MATH 155/GRACEY CH. 10 PRACTICE. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

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1 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 )

2 ) = t +, = 6t + 6, - t ) ) = 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)

3 Gaph the pai of paametic equations with the aid of a gaphing calculato. 9) = (t - sin t), = ( - cos t), 0 t π 9) ) = 6 cos t + cos t, = 6 sin t - sint, 0 t π ) Plot the point whose pola coodinates ae given. ) (, -π/) ) - -

4 ) (, π/) ) - - ) (-, 0) ) - - ) (-, π/) ) - - Detemine if the given pola coodinates epesent the same point. ) (, π/), (-, π/) ) 6) (8, π/), (-8, π/) 6)

5 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 )

6 Gaph the pola equation. 6) = ( + sin θ) 6) ) = - cos θ 7) ) = sin θ 8)

7 9) = cos θ 9) ) = ( -sin θ) 0) ) = θ )

8 ) = - - 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

9 Gaph the paabola. ) = ) ) = ) Find the focus and diecti of the paabola. 6) - = 6) 7) - 8 = 0 7) 9

10 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) A) = - B) = C) = D) = SHORT ANSWER. Wite the wod o phase that best completes each statement o answes the question. Gaph. 9) + = 9) ) + 6 = 6 60) - -

11 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) 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)

12 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) ) = 0 + sin θ 7)

13 8 7) = - cos θ 7)

14 Answe Ke Testname: M_E_PRACTICE ) = - + ) = π - π + π ) + = Counteclockwise fom (-, 0) to (, 0) ) = Entie paabola, bottom to top (fom fouth quadant to oigin to fist quadant)

15 Answe Ke Testname: M_E_PRACTICE ) = 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)

16 Answe Ke Testname: M_E_PRACTICE ) 8 6 ) )

17 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

18 Answe Ke Testname: M_E_PRACTICE ) = ) + ( - 8) = ) ( - ) + = 0 ) ( - ) + ( + ) = 6) ) )

19 Answe Ke Testname: M_E_PRACTICE 9) ) )

20 Answe Ke Testname: M_E_PRACTICE ) ) ) α π ) 9π 8 6) 66π ) (π - ) 8) 9) (π - ) 0) (π - 8) ) (π - ) ) 7π ) (π + ) 0

21 Answe Ke Testname: M_E_PRACTICE ) ) ) (0, -); = 7) (7, 0); = -7 8) B 9)

22 Answe Ke Testname: M_E_PRACTICE 60) - - 6) Vetices: (±, 0); Foci: (±, 0) 6) 9 + = 6) = 6) ) ) Foci: (-, 0), (, 0); Asmptotes: =, = - 67) Vetices: (0, ), (0, -); Asmptotes: = ±

23 Answe Ke Testname: M_E_PRACTICE 68) - = 69) 6-0 = 70) = 7) ) )

Section 8.2 Polar Coordinates

Section 8.2 Polar Coordinates Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal