MATH /GRACEY CH. PRACTICE Name SHORT ANSWER. Wite the wod o phase that best completes each statement o answes the question. At the given point, find the line that is nomal to the cuve at the given point. ) + = +, nomal at (0, ) ) ) - π cos = π, nomal at (, π) ) Paametic equations and a paamete inteval fo the motion of a paticle in the -plane ae given. Identif the paticle's path b finding a Catesian equation fo it. Gaph the Catesian equation. Indicate the potion of the gaph taced b the paticle and the diection of motion. ) = cos t, = sin t, π t π ) - - - - - - - - - - ) = 6t, = 6t, - t ) - - - - - - - - - -
) = t +, = 6t + 6, - t ) - - - - - - - - - - 6) = sin t, = cos t, 0 t π 6) - - - - - - - - - - Find a paametization fo the cuve. 7) The a (half line) with initial point (-, -9) that passes though the point (, -6) 7) 8) The uppe half of the paabola + 7 = 8)
Gaph the pai of paametic equations with the aid of a gaphing calculato. 9) = (t - sin t), = ( - cos t), 0 t π 9) 8 0 0 0 0 60 ) = 6 cos t + cos t, = 6 sin t - sint, 0 t π ) 8 6 - -8-6 - - 6 8 - - -6-8 - Plot the point whose pola coodinates ae given. ) (, -π/) ) - -
) (, π/) ) - - ) (-, 0) ) - - ) (-, π/) ) - - Detemine if the given pola coodinates epesent the same point. ) (, π/), (-, π/) ) 6) (8, π/), (-8, π/) 6)
7) (, π/), (, π/) 7) 8) (, θ), (-, θ + π) 8) 9) (6, π/6), (6, 7π/6) 9) Find the Catesian coodinates of the given point. 0) -, π 0) ) (, π/6) ) ) (-, -π/) ) ) (-, π) ) ), π ) ) -, π/ ) 6) 8, - π 6) Replace the pola equation with an equivalent Catesian equation. 7) cos θ = 7) 8) cos θ + 9 sin θ = 8) 9) = cos θ - sin θ 9) 0) = 8 cot θ csc θ 0) ) = - csc θ ) ) sin θ = ) ) = 6 sin θ ) ) = 0 cos θ ) ) = cos θ - 6 sin θ - 9 )
Gaph the pola equation. 6) = ( + sin θ) 6) - - - - 7) = - cos θ 7) - - - - 8) = sin θ 8) - - - - 6
9) = cos θ 9) - - - - 0) = ( -sin θ) 0) - - - - ) = θ ) 0-0 - 0 - -0 7
) = - - sin θ ) - - - - - - - - ) = + cos θ ) - - - - - - - - Find the aea of the specified egion. ) Inside the cadioid = α( + sin θ), α > 0 ) ) Inside one leaf of the fou-leaved ose = 7 sin θ ) 6) Inside the limacon = 8 + sin θ 6) 7) Inside the smalle loop of the limacon = + sin θ 7) 8) Inside the cicle = sin θ and outside the cadioid = + cos θ 8) 9) Shaed b the cicles = and = cos θ 9) 0) Shaed b the cicle = and the cadioid = ( + sin θ) 0) ) Inside the cicle = 6 and to the ight of the line = sec θ ) ) Inside the cicle = 8 cos θ + sin θ ) ) Inside the oute loop and outside the inne loop of the limacon = sin θ - ) 8
Gaph the paabola. ) = ) - - - - ) = ) - - - Find the focus and diecti of the paabola. 6) - = 6) 7) - 8 = 0 7) 9
MULTIPLE CHOICE. Choose the one altenative that best completes the statement o answes the question. Choose the equation that matches the gaph. 8) 0 8) - - - -0 A) = - B) = C) = D) = SHORT ANSWER. Wite the wod o phase that best completes each statement o answes the question. Gaph. 9) + = 9) - - - - 60) + 6 = 6 60) - -
Find the vetices and foci of the ellipse. 6) 6 + = 96 6) Find the standad-fom equation of the ellipse centeed at the oigin and satisfing the given conditions. 6) An ellipse with length of majo ais and -intecepts (0, ±) 6) 6) An ellipse with vetices (0, ±9) and foci at (0, ± ). 6) Gaph. 6) 6-9 = 6) - - - - 6) 6 - = 6 6) - - - - Solve the poblem. 66) Find the foci and asmptotes of the following hpebola: 6-6 = 66) 67) Find the vetices and asmptotes of the following hpebola: 0-6 = 67)
Find the standad-fom equation of the hpebola centeed at the oigin which satisfies the given conditions. 68) Vetices at (, 0) and (-, 0); foci at (, 0) and (-, 0) 68) 69) Vetices at (0, 6) and (0, -6); asmptotes = and = - 69) Find an equation of the following cuve. 70) An ellipse centeed at the oigin having vete at (0, -) and eccenticit equal to 70) Gaph the paabola o ellipse. Include the diecti that coesponds to the focus at the oigin. 0 7) = + cos θ 7) 6-6 - - 6 - - -6 7) = 0 + sin θ 7) 6-6 - - 6 - - -6
8 7) = - cos θ 7) - - - -
Answe Ke Testname: M_E_PRACTICE ) = - + ) = π - π + π ) + = 9 - - - - - - - - - - Counteclockwise fom (-, 0) to (, 0) ) = - - - - - - - - - - Entie paabola, bottom to top (fom fouth quadant to oigin to fist quadant)
Answe Ke Testname: M_E_PRACTICE ) = - 6 - - - - - - - - - - Entie line, fom left to ight 6) 6 + = - - - - - - - - - - Counteclockwise fom (0, ) to (0, ), one otation 7) Answes will va. Possible answe: = - + t, = -9-7t, t 0 8) Answes will va. Possible answe: = t + 7, = t, t 0 9) 8 0 0 0 0 60
Answe Ke Testname: M_E_PRACTICE ) 8 6 ) - -8-6 - - 6 8 - - -6-8 - - ) - - - 6
Answe Ke Testname: M_E_PRACTICE ) - ) - - ) No 6) Yes 7) No 8) Yes 9) No 0) (0, -) ), ) (-, ) ) (, 0) ) (, ) ) -, 6) (9, -9 ) 7) = 8) + 9 = 9) - = 0) = 8 ) = - - 7
Answe Ke Testname: M_E_PRACTICE ) = ) + ( - 8) = ) ( - ) + = 0 ) ( - ) + ( + ) = 6) - - - - 7) - - - - 8) - - - - 8
Answe Ke Testname: M_E_PRACTICE 9) - - - - 0) - - - - ) 0-0 - 0 - -0 9
Answe Ke Testname: M_E_PRACTICE ) ) - - - - - - - - - - - - - ) α π ) 9π 8 6) 66π - - - 7) (π - ) 8) 9) (π - ) 0) (π - 8) ) (π - ) ) 7π ) (π + ) 0
Answe Ke Testname: M_E_PRACTICE ) - - - - ) - - - 6) (0, -); = 7) (7, 0); = -7 8) B 9) - - - - -
Answe Ke Testname: M_E_PRACTICE 60) - - 6) Vetices: (±, 0); Foci: (±, 0) 6) 9 + = 6) 69 + 8 = 6) - - - - 6) - - - - 66) Foci: (-, 0), (, 0); Asmptotes: =, = - 67) Vetices: (0, ), (0, -); Asmptotes: = ±
Answe Ke Testname: M_E_PRACTICE 68) - = 69) 6-0 = 70) 9 + 8 = 7) 6-6 - - 6 - - 7) -6 6-6 - - 6 - - 7) -6 - - - -