{"id":10140,"date":"2017-06-01t13:45:32","date_gmt":"2017-06-01t20:45:32","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=10140"},"modified":"2017-05-31t08:46:10","modified_gmt":"2017-05-31t15:46:10","slug":"ap-calculus-bc-review-polar-functions","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-bc-review-polar-functions\/","title":{"rendered":"ap calculus bc review: polar functions"},"content":{"rendered":"

polar functions show up on the ap calculus bc exam. while this topic shows up in only a handful of problems on any given ap exam, it is worth your while to learn about polar functions in order to maximize your score. <\/p>\n

remember, the higher your score on the ap calculus bc exam, the better chance you might have to receive college credits!<\/p>\n

\n
\"polar
this is a polar bear<\/em>, not a polar function<\/em>. knowing the difference could save your life or at least a few college credits. image by ansgar walk<\/a>.<\/figcaption><\/figure>\n<\/div>\n

polar coordinates<\/h2>\n

when you first learn graphing, you usually plot points (x<\/em>, y<\/em>) on a grid by starting at the origin, and then moving x<\/em> units to the right (or left if x<\/em> < 0) and y<\/em> units up (or down if y<\/em> < 0).<\/p>\n

this grid of x<\/em>– and y<\/em>-coordinates goes by the fancy name, cartesian plane<\/strong>.<\/p>\n

\"cartesian
the cartesian plane<\/figcaption><\/figure>\n

the cartesian plane is something like a map of city streets. all of the grid lines are straight, equally spaced, and meet at right angles.<\/p>\n

the polar plane<\/h3>\n

now imagine you’re at a research station in antarctica with no streets in sight. all you have for reference is your base camp and a particular direction that you decided to call angle 0, perhaps east on a compass.<\/p>\n

without streets to help you locate points, now you must rely on how far you are from base camp (call that r<\/em> units), and at what angle to the chosen angle 0 (say, θ<\/em> radians). <\/p>\n

the pair of numbers (r<\/em>, θ<\/em>) locates any particular point in the plane. polar coordinates for a polar research station!<\/p>\n

\"polar
polar graph paper. each circle represents a constant distance r<\/em> from the origin. each line represents an angle θ<\/em>.<\/figcaption><\/figure>\n

for example, the polar point (3.2, π\/2) means that you are exactly r<\/em> = 3.2 units away from base camp, in the direction of θ<\/em> = π\/2. we should mention here that the angle is always measured counterclockwise from the θ<\/em> = 0 line. therefore, if θ<\/em> = 0 corresponds to east, then π\/2 is north.<\/p>\n

in this example, it’s easy to see that (3.2, π\/2) in polar coordinates would correspond to (0, 3.2) in cartesian.<\/p>\n

conversion formulas<\/h3>\n

there are conversion formulas that help to change polar (r<\/em>, θ<\/em>) into cartesian (x<\/em>, y<\/em>), and vice versa. <\/p>\n

\"polar<\/p>\n

these formulas are based on a little trigonometry.<\/p>\n

\"polar
polar coordinates are related to cartesian coordinates (x, y) through simple trigonometric formulas (image by p. wormer<\/a>)<\/figcaption><\/figure>\n

plotting in polar<\/h2>\n

a polar function<\/strong> is an equation of the form r<\/em> = f<\/em>(θ<\/em>). for every θ<\/em>-value in the domain of f<\/em>, you find the corresponding r<\/em>-value by plugging θ<\/em> into the function.<\/p>\n

it’s really the same idea as plugging in various x<\/em>-values into a typical (cartesian) function to find the y<\/em>-values. the only difference is that for a polar function, your next step would be to plot the polar<\/em> points, (r<\/em>, θ<\/em>).<\/p>\n

example — graphing a polar function<\/h3>\n

let’s start with a nice easy polar function, r<\/em> = 1 + cos θ<\/em>. first observe that you only need to work out what happens when θ<\/em> is in the interval [0, 2π]. this is because the cosine function is periodic<\/em> with period 2π.<\/p>\n

let’s build a table of values. for θ<\/em>, i typically choose angles that are easy to work with.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
θ<\/em><\/th>\nr<\/em> = 1 + cos θ<\/em><\/th>\npolar point (r<\/em>, θ<\/em>)<\/th>\n<\/tr>\n<\/thead>\n
0<\/td>\n2<\/td>\n(2, 0)<\/td>\n<\/tr>\n
π\/6<\/td>\n1.866<\/td>\n(1.866, π\/6)<\/td>\n<\/tr>\n
π\/4<\/td>\n1.707<\/td>\n(1.707, π\/4)<\/td>\n<\/tr>\n
π\/3<\/td>\n1.5<\/td>\n(1.5, π\/3)<\/td>\n<\/tr>\n
π\/2<\/td>\n1<\/td>\n(1, π\/2)<\/td>\n<\/tr>\n
2π\/3<\/td>\n0.5<\/td>\n(0.5, 2π\/3)<\/td>\n<\/tr>\n
3π\/4<\/td>\n0.293<\/td>\n(0.293, 3π\/4)<\/td>\n<\/tr>\n
5π\/6<\/td>\n0.134<\/td>\n(0.134, 5π\/6)<\/td>\n<\/tr>\n
π<\/td>\n0<\/td>\n(0, π)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

<\/p>\n

notice that my θ<\/em> did not go all the way to 2π. we’ll talk about why in a moment. for now though, let’s plot those points.<\/p>\n

\"sample
plot of sample points for r<\/em> = 1 + cos θ<\/em> in the domain [0, π]<\/figcaption><\/figure>\n

the general shape should be clear. we’ll draw a smooth curve starting at polar point (2, 0) and connecting the dots in a counterclockwise direction.<\/p>\n

\"polar<\/p>\n

now to finish our graph we could list out the values of r<\/em> for π < θ<\/em> ≤ 2π repeating the same procedure as above.<\/p>\n

or<\/em>, we could use our knowledge of the cosine function and save a lot of work!<\/p>\n

remember that cos x<\/em> has mirror symmetry about x<\/em> = π. this means that the values of cos x<\/em> in the interval [π, 2π] will simply rise back up like a mirror image of those in [0, π].<\/p>\n

it may be harder to see what happens in the polar plot, but just imagine the curve bouncing back outward as you complete the journey around the full circle. take a look at the complete graph below.<\/p>\n

\"graph
this graph is called a cardioid<\/strong> becausee of its resemblance to a heart.<\/figcaption><\/figure>\n

what about the calculator?<\/h3>\n

graphing by hand is, of course, very time consuming. however if you’re working in a section of the exam that allows a graphing calculator, then i have good news for you.<\/p>\n

your calculator understands polar functions!<\/p>\n

on most graphing calculators there is setting that puts you into polar mode. then whatever you graph will be interpreted as a polar function.<\/p>\n

derivatives of polar functions<\/h2>\n

this wouldn’t be calculus unless we started talking about derivatives<\/em>!<\/p>\n

suppose you want to find the slope of a polar curve. then the following derivative formula is what you need.<\/p>\n

\"polar<\/p>\n

example — slope of a polar function<\/h3>\n

consider the cardioid function, r<\/em> = 1 + cos θ<\/em>. what is the slope at θ<\/em> = π\/4?<\/p>\n

let’s use the formula to find out. here, f<\/em>(θ<\/em>) = 1 + cos θ<\/em>.<\/p>\n

\"polar<\/p>\n

polar area formula<\/h2>\n

finally, you can use the following formula to work out the area within a polar curve.<\/p>\n

\"polar<\/p>\n

typically on the ap calculus bc exam, a question may ask for the proper setup of the area integral. on the other hand, if you are in a calculator-permitted section, then you can easily find the area by numerical integration.<\/p>\n

example — area of the cardioid<\/h3>\n

let’s use our running example and find the area within the cardioid. remember, f<\/em>(θ<\/em>) = 1 + cos θ<\/em> describes the cardioid for 0 ≤ θ<\/em> ≤ 2π.<\/p>\n

\"polar<\/p>\n

summary<\/h2>\n

here are a few points to remember about polar functions.<\/p>\n

    \n
  • this topic only shows up on the ap calculus bc exam.<\/li>\n
  • know how to plot polar points (r<\/em>, θ<\/em>) as well as sketch polar functions r<\/em> = f<\/em>(θ<\/em>).\n<\/li>\n
  • know the polar derivative formula (for finding slope).<\/li>\n
  • know how to setup the polar area formula.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"

    polar functions show up on the ap calculus bc exam. learn about polar functions and maximize your score on the the exam by reading this review!<\/p>\n","protected":false},"author":223,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[240],"tags":[241],"ppma_author":[24932],"acf":[],"yoast_head":"\nap calculus bc review: polar functions - magoosh blog | high school<\/title>\n<meta name=\"description\" content=\"polar functions show up on the ap calculus bc exam. learn about polar functions and maximize your score on the the exam by reading this review!\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-bc-review-polar-functions\/\" \/>\n<meta property=\"og:locale\" content=\"en_us\" \/>\n<meta property=\"og:type\" content=\"article\" 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