{"id":10140,"date":"2017-06-01T13:45:32","date_gmt":"2017-06-01T20:45:32","guid":{"rendered":"https:\/\/magoosh.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":"https:\/\/magoosh.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],"class_list":["post-10140","post","type-post","status-publish","format-standard","hentry","category-ap","tag-ap-calculus"],"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. 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