a) d c) 6 8 d. [No Calculator] Evaluate
More informationChapter 4 Applications of Derivatives. Section 4.1 Extreme Values of Functions (pp ) Section Quick Review 4.1
Section. 6 8. Continued (e) vt () t > 0 t > 6 t > 8. (a) d d e u e u du where u d (b) d d d d e + e e e e e e + e e + e (c) y(). (d) m e e y (). 7 y. 7( ) +. y 7. + 0. 68 0. 8 m. 7 y0. 8( ) +. y 0. 8+.
More information+ 4 Ex: y = v = (1, 4) x = 1 Focus: (h, k + ) = (1, 6) L.R. = 8 units We can have parabolas that open sideways too (inverses) x = a (y k) 2 + h
Unit 7 Notes Parabolas: E: reflectors, microphones, (football game), (Davinci) satellites. Light placed where ras will reflect parallel. This point is the focus. Parabola set of all points in a plane that
More informationMATH Final Review
MATH 1592 - Final Review 1 Chapter 7 1.1 Main Topics 1. Integration techniques: Fitting integrands to basic rules on page 485. Integration by parts, Theorem 7.1 on page 488. Guidelines for trigonometric
More information(6, 4, 0) = (3, 2, 0). Find the equation of the sphere that has the line segment from P to Q as a diameter.
Solutions Review for Eam #1 Math 1260 1. Consider the points P = (2, 5, 1) and Q = (4, 1, 1). (a) Find the distance from P to Q. Solution. dist(p, Q) = (4 2) 2 + (1 + 5) 2 + (1 + 1) 2 = 4 + 36 + 4 = 44
More informationLater in this chapter, we are going to use vector functions to describe the motion of planets and other objects through space.
10 VECTOR FUNCTIONS VECTOR FUNCTIONS Later in this chapter, we are going to use vector functions to describe the motion of planets and other objects through space. Here, we prepare the way by developing
More informationCHAPTER 4 DIFFERENTIAL VECTOR CALCULUS
CHAPTER 4 DIFFERENTIAL VECTOR CALCULUS 4.1 Vector Functions 4.2 Calculus of Vector Functions 4.3 Tangents REVIEW: Vectors Scalar a quantity only with its magnitude Example: temperature, speed, mass, volume
More informationDepartment of Mathematical and Statistical Sciences University of Alberta
MATH 4 (R) Winter 8 Intermediate Calculus I Solutions to Problem Set #5 Completion Date: Frida Februar 5, 8 Department of Mathematical and Statistical Sciences Universit of Alberta Question. [Sec.., #
More informationSummary, Review, and Test
944 Chapter 9 Conic Sections and Analtic Geometr 45. Use the polar equation for planetar orbits, to find the polar equation of the orbit for Mercur and Earth. Mercur: e = 0.056 and a = 36.0 * 10 6 miles
More informationD In, RS=10, sin R
FEBRUARY 7, 08 Invitational Sickles The abbreviation NOTA means None of These Answers and should be chosen if choices A, B, C and D are not correct Solve for over the Real Numbers: ln( ) ln() The trigonometric
More information1985 AP Calculus AB: Section I
985 AP Calculus AB: Section I 9 Minutes No Calculator Notes: () In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e). () Unless otherwise specified, the domain of
More informationLecture for Week 6 (Secs ) Derivative Miscellany I
Lecture for Week 6 (Secs. 3.6 9) Derivative Miscellany I 1 Implicit differentiation We want to answer questions like this: 1. What is the derivative of tan 1 x? 2. What is dy dx if x 3 + y 3 + xy 2 + x
More informationTangent and Normal Vector - (11.5)
Tangent and Normal Vector - (.5). Principal Unit Normal Vector Let C be the curve traced out bythe vector-valued function r!!t # f!t, g!t, h!t $. The vector T!!t r! % r! %!t!t is the unit tangent vector
More informationVectors, dot product, and cross product
MTH 201 Multivariable calculus and differential equations Practice problems Vectors, dot product, and cross product 1. Find the component form and length of vector P Q with the following initial point
More informationReview Sheet for Exam 1 SOLUTIONS
Math b Review Sheet for Eam SOLUTIONS The first Math b midterm will be Tuesday, February 8th, 7 9 p.m. Location: Schwartz Auditorium Room ) The eam will cover: Section 3.6: Inverse Trig Appendi F: Sigma
More informationb) Use your result from part (a) to find the slope of the tangent line to the parametric 2 4
AP Calculus BC Lesson 11. Parametric and Vector Calculus dy dy 1. By the chain rule = a) Solve the chain rule equation above for dy b) Use your result from part (a) to find the slope of the tangent line
More informationTechnical Calculus I Homework. Instructions
Technical Calculus I Homework Instructions 1. Each assignment is to be done on one or more pieces of regular-sized notebook paper. 2. Your name and the assignment number should appear at the top of the
More informationCalculus III: Practice Final
Calculus III: Practice Final Name: Circle one: Section 6 Section 7. Read the problems carefully. Show your work unless asked otherwise. Partial credit will be given for incomplete work. The exam contains
More informationVector Geometry Final Exam Review
Vector Geometry Final Exam Review Problem 1. Find the center and the radius for the sphere x + 4x 3 + y + z 4y 3 that the center and the radius of a sphere z 7 = 0. Note: Recall x + ax + y + by + z = d
More informationCopyright 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Chapter 5 Review 95 (c) f( ) f ( 7) ( 7) 7 6 + ( 6 7) 7 6. 96 Chapter 5 Review Eercises (pp. 60 6). y y ( ) + ( )( ) + ( ) The first derivative has a zero at. 6 Critical point value: y 9 Endpoint values:
More informationCHAPTER P Preparation for Calculus
CHAPTER P Preparation for Calculus Section P. Graphs and Models...................... Section P. Linear Models and Rates of Change............ Section P. Functions and Their Graphs................. Section
More informationMath 153 Calculus III Notes
Math 153 Calculus III Notes 10.1 Parametric Functions A parametric function is a where x and y are described by a function in terms of the parameter t: Example 1 (x, y) = {x(t), y(t)}, or x = f(t); y =
More information13.3 Arc Length and Curvature
13 Vector Functions 13.3 Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. We have defined the length of a plane curve with parametric equations x = f(t),
More information3 = arccos. A a and b are parallel, B a and b are perpendicular, C a and b are normalized, or D this is always true.
Math 210-101 Test #1 Sept. 16 th, 2016 Name: Answer Key Be sure to show your work! 1. (20 points) Vector Basics: Let v = 1, 2,, w = 1, 2, 2, and u = 2, 1, 1. (a) Find the area of a parallelogram spanned
More informationExercises of Mathematical analysis II
Eercises of Mathematical analysis II In eercises. - 8. represent the domain of the function by the inequalities and make a sketch showing the domain in y-plane.. z = y.. z = arcsin y + + ln y. 3. z = sin
More informationCHAPTER 2 Differentiation
CHAPTER Differentiation Section. The Derivative and the Slope of a Graph............. 6 Section. Some Rules for Differentiation.................. 69 Section. Rates of Change: Velocit and Marginals.............
More informationBE SURE TO READ THE DIRECTIONS PAGE & MAKE YOUR NOTECARDS FIRST!! Part I: Unlimited and Continuous! (21 points)
BE SURE TO READ THE DIRECTIONS PAGE & MAKE YOUR NOTECARDS FIRST!! Part I: United and Continuous! ( points) For #- below, find the its, if they eist.(#- are pt each) ) 7 ) 9 9 ) 5 ) 8 For #5-7, eplain why
More informationCHAPTER 5 Logarithmic, Exponential, and Other Transcendental Functions
CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions Section 5. The Natural Logarithmic Function: Dierentiation.... Section 5. The Natural Logarithmic Function: Integration...... Section
More informationlim x c) lim 7. Using the guidelines discussed in class (domain, intercepts, symmetry, asymptotes, and sign analysis to
Math 7 REVIEW Part I: Problems Using the precise definition of the it, show that [Find the that works for any arbitrarily chosen positive and show that it works] Determine the that will most likely work
More informationCHAPTER 3 Polynomial Functions
CHAPTER Polnomial Functions Section. Quadratic Functions and Models............. 7 Section. Polnomial Functions of Higher Degree......... 7 Section. Polnomial and Snthetic Division............ Section.
More informationMultiple Choice. 1.(6 pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.
Multiple Choice.(6 pts) Find smmetric equations of the line L passing through the point (, 5, ) and perpendicular to the plane x + 3 z = 9. (a) x = + 5 3 = z (c) (x ) + 3( 3) (z ) = 9 (d) (e) x = 3 5 =
More informationCalculus - Chapter 2 Solutions
Calculus - Chapter Solutions. a. See graph at right. b. The velocity is decreasing over the entire interval. It is changing fastest at the beginning and slowest at the end. c. A = (95 + 85)(5) = 450 feet
More informationCHAPTER 1 Functions, Graphs, and Limits
CHAPTER Functions, Graphs, and Limits Section. The Cartesian Plane and the Distance Formula.......... Section. Graphs of Equations........................ 8 Section. Lines in the Plane and Slope....................
More informationTMTA Calculus and Advanced Topics Test 2010
. Evaluate lim Does not eist - - 0 TMTA Calculus and Advanced Topics Test 00. Find the period of A 6D B B y Acos 4B 6D, where A 0, B 0, D 0. Solve the given equation for : ln = ln 4 4 ln { } {-} {0} {}
More informationReview problems for the final exam Calculus III Fall 2003
Review problems for the final exam alculus III Fall 2003 1. Perform the operations indicated with F (t) = 2t ı 5 j + t 2 k, G(t) = (1 t) ı + 1 t k, H(t) = sin(t) ı + e t j a) F (t) G(t) b) F (t) [ H(t)
More informationTest # 1 Review Math MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Test # 1 Review Math 13 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the slope of the curve at the given point P and an equation of the
More informationMATH Fall 08. y f(x) Review Problems for the Midterm Examination Covers [1.1, 4.3] in Stewart
MATH 121 - Fall 08 Review Problems for te Midterm Eamination Covers [1.1, 4.3] in Stewart 1. (a) Use te definition of te derivative to find f (3) wen f() = π 1 2. (b) Find an equation of te tangent line
More informationFind the volume of the solid generated by revolving the shaded region about the given axis. Use the disc/washer method 1) About the x-axis
Final eam practice for Math 6 Disclaimer: The actual eam is different Find the volume of the solid generated b revolving the shaded region about the given ais. Use the disc/washer method ) About the -ais
More informationMat 267 Engineering Calculus III Updated on 9/19/2010
Chapter 11 Partial Derivatives Section 11.1 Functions o Several Variables Deinition: A unction o two variables is a rule that assigns to each ordered pair o real numbers (, ) in a set D a unique real number
More informationChapter 1 Prerequisites for Calculus
Section. Chapter Prerequisites for Calculus Section. Lines (pp. ) Quick Review.. + ( ) + () +. ( +). m. m ( ) ( ). (a) ( )? 6 (b) () ( )? 6. (a) 7? ( ) + 7 + Yes (b) ( ) + 9 No Yes No Section. Eercises.
More informationMath 2414 Activity 1 (Due by end of class Jan. 26) Precalculus Problems: 3,0 and are tangent to the parabola axis. Find the other line.
Math Activity (Due by end of class Jan. 6) Precalculus Problems: 3, and are tangent to the parabola ais. Find the other line.. One of the two lines that pass through y is the - {Hint: For a line through
More informationMATH107 Vectors and Matrices
School of Mathematics, KSU 20/11/16 Vector valued functions Let D be a set of real numbers D R. A vector-valued functions r with domain D is a correspondence that assigns to each number t in D exactly
More informationAP Calculus (BC) Summer Assignment (169 points)
AP Calculus (BC) Summer Assignment (69 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion
More informationHEAT-3 APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA MAX-MARKS-(112(3)+20(5)=436)
HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA TIME-(HRS) Select the correct alternative : (Only one is correct) MAX-MARKS-(()+0(5)=6) Q. Suppose & are the point of maimum and the point of minimum
More informationAPPLICATIONS OF DERIVATIVES UNIT PROBLEM SETS
APPLICATIONS OF DERIVATIVES UNIT PROBLEM SETS PROBLEM SET #1 Related Rates ***Calculators Allowed*** 1. An oil tanker spills oil that spreads in a circular pattern whose radius increases at the rate of
More informationTangent and Normal Vector - (11.5)
Tangent and Normal Vector - (.5). Principal Unit Normal Vector Let C be the curve traced out by the vector-valued function rt vector T t r r t t is the unit tangent vector to the curve C. Now define N
More informationax, From AtoB bx c, From BtoC
Name: Date: Block: Semester Assessment Revision 3 Multiple Choice Calculator Active NOTE: The eact numerical value of the correct answer may not always appear among the choices given. When this happens,
More informationCHAPTER 1 Functions and Their Graphs
PART I CHAPTER Functions and Their Graphs Section. Lines in the Plane....................... Section. Functions........................... Section. Graphs of Functions..................... Section. Shifting,
More information1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is
1. The value of the double integral (a) 15 26 (b) 15 8 (c) 75 (d) 105 26 5 4 0 1 1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 2. What is the value of the double integral interchange the order
More informationMATH 150/GRACEY PRACTICE FINAL. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
MATH 0/GRACEY PRACTICE FINAL Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Choose the graph that represents the given function without using
More information