Parametrics and Vectors (BC Only)
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1 Paramerics and Vecors (BC Only) The following relaionships should be learned and memorized. The paricle s posiion vecor is r() x(), y(). The velociy vecor is v(),. The speed is he magniude of he velociy vecor The acceleraion vecor is a() d x, d y. The oal disance raveled by he posiion vecor is given by. The following formulas for paramerics will be used on he AP Calculus exam: The firs derivaive (he change in y wih respec o x) is. d y d The second derivaive of y wih respec o x is or d y d Sudens should be able o: Deermine he slope of he curve a any poin Wrie an equaion of he line angen o he curve a a given poin Calculae he lengh of a curve beween wo poins by using he arc lengh formula L Deermine posiion, velociy, and acceleraion vecors Calculae magniude and speed Calculae displacemen and oal disance raveled Copyrigh 4 Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a
2 Muliple Choice Paramerics and Vecors (BC only). (calculaor no allowed) The lengh of he pah described by he parameric equaions x and y, where, is given by (A) (B) (C) (D) (E) (calculaor no allowed) A paricle moves on a plane curve so ha a any ime is x- coordinae is and is y-coordinae is ( ). The acceleraion vecor of he paricle a is (A), (B), (C), 6 (D) 6, (E) 6, 4. (calculaor no allowed) A ime, a paricle moving in he xy plane has velociy vecor given by Wha is he acceleraion vecor of he paricle a ime? v (),5. (A) 45 9, (B) 6, 5 (C), (D) 6 (E) 6 Copyrigh 4 Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a
3 Paramerics and Vecors (BC only) 4. (calculaor no allowed) 4 A curve in he plane is defined paramerically by he equaions x and y. An equaion of he line angen o he curve a is (A) y x (B) y 8x (C) y x (D) y 4x 5 (E) y 8x 5. (calculaor no allowed) In he xy plane, he graph of he parameric equaions x 5 and y, for, is a line segmen wih slope (A) 5 5 (B) (C) (D) 5 (E) 6. (calculaor no allowed) In he xy plane, a paricle moves along he parabola y x x wih a consan speed of unis per second. If, wha is he value of when he paricle is a he poin,? (A) (B) (C) (D) 6 (E) 6 Copyrigh 4 Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a
4 Paramerics and Vecors (BC only) 7. (calculaor no allowed) The posiion of a paricle moving in he xy plane is given by he parameric equaions x and (A) only (B) only (C) only (D) and only (E), and y. For wha values of is he paricle a res? 8. (calculaor allowed) If f is a vecor-valued funcion defined by f () ( e,cos), hen f () (A) e sin (B) e cos e, sin (D) e,cos (C) (E) e, cos 9. (calculaor allowed) A paricle moves in he xy plane so ha is posiion a any ime is given by and y() sin(4). Wha is he speed of he paricle when? (A).99 (B).6 (C) (D).6 (E) 47.9 x() Copyrigh 4 Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a
5 Free Response Paramerics and Vecors (BC only). (calculaor allowed) An objec moving along a curve in he xy-plane is a posiion x(), y() a ime, where ane and A ime, he objec is a posiion (, ). sece for. (a) Wrie an equaion for he line angen o he curve a posiion (, ). (b) Find he acceleraion vecor and he speed of he objec a ime. (c) Find he oal disance raveled by he objec over he ime inerval. (d) Is here a ime a which he objec is on he y-axis? Explain why or why no. Copyrigh 4 Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a
6 . (calculaor allowed) Paramerics and Vecors (BC only) A paricle sars a poin A on he posiive x-axis a ime and ravels along he curve from A o B o C o D, as shown above. The coordinaes of he paricle s posiion x(), y() are differeniable funcions of, where x () 9cos sin 6 and y() is no explicily given. A ime = 9, he paricle reaches is final posiion D on he posiive x-axis. (a) A poin C, is posiive? A poin C, is posiive? Give a reason for each answer. (b) The slope of he curve is undefined a poin B. A wha ime is he paricle a poin B? Copyrigh 4 Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a
7 (c) The line angen o he curve a he poin x(8), y (8) has equaion velociy vecor and he speed of he paricle a his poin. Paramerics and Vecors (BC only) 5 y x. Find he 9 (d) How far apar are poins A and D, he iniial and final posiions, respecively, of he paricle? Copyrigh 4 Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a
8 . (calculaor allowed) Paramerics and Vecors (BC only) The velociy vecor of a paricle moving in he xy-plane has componens given by 4cos sine and sin, for.5. A ime, he posiion of he paricle is (-, ). (a) For.5, find all values of a which he line angen o he pah of he paricle is verical. (b) Wrie an equaion for he line angen o he pah of he paricle a =. (c) Find he speed of he paricle a =. (d) Find he acceleraion vecor of he paricle a =. Copyrigh 4 Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a
9 . (calculaor allowed) Paramerics and Vecors (BC only) A ime, a paricle moving in he xy-plane is a posiion x(), y() where x() and y() are no explicily given. For, 4 and sin. A ime, x() and y() 4. (a) Find he speed of he paricle a ime =, and find he acceleraion vecor of he paricle a ime =. (b) Find he slope of he line angen o he pah of he paricle a ime =. (c) Find he posiion of he paricle a ime =. (d) Find he oal disance raveled by he paricle over he ime inerval. Copyrigh 4 Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a
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