{"id":8390,"date":"2017-01-10t13:38:27","date_gmt":"2017-01-10t21:38:27","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=8390"},"modified":"2017-01-04t21:02:11","modified_gmt":"2017-01-05t05:02:11","slug":"ap-calculus-review-newtons-method","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-newtons-method\/","title":{"rendered":"ap calculus review: newton\u2019s method"},"content":{"rendered":"

often when we are trying to find the roots of a function, the algebraic methods we learned in earlier math classes are either tedious or impossible. newton’s method allows us to overcome this. imagine trying to find the roots of f(x) = x4<\/sup> – 3x2<\/sup> + 2x – 1.<\/em> we know that the equation has either 0, 2, or 4 real roots, although just looking at it, this would not be obvious.<\/p>\n

\"ap<\/p>\n

the graph shows us that the equation indeed has 2 roots, but we are still not sure what these roots are (although our graphing calculator can solve this for us; see our post on calculator strategies for the ap calculus exam<\/a>\u00a0for more).<\/p>\n

newton\u2019s method is an iterative method to find approximate roots of equations.<\/h2>\n

newton\u2019s method usually does not give the exact answer, but will allow us to find very exact approximations. the fist requirement for newton\u2019s method is that we know the derivative of the function.<\/p>\n

let\u2019s walk through an example to show where newton\u2019s method comes from.<\/p>\n

\"screen-shot-2016-12-30-at-1-07-01-pm\"<\/p>\n

first step:<\/strong> take a random guess as to what the root might be. let us choose x = 2 for this first guess. we’ll call this x0<\/sub>. depending on our initial guess, our method might find the first or the second root.<\/p>\n

second step:<\/strong>\u00a0find the equation<\/a> of a line tangent to the curve at the point x0<\/sub>.<\/p>\n

\"screen-shot-2016-12-30-at-1-13-12-pm\" \"screen-shot-2016-12-30-at-1-13-16-pm\"<\/p>\n

\"screen-shot-2016-12-29-at-2-07-32-pm\"<\/p>\n

notice our tangent line has its own root\u00a0close to the root of our original equation. it is easy to find the root of a linear function. if we take the root of y = 22x-37, we get 37\/22, which is about 1.682. this isn\u2019t a perfect approximation, but it\u2019s close. if we were to repeat this entire method, using x = 1.682 instead of x =2, we would get an even closer approximation.<\/p>\n

now there is a quick formula that you can derive that gives us our sequence of increasingly accurate approximations.<\/p>\n

\"screen-shot-2016-12-30-at-1-07-16-pm\"<\/p>\n

plugging in x0<\/sub> = 2, we get x1<\/sub> = 1.682, exactly what we found above. if we do this a few times, we see that we get increasingly close to our root:<\/p>\n

x0<\/sub> = 1.682
\nx1<\/sub> = 1.51
\nx2<\/sub> = 1.454
\nx3<\/sub> = 1.448<\/p>\n

each step brings us closer and closer to the root. we\u2019ll never get perfectly there in this example, but within 4 steps, we\u2019ve within .001. a few more steps, and we\u2019d be within millionths of the correct answer.<\/p>\n

newton\u2019s method is an extremely efficient way of finding approximate roots to equations. it works even if the equation is incredibly complicated or would be impossible or difficult to algebraically find exact roots. it rarely gets an exact correct answer, but allows us to get very close. numerical methods, such as newton\u2019s method, for finding roots are the way many computer programs (including many graphing calculators) find answers to equations. the requirement for newton\u2019s method is that you know the derivative of the function.<\/p>\n

now let\u2019s practice:<\/p>\n

take f(x) = x2<\/sup> – 9<\/em>. you know the answer to this equation is +\/- 3. try newton\u2019s method with this equation to see how many iterations it takes to get within a few thousands of the correct answer.<\/p>\n","protected":false},"excerpt":{"rendered":"

newton’s method is a fantastic process for approximating roots of equations. click here to see where it comes from, and how to use it on the ap calc exam.<\/p>\n","protected":false},"author":224,"featured_media":8391,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[240],"tags":[241],"ppma_author":[24930],"acf":[],"yoast_head":"\nap calculus review: newton\u2019s method - magoosh blog | high school<\/title>\n<meta name=\"description\" content=\"newton's method is a fantastic process for approximating roots of equations. click here to see where it comes from, and how to use it on the ap calc exam.\" \/>\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-review-newtons-method\/\" \/>\n<meta property=\"og:locale\" content=\"en_us\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"ap calculus review: newton\u2019s method\" \/>\n<meta property=\"og:description\" content=\"newton's method is a fantastic process for approximating roots of equations. click here to see where it comes from, and how to use it on the ap calc exam.\" \/>\n<meta property=\"og:url\" content=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-newtons-method\/\" \/>\n<meta property=\"og:site_name\" content=\"magoosh blog | high school\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/magooshsat\/\" \/>\n<meta property=\"article:published_time\" content=\"2017-01-10t21:38:27+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2017-01-05t05:02:11+00:00\" \/>\n<meta property=\"og:image\" content=\"\/\/www.catharsisit.com\/hs\/files\/2016\/12\/screen-shot-2016-12-29-at-1.54.13-pm.png\" \/>\n\t<meta property=\"og:image:width\" content=\"379\" \/>\n\t<meta property=\"og:image:height\" content=\"449\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/png\" \/>\n<meta name=\"author\" content=\"zachary\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@magooshsat_act\" \/>\n<meta name=\"twitter:site\" content=\"@magooshsat_act\" \/>\n<meta name=\"twitter:label1\" content=\"written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"zachary\" \/>\n\t<meta name=\"twitter:label2\" content=\"est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"article\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-newtons-method\/#article\",\"ispartof\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-newtons-method\/\"},\"author\":{\"name\":\"zachary\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/2bb580c8b05f06690589778ec3805636\"},\"headline\":\"ap calculus review: newton\u2019s method\",\"datepublished\":\"2017-01-10t21:38:27+00:00\",\"datemodified\":\"2017-01-05t05:02:11+00:00\",\"mainentityofpage\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-newtons-method\/\"},\"wordcount\":524,\"commentcount\":0,\"publisher\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/#organization\"},\"keywords\":[\"ap calculus\"],\"articlesection\":[\"ap\"],\"inlanguage\":\"en-us\",\"potentialaction\":[{\"@type\":\"commentaction\",\"name\":\"comment\",\"target\":[\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-newtons-method\/#respond\"]}]},{\"@type\":\"webpage\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-newtons-method\/\",\"url\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-newtons-method\/\",\"name\":\"ap calculus review: newton\u2019s method - 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