k ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola.

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1 Stndrd Eqution of Prol with vertex ( h, k ) nd directrix y = k p is ( x h) p ( y k ) = 4. Verticl xis of symmetry Stndrd Eqution of Prol with vertex ( h, k ) nd directrix x = h p is ( y k ) p( x h) = 4. Horizontl xis of symmetry The focus lies on the xis p units (directed distnce) from the vertex. The coordintes of the focus re s follows: ( h, k + p) Verticl xis of symmetry ( h p, k ) + Horizontl xis of symmetry A focl chord is line segment which psses through the focus of prol nd hs endpoints on the prol. The specific focl chord perpendiculr to the xis of the prol is the ltus rectum. A surfce is considered reflective if the tngent line t ny point on the surfce mkes equl ngles with n incoming ry nd the resulting outgoing ry. The ngle corresponding to the incoming ry is the ngle of incidence nd the ngle corresponding to the outgoing ry is the ngle of reflection. Let P e point on prol. The tngent line mkes equl ngles with the following two lines: 1. The line pssing through P nd the focus.. The line pssing through P prllel to the xis of the prol.

2 1. Consider the prol y 4x 4 = 0. Find the vertex, focus, nd directrix of the prol nd sketch its grph.. Find dy/dx t x = 1.. Find n eqution of the prol with directrix: y = ; endpoints of ltus rectum re ( 0, ) nd ( 8, ).

3 Stndrd Eqution of n Ellipse with center ( h, k ) nd mjor nd minor xes of length nd where >, is ( x h) ( y k ) Or + = 1 ( x h) ( y k ) + = 1 Mjor xis is horizontl Mjor xis is verticl The foci lies on the mjor xis, c units from the center, with c =. Reflective Property of n Ellipse: Let P e point on n ellipse. The tngent line to n ellipse t point P mkes equl ngles with the lines through P nd the foci. Eccentricity of n Ellipse: The eccentricity e of n ellipse is given y e c =. This mesures the ovlness of n ellipse. When the eccentricity is very smll nd the foci re close to the center, the ellipse is nerly circulr. When the eccentricity is close to one nd the foci re close to the vertices, the elliple will e elongted. 3. Find the eqution of the ellipse with 1 eccentricity: nd vertices: ( 0, ) nd ( 8, ).

4 4. Consider the ellipse x y x y = 1.. Find the center, foci, nd vertices of the ellipse, nd sketch its grph.. Find the eqution(s) of the i. Tngent line(s) t y = ii. Norml lines t x =

5 Stndrd Eqution of Hyperol with center ( h, k ) is ( x h) ( y k ) Or = 1 ( y k ) ( x h) = 1 Trnsverse xis is horizontl Trnsverse xis is verticl The vertices re units from the center. The foci re c units from the center, with c = +. Asymptotes of Hyperol: For horizontl trnsverse xis, the equtions of the symptotes re y = k + ( x h) nd y = k ( x h). For verticl trnsverse xis, the equtions of the symptotes re y = k + ( x h) nd y = k ( x h). The line segment of length joining the points which re units wy from the center is referred to s the conjugte xis. The eccentricity e is e = c. 5. Find the eqution of the hyperol with 3 symptotes: y = ± x nd focus: ( 10, 0 ). 4

6 6. Consider the hyperol. Find the center, foci, nd vertices of the hyperol, nd sketch its grph using symptotes s n id. x y = Find equtions for the i. Tngent lines t x = 10. ii. Norml lines t x = 10.

7 7. Clssify the grph of the eqution s circle, prol, n ellipse, or hyperol... c. d. e. 1x + x + y = 15 x x + y 5y = 1+ x x x + y = y + x = y x 8x + y + 5y = 6 8. Consider the prmetric eqution x = = tn θ nd y sec θ.. Eliminte the prmeter nd grph the prmetric eqution y hnd, indicting the orienttion.. Evlute dy dx.

8 PARAMETRIC FORM OF THE DERIVATIVE If smooth curve C is given y the equtions x f ( t) y g ( t) slope t C t ( x, y ) is dy dy dx = dt, 0. dx dx dt dt = nd =, then the HIGHER ORDER DERIVATIVES OF PARAMETRIC EQUATIONS If smooth curve C is given y the equtions x f ( t) y g ( t) slope t C t ( x, y ) is d dy d y dt dx dx =, 0. dx dx dt dt In generl we hve, n d y dx n = nd =, then the n 1 d d y n 1 dt dx dx =, 0. dx dt dt

9 π x = cos θ nd y = sin θ t θ =. 9. Consider the prmetric equtions, 4 dy. Find dx.. Find d y dx. c. Find the slope nd concvity t the given vlue of the prmeter.

10 10. Consider the curve x t 1, y t 3t = + = +.. Find ll points of horizontl tngency to the curve.. Find ll points of verticl tngency to the curve ARC LENGTH IN PARAMETRIC FORM If smooth curve C is given y the equtions x f ( t) y g ( t) tht C does not intersect itself on the intervl t = nd =, such (except possily t the endpoints), then the rc length of C over the intervl is given y dx dy s = dt f ( t) g ( t) dt. + = + dt dt NOTE: When pplying the rc length formul to curve, e sure tht the curve is trced only once on the intervl of integrtion.

11 11. Find the rc length of the curve 1 t. t t 5 =, = + 3 on the intervl x t y AREA OF A SURFACE OF REVOLUTION = nd =, does not If smooth curve C is given y the equtions x f ( t) y g ( t) cross itself on the intervl t, then the re S of the surfce of revolution formed y revolving C out the coordinte xes is given y the following. dx dy 1. S = π g ( t) dt Revolution out the x xis: g ( t) 0 + dt dt dx dy. S = π f ( t) dt Revolution out the y xis: f ( t ) 0 + dt dt

12 1. Find the re of the surfce generted y revolving the curve out the y- xis. π x = 4cos θ nd y = 4sin θ on the intervl 0 θ SLOPE IN POLAR FORM If f is differentile function ofθ, then the slope of the tngent line to the grph of r = f ( θ ) t the point (, ) r θ is dy dy dθ f ( θ ) cosθ + f ( θ ) sinθ dx = =, 0 t ( r, θ ). dx dx f ( θ ) sinθ + f ( θ ) cosθ dθ dθ TANGENT LINES AT THE POLE If f ( ) f ( ) α = 0 nd α 0, then the line θ = α is tngent t the pole to the grph of r = f ( θ ).

13 AREA IN POLAR COORDINATES α, β, 0 < β α π, If f is continuous nd nonnegtive on the intervl [ ] then the re of the region ounded y the grph of r f ( θ ) rdil lines of θ = α nd θ β = is given y 1 β A = f ( ) d θ θ α 1 = ARC LENGTH IN POLAR FORM β r d α θ. = etween the α θ β The Let f e function whose derivtive is continuous on n intervl. length of the grph of r f ( θ ) = from θ = α to θ β = is β β dr s = f ( ) + f ( ) d = r + d α α θ θ θ θ. dθ AREA OF A SURFACE OF REVOLUTION Let f e function whose derivtive is continuous on n intervl α θ β. re of the surfce formed y revolving the grph of r f ( θ ) to θ = β out the indicted line is s follows. β ( ) ( ) ( ) The = from θ = α 1. S = π f θ sin θ f θ + f θ dθ (out the polr xis) α β π. S = π f ( θ ) cos θ f ( θ ) + f ( θ ) dθ out the line θ = α

14 13. Find two sets of polr coordintes for the rectngulr coordinte ( 3, 3). 14. Consider the polr eqution r = 1+ sinθ.. Sketch grph of the polr eqution y hnd.

15 . Find ll points of horizontl nd verticl tngency. c. Find the tngents t the pole. d. Find the re of the interior.

16 e. Find the rc length of the curve over the intervl 0 θ π. f. Find the re of the surfce formed y revolving the curve out the polr xis over the intervl 0 θ π.

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