[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572656-105":53,"doc-detail-572656-en":127},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","harolds-business-calculus-cheat-sheet-algebra-reference","Harold’s Business Calculus Cheat Sheet - Algebra Reference","","Harold’s Business Calculus Cheat Sheet compiles key algebra and calculus references for quick problem solving. It summarizes exponent rules, including multiplication, power-to-a-power, distributive behavior, zero and negative power handling, plus common logarithm identities. The sheet also covers radicals-to-power conversions and detailed logarithm conversions. For calculus, it provides limit definitions, existence conditions, limit rules, continuity criteria on points and closed intervals, and foundational rate of change formulas, including the derivative definition via tangent line and limit of difference quotients.",{"@graph":63,"@context":119},[64,80,102],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/harolds-business-calculus-cheat-sheet-algebra-reference/572656/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/harolds-business-calculus-cheat-sheet-algebra-reference/572656.png","ImageObject",442,249,{"name":88,"@type":89},"DoDO","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-21","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",8,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"What exponent rules does the cheat sheet summarize?","Question",{"text":109,"@type":110},"It lists exponent multiplication rules, power-to-a-power, distributive forms, zero power, and how signs change for negative powers.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"How does it define a limit and explain when limits do not exist?",{"text":114,"@type":110},"It defines the limit using values of the function approaching a target from both sides and then enumerates cases where the limit fails to exist, such as infinite divergence or mismatched one-sided limits.",{"name":116,"@type":107,"acceptedAnswer":117},"What conditions define continuity at a point and continuity on a closed interval?",{"text":118,"@type":110},"Continuity at a point requires the function value to be defined, the limit to exist, and the limit to equal the function value. Continuity on a closed interval requires continuity on the open interval plus right-continuity at the left endpoint and left-continuity at the right endpoint.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},572656,1792009047,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":136,"language":137,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":138,"faqs":139,"seo_title":140,"seo_description":61,"update_tm":141,"read_time":142},962093270745,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Harold’s Business Calculus Cheat Sheet  \n20 May 2026  \nAlgebra Reference  \n\n| Exponents |  |  |  |\n| --- | --- | --- | --- |\n| Multiplication | 􀜽􀯡 􀜽􀯠 = 􀜽􀯡+􀯠 | 􀜽􀜽􀯡􀯠 = 􀜽􀯡−􀯠 = 􀜽1􀯠−􀯡 |  |\n| Power to a Power | (􀜽􀯡 )􀯠 = 􀜽􀯡􀯠 |  |  |\n| Distributive | (􀜽􀜾)􀯡 = 􀜽􀯡 􀜾 􀯡 | (􀜽􀜾)􀯡 = 􀜽􀜾􀯡􀯡 |  |\n| Zero Power | 􀜽 0 = 1 􀝅􀝂 􀜽 ≠ 0 |  |  |\n| Power Sign Change | 􀜽 −􀯡 = 􀜽1􀯡 | 1\u003Cbr>􀜽 −􀯡 | = 􀜽􀯡 |\n| Negative Powers | (􀜽􀜾)−􀯡 = (􀜾􀜽)􀯡 = 􀜾􀜽􀯡􀯡 | 􀜽􀯡~~􀯠~~ = (􀜽1~~􀯠~~)􀯡 = (􀜽􀯡 )1~~􀯠~~ |  |\n\n\n| Radicals |  |  |\n| --- | --- | --- |\n| Convert to Power |    1\u003Cbr>􀳙√􀜽 = 􀜽􀯡 | \u003Cbr>􀳙√􀜽􀜾 = 􀳙√􀜽 􀳙√􀜾 |\n| Root of a Root | 􀳘√􀳙√􀜽 = 􀳙􀳘√􀜽 | 􀳙√􀜽􀜾 = 􀜽􀜾 |\n\n\n| Logarithms |  |  |  |\n| --- | --- | --- | --- |\n| Definition | Log ≡ Exponential |  |  |\n|  | 􀝈􀝋􀝃􀯕 􀝔 = 􀝕 ≡ 􀝔 = 􀜾 􀯬 |  |  |\n| Example | 􀝈􀝋􀝃 5 125 = 3 ≡ 125 = 53 |  |  |\n| Common Log | 􀝈􀝋􀝃 􀝔 = 􀝈􀝋􀝃10 􀝔 | Base e assumed in pre-1955 math textbooks. Base 2 assumed in computer science textbooks. |  |\n| Natural Log | 􀝈􀝊 􀝔 = 􀝈􀝋􀝃 􀯘 􀝔 |  | where 􀝁 ≈ 2.718281828. .. |\n| Powers (x2) | 􀝈􀝋􀝃􀯕 (􀝔􀯥 ) = 􀝎 􀝈􀝋􀝃􀯕 􀝔 |  | 􀝈􀝊 􀝔 􀯥 = 􀝎 􀝈􀝊 􀝔 |\n| Multiplication (×) | 􀝈􀝋􀝃􀯕 (􀝔􀝕) = 􀝈􀝋􀝃􀯕 􀝔 + 􀝈􀝋􀝃􀯕 􀝕 |  | 􀝈􀝊 (􀝔􀝕) = 􀝈􀝊 􀝔 + 􀝈􀝊 􀝕 |\n| Division (÷) | 􀝈􀝋􀝃􀯕 (􀝔􀝕) = 􀝈􀝋􀝃􀯕 􀝔 − 􀝈􀝋􀝃􀯕 􀝕 |  | 􀝈􀝊 (􀝔􀝕) = 􀝈􀝊 􀝔 − 􀝈􀝊 􀝕 |\n| Zero (0) and One (1) | 􀝈􀝋􀝃􀯕 1 = 0 |  | 􀝈􀝋􀝃􀯕 􀜾 = 1 |\n| Inverse Functions | 􀝈􀝋􀝃􀯕 􀜾 􀯫 = 􀝔 |  | 􀜾 􀯟􀯢􀯚 􀳍 􀯫 = 􀝔 |\n| Change of Base | 􀝈􀝋􀝃􀯕 􀝔 = 􀝈􀝈􀝋􀝋􀝃􀝃􀯔􀯔~~ ~~􀝔􀜾 = 􀝈􀝈􀝊􀝊~~ ~~􀝔􀜾 |  | TI-84:\u003Cbr>[MATH] + [A: logBASE(] → log ( ) |\n\n3.1 Limits  \n\n| Property | (Map to Larson’s 1-pager of common derivatives) |  |\n| --- | --- | --- |\n| Definition of Limit | Let 􀝂 be a function and let 􀜽 and 􀜮 be real numbers. If\u003Cbr>[1. as](1. as) 􀝔 takes values closer and closer (but not equal) to 􀜽 on both sides of 􀜽, the corresponding values of 􀝂(􀝔) get closer and closer (and perhaps equal) to 􀜮; and\u003Cbr>2. the value of 􀝂(􀝔) can be made as close to 􀜮 as desired by taking values of 􀝔 close enough to 􀜽;\u003Cbr>then 􀜮 is the limit of 􀝂(􀝔) as x approaches 􀜽, written\u003Cbr>􀝈􀝅􀝉 􀝂 (􀝔) = 􀜮\u003Cbr>􀯫→􀯔 |  |\n| Existence of Limits | The limit of 􀝂 as 􀝔 approaches 􀜽 may not exist.\u003Cbr>1. If 􀝂(􀝔) becomes infinitely large in magnitude (positive or negative) as 􀝔 approaches the number 􀜽 from either side, we write\u003Cbr>􀝈􀝅􀝉 􀝂(􀝔) = ∞\u003Cbr>􀯫→􀯔\u003Cbr>or 􀝈􀝅􀝉 􀝂(􀝔) = −∞\u003Cbr>􀯫→􀯔\u003Cbr>In either case the limit does not exist.\u003Cbr>2. If 􀝂(􀝔) becomes infinitely large in magnitude (positive) as 􀝔 approaches 􀜽 from one side and infinitely large in magnitude (negative) as 􀝔 approaches 􀜽 from the other side, then 􀝈􀝅􀝉 􀝂(􀝔)\u003Cbr>􀯫→􀯔\u003Cbr>does not exist.\u003Cbr>3. If 􀝈􀝅􀝉 􀝂 (􀝔) = 􀜮 and 􀝈􀝅􀝉 􀝂 (􀝔) = 􀜯 , and 􀜮 ≠ 􀜯, then 􀯫→􀯔 − 􀯫→􀯔+\u003Cbr>􀝈􀝅􀝉 􀝂(􀝔) does not exist.\u003Cbr>􀯫→􀯔 |  |\n| Limits at Infinity | 􀝈􀯫 􀝔1􀯡 = 0 | 􀯫 􀝔1􀯡 = 0 |\n| Finding Limits at Infinity | If 􀝂(􀝔) = 􀯣􀯤((􀯫􀯫)), for polynomials 􀝌 (􀝔) and 􀝍 (􀝔), 􀝍 (􀝔) ≠ 0,􀝈􀝅􀝉 􀝂 (􀝔) and 􀝈􀝅􀝉 􀝂 (􀝔) can be found as follows.\u003Cbr>􀯫→+∞ 􀯫→ − ∞\u003Cbr>1. Divide 􀝌 (􀝔) and 􀝍 (􀝔) by the highest power of 􀝔 in 􀝍 (􀝔) .\u003Cbr>2. Use the rules for limits, including the rules for limits at infinity,􀝈􀝅􀝉  1  = 0\u003Cbr>􀯫→∞ 􀝔 􀯡\u003Cbr>and 􀝈􀝅􀝉  1  = 0\u003Cbr>􀯫→ − ∞ 􀝔 􀯡\u003Cbr>to find the limit of the result from Step 1. |  |\n\nRules for Limits  \n\n| Rule | (Map to Larson’s 1-pager of common derivatives) |\n| --- | --- |\n| Given | Let 􀜽, 􀜣 , and 􀜤 be real numbers, and let 􀝂 and 􀝃 be functions such that\u003Cbr>􀝈􀝅􀝉 􀝂 (􀝔) = 􀜣\u003Cbr>􀯫→􀯔\u003Cbr>and\u003Cbr>􀝈􀝅􀝉 􀝃 (􀝔) = 􀜤 .\u003Cbr>􀯫→􀯔 |\n| 1. Constant (c) | If 􀜿 is a constant, then\u003Cbr>􀝈􀝅􀝉 􀜿 = 􀜿\u003Cbr>􀯫→􀯔\u003Cbr>and\u003Cbr>􀝈􀝅􀝉 [􀜿 ∙ 􀝂 (􀝔)] = 􀜿 ∙ 􀝈􀝅􀝉 􀝂 (􀝔) = 􀜿 ∙ 􀜣\u003Cbr>􀯫→􀯔 􀯫→􀯔 |\n| 2. Sum or Difference (+, −) | The limit of a sum or difference is the sum or difference of the limits.\u003Cbr>􀝈􀝅􀝉 [􀝂(􀝔) ± 􀝃 (􀝔)] = 􀝈􀝅􀝉 􀝂 (􀝔) ± 􀝈􀝅􀝉 􀝃 (􀝔) = 􀜣 ± 􀜤\u003Cbr>􀯫→􀯔 􀯫→􀯔 􀯫→􀯔 |\n| 3. Product (×) | The limit of products is the product of the limits.\u003Cbr>􀝈􀝅􀝉 [􀝂 (􀝔) ∙ 􀝃 (􀝔)] = 􀝈􀝅􀝉 􀝂 (􀝔) ∙ 􀝈􀝅􀝉 􀝃 (􀝔) = 􀜣 ∙ 􀜤\u003Cbr>􀯫→􀯔 􀯫→􀯔 􀯫→􀯔 |\n| 4. Quotient (÷) | The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero.\u003Cbr>􀝈􀝅􀝉 􀝂 (􀝔) = 􀝈􀯫 􀝂 (􀝔) = 􀜣\u003Cbr>􀯫→􀯔 􀝃 (􀝔) 􀝈􀝅􀝉 􀝃 (􀝔) 􀜤\u003Cbr>􀯫→􀯔\u003Cbr>if 􀜤 ≠ 0. |\n| 5. Polynomial (􀡼(􀢞)) | If 􀝌(􀝔) is a polynomial, then\u003Cbr>􀝈􀝅􀝉 􀝌 (􀝔) = 􀝌 (􀜽)\u003Cbr>􀯫→􀯔 |\n| ","cbCaihmhesLXnaUi","https://ap.wps.com/l/cbCaihmhesLXnaUi","pdf",1005381,19,"English","# Algebra Reference\n## Exponents\n## Radicals\n## Logarithms\n# Limits (3.1)\n# Limits Rules\n# Continuity (3.2)\n# Rates of Change (3.3)\n# Definition of the Derivative (3.4)","[{\"question\":\"What exponent rules does the cheat sheet summarize?\",\"answer\":\"It lists exponent multiplication rules, power-to-a-power, distributive forms, zero power, and how signs change for negative powers.\"},{\"question\":\"How does it define a limit and explain when limits do not exist?\",\"answer\":\"It defines the limit using values of the function approaching a target from both sides and then enumerates cases where the limit fails to exist, such as infinite divergence or mismatched one-sided limits.\"},{\"question\":\"What conditions define continuity at a point and continuity on a closed interval?\",\"answer\":\"Continuity at a point requires the function value to be defined, the limit to exist, and the limit to equal the function value. Continuity on a closed interval requires continuity on the open interval plus right-continuity at the left endpoint and left-continuity at the right endpoint.\"}]","Harold’s Business Calculus Cheat Sheet - Algebra Reference | PDF",1791928736,7]