{"id":12430,"date":"2018-08-03t17:18:37","date_gmt":"2018-08-04t00:18:37","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=12430"},"modified":"2018-08-03t17:18:37","modified_gmt":"2018-08-04t00:18:37","slug":"linear-approximation-ap-calculus-exam-review","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/","title":{"rendered":"linear approximation: ap calculus exam review"},"content":{"rendered":"

what is linear approximation? basically, it’s a method from calculus used to “straighten out” the graph of a function near a particular point. scientists often use linear approximation to understand complicated relationships among variables.<\/p>\n

in this review article, we’ll explore the methods and applications of linear approximation. we’ll also take a look at plenty of examples along the way to better prepare you for the ap calculus exams.<\/p>\n

linear approximation and tangent lines<\/h2>\n

by definition the linear approximation<\/strong> for a function f<\/em>(x<\/em>) at a point x<\/em> = a<\/em> is simply the equation of the tangent line<\/em> to the curve at that point. and that means that derivatives<\/em> are key! (check out how to find the slope of a line tangent to a curve<\/a> or is the derivative of a function the tangent line?<\/a> for some background material.)<\/p>\n

\"three
three different tangent lines for a curve y<\/em> = f<\/em>(x<\/em>).<\/figcaption><\/figure>\n

formula for the linear approximation<\/h3>\n

given a point x<\/em> = a<\/em> and a function f<\/em> that is differentiable at a<\/em>, the linear approximation<\/strong> l<\/em>(x<\/em>) for f<\/em> at x<\/em> = a<\/em> is:<\/p>\n

l<\/em>(x<\/em>) = f<\/em>(a<\/em>) + f<\/em> '(a<\/em>)(x<\/em> – a<\/em>)<\/p>\n

the main idea behind linearization is that the function l<\/em>(x<\/em>) does a pretty good job approximating values of f<\/em>(x<\/em>), at least when x<\/em> is near a<\/em>.<\/p>\n

in other words, l<\/em>(x<\/em>) ≈ f<\/em>(x<\/em>) whenever x<\/em> ≈ a<\/em>.<\/p>\n

example 1 — linearizing a parabola<\/h3>\n

find the linear approximation of the parabola f<\/em>(x<\/em>) = x<\/em>2<\/sup> at the point x<\/em> = 1.<\/p>\n

  a. x<\/em>2<\/sup> + 1<\/p>\n

  b. 2x<\/em> + 1<\/p>\n

  c. 2x<\/em> – 1<\/p>\n

  d. 2x<\/em> – 2<\/p>\n

solution<\/h4>\n

c<\/strong>. <\/p>\n

note that f<\/em> '(x<\/em>) = 2x<\/em> in this case. using the formula above with a<\/em> = 1, we have:<\/p>\n

l<\/em>(x<\/em>) = f<\/em>(1) + f<\/em> '(1)(x<\/em> – 1)<\/p>\n

l<\/em>(x<\/em>) = 12<\/sup> + 2(1)(x<\/em> – 1) = 2x<\/em> – 1<\/p>\n

follow-up: interpreting the results<\/h4>\n

clearly, the graph of the parabola f<\/em>(x<\/em>) = x<\/em>2<\/sup> is not a straight line. however, near any particular point, say x<\/em> = 1, the tangent line does a pretty good job following the direction of the curve. <\/p>\n

\"graph
the parabola y<\/em> = x<\/em>2<\/sup> can be approximated by the line y<\/em> = 2x<\/em> – 1 when x<\/em> ≈ 1.<\/figcaption><\/figure>\n

how good is this approximation? well, at x<\/em> = 1, it’s exact! l<\/em>(1) = 2(1) – 1 = 1, which is the same as f<\/em>(1) = 12<\/sup> = 1.<\/p>\n

but the further away you get from 1, the worse the approximation becomes.<\/p>\n\n\n\n\n\n\n\n\n
x<\/em><\/th>\nf<\/em>(x<\/em>) = x<\/em>2<\/sup><\/th>\nl<\/em>(x<\/em>) = 2x<\/em> – 1<\/th>\n<\/tr>\n<\/thead>\n
1.1<\/td>\n1.21<\/td>\n1.2<\/td>\n<\/tr>\n
1.2<\/td>\n1.44<\/td>\n1.4<\/td>\n<\/tr>\n
1.5<\/td>\n2.25<\/td>\n2<\/td>\n<\/tr>\n
2<\/td>\n4<\/td>\n3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

<\/p>\n

approximating using differentials<\/h2>\n

the formula for linear approximation can also be expressed in terms of differentials<\/strong>. basically, a differential is a quantity that approximates a (small) change in one variable due to a (small) change in another. the differential of x<\/em> is dx<\/em>, and the differential of y<\/em> is dy<\/em>. <\/p>\n

based upon the formula dy<\/em>\/dx<\/em> = f<\/em> '(x<\/em>), we may identify:<\/p>\n

dy<\/em> = f<\/em> '(x<\/em>) dx<\/em><\/p>\n

the related formula allows one to approximate near a particular fixed point:<\/p>\n

f<\/em>(x<\/em> + dx<\/em>) ≈ y<\/em> + dy<\/em><\/p>\n

example 2 — using differentials with limited information<\/h3>\n

suppose g<\/em>(5) = 30 and g<\/em> '(5) = -3. estimate the value of g<\/em>(7).<\/p>\n

  a. 24<\/p>\n

  b. 27<\/p>\n

  c. 28<\/p>\n

  d. 33<\/p>\n

solution<\/h4>\n

a<\/strong>. <\/p>\n

in this example, we do not know the expression for the function g<\/em>. fortunately, we don’t need to know!<\/p>\n

first, observe that the change in x<\/em> is dx<\/em> = 7 – 5 = 2.<\/p>\n

next, estimate the change in y<\/em> using the differential formula.<\/p>\n

dy<\/em> = g<\/em> '(x<\/em>) dx<\/em> = g<\/em> '(5) · 2 = (-3)(2) = -6.<\/p>\n

finally, put it all together:<\/p>\n

g<\/em>(5 + 2) ≈ y<\/em> + dy<\/em> = g<\/em>(5) + (-6) = 30 + (-6) = 24<\/p>\n

example 3 — using differentials to approximate a value<\/h3>\n

approximate \"cube_root_1.1\" using differentials. express your answer as a decimal rounded to the nearest hundred-thousandth.<\/p>\n

solution<\/h4>\n

1.03333<\/strong>. <\/p>\n

here, we should realize that even though the cube root of 1.1 is not easy to compute without a calculator, the cube root of 1 is trivial. so let’s use a<\/em> = 1 as our basis for estimation.<\/p>\n

consider the function \"cube_root_x\". find its derivative (we’ll need it for the approximation formula).<\/p>\n

\"derivative_of_cube_root_x\"<\/p>\n

then, using the differential, \"differential_of_cube_root_x\", we can estimate the required quantity.<\/p>\n

\"estimating_cube_root_1.1\"<\/p>\n

exact change versus approximate change<\/h2>\n

sometimes we are interested in the exact<\/em> change of a function’s values over some interval. suppose x<\/em> changes from x<\/em>1<\/sub> to x<\/em>2<\/sub>. then the exact change<\/strong> in f<\/em>(x<\/em>) on that interval is:<\/p>\n

δy<\/em> = f<\/em>(x<\/em>2<\/sub>) – f<\/em>(x<\/em>1<\/sub>)<\/p>\n

we also use the “delta” notation for change in x<\/em>. in fact, δx<\/em> and dx<\/em> typically mean the same thing:<\/p>\n

δx<\/em> = dx<\/em> = x<\/em>2<\/sub> – x<\/em>1<\/sub><\/p>\n

however, while δy<\/em> measures the exact change in the function’s value, dy<\/em> only estimates the change based on a derivative value.<\/p>\n

\"linear<\/p>\n

example 4 — comparing exact and approximate values<\/h3>\n

let f<\/em>(x<\/em>) = cos(3x), and let l<\/em>(x<\/em>) be the linear approximation to f<\/em> at x<\/em> = π\/6. which expression represents the absolute error in using l<\/em> to approximate f<\/em> at x<\/em> = π\/12?<\/p>\n

  a. π\/6 – √2<\/span>\/2<\/p>\n

  b. π\/4 – √2<\/span>\/2<\/p>\n

  c. √2<\/span>\/2 – π\/6<\/p>\n

  d. √2<\/span>\/2 – π\/4<\/p>\n

solution<\/h4>\n

b.<\/strong><\/p>\n

absolute error<\/strong> is the absolute difference between the approximate and exact values, that is, e<\/em> = | f<\/em>(a<\/em>) – l<\/em>(a<\/em>) |.<\/p>\n

equivalently, e<\/em> = | δy<\/em> – dy<\/em> |.<\/p>\n

let’s compute dy<\/em> ( = f<\/em> '(x<\/em>) dx<\/em> ). here, the change in x<\/em> is negative. dx<\/em> = π\/12 – π\/6 = -π\/12. note that by the chain rule, we obtain: f<\/em> '(x<\/em>) = -3 sin(3x<\/em>). putting it all together,<\/p>\n

dy<\/em> = -3 sin(3 π\/6 ) (-π\/12) = -3 sin(π\/2) (-π\/12) = 3π\/12 = π\/4<\/p>\n

ok, next we compute the exact change.<\/p>\n

δy<\/em> = f<\/em>(π\/12) – f<\/em>(π\/6) = cos(π\/4) – cos(π\/2) = √2<\/span>\/2<\/p>\n

lastly, we take the absolute difference to compute the error, <\/p>\n

e<\/em> = | √2<\/span>\/2 – π\/4 | = π\/4 – √2<\/span>\/2.<\/p>\n

application — finding zeros<\/h2>\n

linear approximations also serve to find zeros of functions. in fact newton’s method<\/em> (see ap calculus review: newton’s method<\/a> for details) is nothing more than repeated linear approximations to target on to the nearest root of the function.<\/p>\n

the method is simple. given a function f<\/em>, suppose that a zero for f<\/em> is located near x<\/em> = a<\/em>. just linearize f<\/em> at x<\/em> = a<\/em>, producing a linear function l<\/em>(x<\/em>). then the solution to l<\/em>(x<\/em>) = 0 should be fairly close to the true zero of the original function f<\/em>.<\/p>\n

example 5 — estimating zeros<\/h3>\n

estimate the zero of the function \"polynomial_function\" using a tangent line approximation at x<\/em> = -1.<\/p>\n

  a. -1.48<\/p>\n

  b. -1.53<\/p>\n

  c. -1.62<\/p>\n

  d. -1.71<\/p>\n

solution<\/h4>\n

d<\/strong>.<\/p>\n

remember, the purpose of this question is to estimate<\/em> the zero. first of all, the tangent line approximation is nothing more than a linearization. we’ll need to know the derivative:<\/p>\n

\"polynomial_function_derivative\"<\/p>\n

then find the expression for l<\/em>(x<\/em>). note that g<\/em>(-1) ≈ 2.13 and g<\/em> '(-1) = 3.<\/p>\n

l<\/em>(x<\/em>) = 2.13 + 3(x<\/em> – (-1)) = 5.13 + 3x<\/em>.<\/p>\n

finally to find the zero, set l<\/em>(x<\/em>) = 0 and solve for x<\/em><\/p>\n

0 = 5.13 + 3x<\/em>  →  x<\/em> = -5.13\/3 = -1.71<\/p>\n","protected":false},"excerpt":{"rendered":"

what is linear approximation? basically, it’s a method from calculus used to “straighten out” the graph of a function near a particular point. scientists often use linear approximation to understand complicated relationships among variables. in this review article, we’ll explore the methods and applications of linear approximation. we’ll also take a look at plenty of […]<\/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-12430","post","type-post","status-publish","format-standard","hentry","category-ap","tag-ap-calculus"],"acf":[],"yoast_head":"\nlinear approximation: ap calculus exam review - magoosh blog | high school<\/title>\n<meta name=\"description\" content=\"in this review article, we'll take a look at plenty of examples of linear approximation to better prepare you for the ap calculus exams.\" \/>\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\/linear-approximation-ap-calculus-exam-review\/\" \/>\n<meta property=\"og:locale\" content=\"en_us\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"linear approximation: ap calculus exam review\" \/>\n<meta property=\"og:description\" content=\"in this review article, we'll take a look at plenty of examples of linear approximation to better prepare you for the ap calculus exams.\" \/>\n<meta property=\"og:url\" content=\"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/\" \/>\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=\"2018-08-04t00:18:37+00:00\" \/>\n<meta property=\"og:image\" content=\"\/\/www.catharsisit.com\/hs\/files\/2017\/01\/graph_with_tangent_lines.png\" \/>\n<meta name=\"author\" content=\"shaun ault\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@shaunaultmath\" \/>\n<meta name=\"twitter:site\" content=\"@magooshsat_act\" \/>\n<meta name=\"twitter:label1\" content=\"written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"shaun ault\" \/>\n\t<meta name=\"twitter:label2\" content=\"est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"6 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"article\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/#article\",\"ispartof\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/\"},\"author\":{\"name\":\"shaun ault\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f\"},\"headline\":\"linear approximation: ap calculus exam review\",\"datepublished\":\"2018-08-04t00:18:37+00:00\",\"datemodified\":\"2018-08-04t00:18:37+00:00\",\"mainentityofpage\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/\"},\"wordcount\":1147,\"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\/linear-approximation-ap-calculus-exam-review\/#respond\"]}]},{\"@type\":\"webpage\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/\",\"url\":\"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/\",\"name\":\"linear approximation: ap calculus exam review - magoosh blog | high school\",\"ispartof\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/#website\"},\"datepublished\":\"2018-08-04t00:18:37+00:00\",\"datemodified\":\"2018-08-04t00:18:37+00:00\",\"description\":\"in this review article, we'll take a look at plenty of examples of linear approximation to better prepare you for the ap calculus exams.\",\"breadcrumb\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/#breadcrumb\"},\"inlanguage\":\"en-us\",\"potentialaction\":[{\"@type\":\"readaction\",\"target\":[\"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/\"]}]},{\"@type\":\"breadcrumblist\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/#breadcrumb\",\"itemlistelement\":[{\"@type\":\"listitem\",\"position\":1,\"name\":\"home\",\"item\":\"\/\/www.catharsisit.com\/hs\/\"},{\"@type\":\"listitem\",\"position\":2,\"name\":\"linear approximation: ap calculus exam review\"}]},{\"@type\":\"website\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#website\",\"url\":\"\/\/www.catharsisit.com\/hs\/\",\"name\":\"magoosh blog | high school\",\"description\":\"act, sat, college admissions, life\",\"publisher\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/#organization\"},\"potentialaction\":[{\"@type\":\"searchaction\",\"target\":{\"@type\":\"entrypoint\",\"urltemplate\":\"\/\/www.catharsisit.com\/hs\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inlanguage\":\"en-us\"},{\"@type\":\"organization\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#organization\",\"name\":\"magoosh\",\"url\":\"\/\/www.catharsisit.com\/hs\/\",\"logo\":{\"@type\":\"imageobject\",\"inlanguage\":\"en-us\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/logo\/image\/\",\"url\":\"\/\/www.catharsisit.com\/hs\/files\/2019\/02\/magoosh-logo-purple-60h.png\",\"contenturl\":\"\/\/www.catharsisit.com\/hs\/files\/2019\/02\/magoosh-logo-purple-60h.png\",\"width\":265,\"height\":60,\"caption\":\"magoosh\"},\"image\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/logo\/image\/\"},\"sameas\":[\"https:\/\/www.facebook.com\/magooshsat\/\",\"https:\/\/twitter.com\/magooshsat_act\"]},{\"@type\":\"person\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f\",\"name\":\"shaun ault\",\"image\":{\"@type\":\"imageobject\",\"inlanguage\":\"en-us\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/image\/7c80c2046678e19beb5b9c8401d56613\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/28c36635b8fed6717752755f46cda239?s=96&d=mm&r=g\",\"contenturl\":\"https:\/\/secure.gravatar.com\/avatar\/28c36635b8fed6717752755f46cda239?s=96&d=mm&r=g\",\"caption\":\"shaun ault\"},\"description\":\"shaun earned his ph. d. in mathematics from the ohio state university in 2008 (go bucks!!). he received his ba in mathematics with a minor in computer science from oberlin college in 2002. in addition, shaun earned a b. mus. from the oberlin conservatory in the same year, with a major in music composition. shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!\",\"sameas\":[\"http:\/\/valdosta.academia.edu\/shaunault\",\"https:\/\/twitter.com\/shaunaultmath\"],\"url\":\"\/\/www.catharsisit.com\/hs\/author\/shaunault\/\"}]}<\/script>\n<!-- \/ yoast seo premium plugin. -->","yoast_head_json":{"title":"linear approximation: ap calculus exam review - magoosh blog | high school","description":"in this review article, we'll take a look at plenty of examples of linear approximation to better prepare you for the ap calculus exams.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/","og_locale":"en_us","og_type":"article","og_title":"linear approximation: ap calculus exam review","og_description":"in this review article, we'll take a look at plenty of examples of linear approximation to better prepare you for the ap calculus exams.","og_url":"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/","og_site_name":"magoosh blog | high school","article_publisher":"https:\/\/www.facebook.com\/magooshsat\/","article_published_time":"2018-08-04t00:18:37+00:00","og_image":[{"url":"\/\/www.catharsisit.com\/hs\/files\/2017\/01\/graph_with_tangent_lines.png"}],"author":"shaun ault","twitter_card":"summary_large_image","twitter_creator":"@shaunaultmath","twitter_site":"@magooshsat_act","twitter_misc":{"written by":"shaun ault","est. reading time":"6 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"article","@id":"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/#article","ispartof":{"@id":"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/"},"author":{"name":"shaun ault","@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f"},"headline":"linear approximation: ap calculus exam review","datepublished":"2018-08-04t00:18:37+00:00","datemodified":"2018-08-04t00:18:37+00:00","mainentityofpage":{"@id":"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/"},"wordcount":1147,"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\/linear-approximation-ap-calculus-exam-review\/#respond"]}]},{"@type":"webpage","@id":"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/","url":"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/","name":"linear approximation: ap calculus exam review - magoosh blog | high school","ispartof":{"@id":"\/\/www.catharsisit.com\/hs\/#website"},"datepublished":"2018-08-04t00:18:37+00:00","datemodified":"2018-08-04t00:18:37+00:00","description":"in this review article, we'll take a look at plenty of examples of linear approximation to better prepare you for the ap calculus exams.","breadcrumb":{"@id":"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/#breadcrumb"},"inlanguage":"en-us","potentialaction":[{"@type":"readaction","target":["\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/"]}]},{"@type":"breadcrumblist","@id":"\/\/www.catharsisit.com\/hs\/ap\/linear-approximation-ap-calculus-exam-review\/#breadcrumb","itemlistelement":[{"@type":"listitem","position":1,"name":"home","item":"\/\/www.catharsisit.com\/hs\/"},{"@type":"listitem","position":2,"name":"linear approximation: ap calculus exam review"}]},{"@type":"website","@id":"\/\/www.catharsisit.com\/hs\/#website","url":"\/\/www.catharsisit.com\/hs\/","name":"magoosh blog | high school","description":"act, sat, college admissions, life","publisher":{"@id":"\/\/www.catharsisit.com\/hs\/#organization"},"potentialaction":[{"@type":"searchaction","target":{"@type":"entrypoint","urltemplate":"\/\/www.catharsisit.com\/hs\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inlanguage":"en-us"},{"@type":"organization","@id":"\/\/www.catharsisit.com\/hs\/#organization","name":"magoosh","url":"\/\/www.catharsisit.com\/hs\/","logo":{"@type":"imageobject","inlanguage":"en-us","@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/logo\/image\/","url":"\/\/www.catharsisit.com\/hs\/files\/2019\/02\/magoosh-logo-purple-60h.png","contenturl":"\/\/www.catharsisit.com\/hs\/files\/2019\/02\/magoosh-logo-purple-60h.png","width":265,"height":60,"caption":"magoosh"},"image":{"@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/logo\/image\/"},"sameas":["https:\/\/www.facebook.com\/magooshsat\/","https:\/\/twitter.com\/magooshsat_act"]},{"@type":"person","@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f","name":"shaun ault","image":{"@type":"imageobject","inlanguage":"en-us","@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/image\/7c80c2046678e19beb5b9c8401d56613","url":"https:\/\/secure.gravatar.com\/avatar\/28c36635b8fed6717752755f46cda239?s=96&d=mm&r=g","contenturl":"https:\/\/secure.gravatar.com\/avatar\/28c36635b8fed6717752755f46cda239?s=96&d=mm&r=g","caption":"shaun ault"},"description":"shaun earned his ph. d. in mathematics from the ohio state university in 2008 (go bucks!!). he received his ba in mathematics with a minor in computer science from oberlin college in 2002. in addition, shaun earned a b. mus. from the oberlin conservatory in the same year, with a major in music composition. shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!","sameas":["http:\/\/valdosta.academia.edu\/shaunault","https:\/\/twitter.com\/shaunaultmath"],"url":"\/\/www.catharsisit.com\/hs\/author\/shaunault\/"}]}},"authors":[{"term_id":24932,"user_id":223,"is_guest":0,"slug":"shaunault","display_name":"shaun ault","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/28c36635b8fed6717752755f46cda239?s=96&d=mm&r=g","user_url":"http:\/\/valdosta.academia.edu\/shaunault","last_name":"ault","first_name":"shaun","description":"shaun earned his ph. d. in mathematics from the ohio state university in 2008 (go bucks!!). he received his ba in mathematics with a minor in computer science from oberlin college in 2002. in addition, shaun earned a b. mus. from the oberlin conservatory in the same year, with a major in music composition. shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!"}],"_links":{"self":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts\/12430"}],"collection":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/users\/223"}],"replies":[{"embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/comments?post=12430"}],"version-history":[{"count":0,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts\/12430\/revisions"}],"wp:attachment":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/media?parent=12430"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/categories?post=12430"},{"taxonomy":"post_tag","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/tags?post=12430"},{"taxonomy":"author","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/ppma_author?post=12430"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}