f sin the slope of the tangent line is given by f sin f cos cos sin , but it s also given by 2. So solve the DE with initial condition: sin cos

Transcription

1 Math 414 Activity 1 (Due by end of class August 1) 1 Four bugs are placed at the four corners of a square with side length a The bugs crawl counterclockwise at the same speed and each bug crawls directly toward the next bug at all times They approach the center of the square along spiral paths a) Find the polar equation of a bug s path assuming that the center of the square is the origin Notice that each bug s path is the previous bug s path rotated by radians, so let s just find the path of the bug that starts in the first quadrant Let s call its path r f At the point f cos, f sin the slope of the tangent line is given by dy f sin f cos, but dx f cos f sin it s also given by f sin f sin cos sin So solve the DE with initial condition: sin cos f cos f cos f sin f cos cos sin a ; f f cos f sin sin cos 4 If we cross-multiply, we get f sin f sin cos f sin cos f cos Canceling and combining f cos f sin cos f sin cos f sin common terms on both sides leads to f f now you need to solve the DE f f b) What s lim f? sin cos sin cos 0 So with the initial condition c) Find the total distance traveled by a bug as it makes its way towards the origin a f 4

2 a) Attempt the same problem as #1, except that there are three bugs at the corners of an equilateral triangle of side length a centered at the origin with one vertex along the positive x-axis Notice that each bug s path is the previous bug s path rotated by radians, so let s just find the path of the bug that starts on the x-axis Let s call its path r f At the point f cos, f sin the slope of the tangent line is given by f dy f sin f sin cos f sin, but it s also given by dx f cos f sin f cos f cos sin x y sin xcos y cos xsin y and Simplify using the identities: cos x y cos xcos y sin xsin y, leads to f sin f cos f sin f cos f cos f sin f cos f sin use the initial condition a f 0 to solve Now cross-multiply, simplify, and b) Find the total distance traveled by a bug as it makes its way towards the origin

3 Find the area of the intersection of the two circles r acos and r asin, where a is a positive constant 4 Find the area inside the circle r but outside the circles r cos and r cos {No calculus is required or recommended!}

4 5 A farmer has a fenced circular pasture of radius a and wants to tie a cow to the fence with a rope of length b so as to allow the cow to graze half the pasture How long should the rope be to accomplish this? a b a The length of the rope, b, must be longer than a and shorter than a, ie a b a To find the area of the grazing region, we can use polar coordinates: sin 1 b a

5 1 b 1 sin b sin a a 1 1 The grazing area 4a sin d b d 4a sin d b d 0 1 b 1 sin 0 b sin a a a We want this to equal half the pasture area which is, so we get the equation 1 b sin a a 4a sin d b d If we multiply both sides by a and perform the 0 1 b sin a b 1 b b b b integrations, we arrive at the equation 4 sin 4 a a a a a a) Verify the previous equation b 1x If we let x, we get the simplified equation 4 x sin x 4 x x, and a we re looking for the solution x, with 1x Let s rearrange it into 1x 4 x sin x 4 x x 0 Here s a plot of the left side of the equation with 1x

6 b) Approximate the solution by performing the Bisection Method on the interval 1, : Left Endpoint(sign) Midpoint(sign) Right Endpoint(sign) Error Bound 5 1(-) 4 (+) (+) 5 1(-) 9 8 (-) 5 4 (+) 15 So this means that the length of the rope, b, should approximately equal a times your best estimate of the solution: {Check out the Bisection Worksheet link on the Course Webpage! There s no arcsine in Visual Basic, so use Atn 4Atn 1 } x 1 x, and also 6 Find the area bounded by the loop of the folium of Descartes with equation x y xy Start by converting the equation to polar coordinates and then use the substitution 1 utan tan u {Hint: 4 sin cos sin cos sin cos sin cos 4 sin cos 6 cos sin cos cos tan 1 tan sec sec sec }

7 7 The line r 4sec is tangent to the graph of r a1 cos Find a {Hint: Consider that a can be positive or negative} 8 Find the vertices of the triangle determined by the three lines 0, tan 1, and 9 Find the area of the triangle determined by the three lines tan, tan, and r 4csc 10 Consider the polar equation r sintan a) The polar equation is equivalent to the parametric equations: x sin tan cos sin sin, n y sin tan sin cos Show that the entire graph of r sintan lies within the vertical strip 0x 1 r 4sec b) Show that the vertical line x 1 is a vertical asymptote for the graph of r sintan sin sin {Hint: Consider lim sin and lim, as well as lim sin and lim } cos cos 11 The length of paper on a tightly wound roll needs to be determined The roll of paper has an inner radius of cm, an outer radius of 5 cm, and the thickness of the paper is 1mm A polar function r f whose graph models the coiled paper is given by a spiral whose radius increases by 1 cm for each rotation of radians So r 1 ;0 40 Determine the length of this polar curve to estimate the length of the coiled paper

8 x 1 The formula for the area of an ellipse, a x a cos t equations of the ellipse: y bsin t y b 1, can be derived from the parametric ;0 t The area is given by b t a t dt ab tdt ab tdt ab Or in other words, the sin sin sin 1 cos area of an ellipse is the product of the lengths of the major and minor axes and Interpret the integral to find its value {Hint: If d as the area of a certain ellipse containing the origin in order cos 0 d is the polar integral for the area inside an ellipse containing the cos 0 1 origin, then r and r cos cos This gives you the polar equation of the ellipse, so use it to determine the lengths of the major and minor axes x y x x y x }

9 1 Find as many of the five points of intersection of the two curves r 1 sin and r 1 cos as you can You can easily see four of the intersection points from the graph To find the fifth intersection point, zoom-in at the origin 14 Find the length of the curve r cos {Hint: x sin 1 1 cos x }

10 15 Find the length of the curve r sin Approximate, if necessary(right, Left, Mid, Trap, Simpson) 16 Suppose that the polar curve r f with has f continuous and length L Find the length of the polar curve r cf with in terms of L, for any constant c {Hint: Be careful, c could be negative} 17 Suppose that the polar curve r f with and f continuous encloses a region of area A Find the area of the region enclosed by the polar curve r cf with in terms of A, for any constant c {Hint: Be careful, c could be negative} 18 a) Show that the polar equation r acos bsin, for a and b real numbers not both zero, describes a circle {Hint: Multiply both sides by r and use x rcos, y rsin, and x y r } b) Find formulas for the center and radius of such a circle c) Find a formula for the area enclosed by such a circle 19 a) For two points with rectangular coordinates x, y and, 1 1 distance between the points is d x x1 y y1 for two points with polar coordinates r and r, x y, the formula for the Derive the distance formula 1, 1 {Hint: Use the rectangular distance formula, the fact that x rcos and y rsin, cos cos cos sin sin } and the identity

11 b) Use the distance formula in polar coordinates to find the distance between the two points with polar coordinates 1, and, 0 The parabola y x divides the area of the circle x y into two parts Find the area of the smaller part {Hint: The polar equation of the circle is r, and the polar equation of the parabola is r sec tan} 1 Find the area inside the lemniscate r 4sin, but outside the lemniscate r 4sin

12 Find the volume generated by revolving the area enclosed by the cardioid r 1 cos about the x-axis The graph of the polar coordinate equation r csc 4 is called a conchoid of Nicomedes The graph is unbounded, but it contains a closed loop Find the area of the region inside the loop 5

13 4 Consider the lemniscate r 4cos The curve has two tangent lines at the origin Find the slopes of the tangent lines at the origin as follows: a) Use the parametric formula for dy dx of a polar curve, and find limit as 4 and 4 b) Convert to rectangular coordinates: 4 r 4r cos 4 r cos r sin x y 4 x y, so divide through by y y dy y 0 y to get x y y 41 Notice that lim lim, so let x x x 0 0 dx 0,0 x 0 x x y y xy, 0 in x y y 41 to get the result x x x c) Use implicit differentiation in x y 4 x y to get the result along with dy y 0 y lim lim dx x0 0 0,0 x 0 x x