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Final_review

# Final_review - Final Review Edward Carter 1 1.1 Parametric...

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Final Review Edward Carter May 13, 2005 1 Parametric Equations and Polar Coordinates 1.1 Things to Know If dx dt = 0, dy dx = dy dt dx dt and d 2 y dx 2 = d dt ( dy dx ) dx dt . Higher derivatives are computed in a similar manner. For a parametric curve x = f ( t ), y = g ( t ), a t b , the area under the curve is given by b a g ( t ) f ( t ) dt . One thing to note about this formula is that it gives a number that depends on how many times your curve wraps around the area you’re trying to find, as well as which way (that is, clockwise or counterclockwise). The arclength of a parametric curve where a t b is given by b a dy dt 2 + dx dt 2 dt. If you’re trying to find the length of a closed curve, make sure you aren’t wrapping around multiple times. The surface area obtained by rotating a parametric curve with a t b is given by b a 2 πy dx dt 2 + dy dt 2 dt. Again, make sure you aren’t wrapping around multiple times in the case of a closed curve. Here, there’s also the danger that parts of your curve are symmetric about the x -axis. That can also cause this formula to give you numbers that aren’t what you’re looking for. To convert between polar to cartesian, x = r cos θ and y = sin θ . Use these formulas most of the time. To convert from cartesian to polar, r 2 = x 2 + y 2 and tan θ = y x . Use these formulas if you are given x and y in terms of a third variable such as t and you want to obtain r and θ in terms of t . Polar area, for a curve where a θ b , is given by b a 1 2 r 2 . Polar arclength, for a curve where a θ b , is given by b a r 2 + dr 2 dθ. 1

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1.2 Problems 1. (Practice Midterm 1 for Midterm 1) Describe the planar curve given parametrically by x = cosh t and y = sinh t using a relation between x and y . 2. (Practice Midterm 1 for Midterm 1) Sketch the curve r = 2 + 3 cos θ and find the area enclosed by its inner loop. 3. (Practice Midterm 2 for Midterm 1) Sketch the curve r = cos 2 θ and find the area enclosed by it. 4. (Chapter 10 Review) Sketch the curve x = tan t , y = cot t by eliminating t . 5. (Chapter 10 Review) Sketch the curve x = 1 + e 2 t , y = e t by eliminating t . 6. (Chapter 10 Review) Sketch the curve r = 1 - 3 sin θ and find the area enclosed by its inner loop. 7. (Chapter 10 Review) Find the points of intersection of the curves r = cot θ and r = 2 cos θ . 8. (Chapter 10 Review) Sketch the curve r = sin 4 θ and find the area it encloses. 9. Sketch the curve r 2 = sin 2 θ and find the area it encloses. 10. Sketch the curve r 2 θ = 1. 2 Vectors, Lines, and Planes 2.1 Things to Know a · a = | a | 2 . a · ( b + c ) = a · b + a · c . 0 · a = 0. a · b = b · a . ( c a ) · b = c ( a · b ) = a · ( c b ). If a and b are both nonzero and θ is the angle between them, then a · b = | a || b | cos θ . This fact is used to derive formulas for the scalar and vector projections, which you should also know.
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Final_review - Final Review Edward Carter 1 1.1 Parametric...

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