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It also includes vector calculus, numerical methods like Newton iteration and the trapezoidal rule, plus general strategy recommendations and kinematics formulas for translational, rotational, curvilinear, and rigid-body motion.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/paper-templates/","Paper Templates",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/formulas-for-ipho-quick-reference-july-4-2018/572658/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/formulas-for-ipho-quick-reference-july-4-2018/572658.png","ImageObject",442,249,{"name":42,"@type":43},"Malam Tenang","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-15","2026-10-13",true,{"@type":52,"interactionType":53,"userInteractionCount":22},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"What approximation tools are included under Taylor series?","Question",{"text":62,"@type":63},"The document provides Taylor-series truncation for approximations and lists special linear approximations such as sin x ≈ x, cos x ≈ 1 − x^2/2, and (1 + x)^n ≈ 1 + nx.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"Which vector and geometry relations are provided?",{"text":67,"@type":63},"It includes distributive dot/cross product identities, mixed product (volume) identities, sine/cosine laws, the inscribed angle theorem, triangle surface area and centroid properties.",{"name":69,"@type":60,"acceptedAnswer":70},"What kinematics formulas for different types of motion are covered?",{"text":71,"@type":63},"It gives translational relations using integrals/areas under graphs, rotational analogs using ω and ε, curvilinear motion guidance, rigid-body motion constraints, and notes on non-inertial reference frames.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},572658,1792067152,{"code":4,"msg":81,"data":82},"success",[83,88,93,98,103,108,113,118,122],{"id":84,"doc_module":22,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},11,"Presentations",90,"presentations",{"id":89,"doc_module":22,"doc_module_name":25,"category_name":90,"show_sort_weight":91,"slug":92},12,"Resumes",80,"resumes",{"id":94,"doc_module":22,"doc_module_name":25,"category_name":95,"show_sort_weight":96,"slug":97},14,"Invoices",70,"invoices",{"id":99,"doc_module":22,"doc_module_name":25,"category_name":100,"show_sort_weight":101,"slug":102},15,"Posters",60,"posters",{"id":104,"doc_module":22,"doc_module_name":25,"category_name":105,"show_sort_weight":106,"slug":107},16,"Social Media",50,"social-media",{"id":109,"doc_module":22,"doc_module_name":25,"category_name":110,"show_sort_weight":111,"slug":112},17,"Forms",40,"forms",{"id":114,"doc_module":22,"doc_module_name":25,"category_name":115,"show_sort_weight":116,"slug":117},18,"Letters",30,"letters",{"id":119,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":120,"slug":121},21,5,"papers-templates",{"id":123,"doc_module":22,"doc_module_name":25,"category_name":124,"show_sort_weight":4,"slug":125},158,"General","general-158",{"code":4,"msg":81,"data":127},{"doc_id":78,"user_id":128,"nickname":42,"user_avatar":129,"doc_module":22,"category_id":119,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":22,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":33,"language":135,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":12,"update_tm":139,"read_time":26},962093532863,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Formulas for IPhO  \nVersion: July 4, 2018  \nI Mathematics  \n1. Taylor series (truncate for approximations): F (x) = F(x0 ) +XF (n)(x0 )(x − x0 )n /n!  \nSpecial case—linear approximation:  \nF (x) ≈ F(x0 ) + F0 (x0 )(x − x0 ) Some examples for |x| 􀀜 1:  \nsin x ≈ x, cos x ≈ 1 − x2 /2, ex ≈ 1 + x ln(1 + x) ≈ x, (1 + x)n ≈ 1 + nx  \n2. Perturbation method: find the solution iteratively using the solution to the ”non-perturbed”(directly solvable) problem as the 0th approximation; corrections for the next approximation are calculated on the basis on the previous one.  \n3. Solution of the linear differential equation with constant coefficients ay00 + by0 + cy = 0:  \ny = Aexp(λ1 x) + B exp(λ2 x),  \nwhere λ 1 ,2 is the solution of the characteristic equation aλ2 + bλ + c = 0 if λ 1  λ2 . If the solution of the characteristic equation is complex, while a, b and c are real numbers, then λ 1 ,2 = γ ± iω and  \ny = Ceγx sin(ωx + ϕ0 ) .  \n4. Complex numbers  \nz = a + bi = |z|eiϕ , ¯z = a − ib = |z|e−iϕ|z| 2 = z ¯z = a2 + b2 , ϕ = arg z = arcsin |~~ ~~bz| Rez = (z + ¯z)/2, Imz = (z − ¯z)/2  \n|z1 z2 | = |z1 ||z2 | , arg z 1 z2 = arg z 1 + arg z2 eiϕ = cos ϕ + isin ϕ  \ncos ϕ = ~~e~~iϕ~~e~~−iϕ , sin ϕ = ~~e~~iϕ~~e~~i−iϕ  \n5. Cross and dot products of vectors are distributive: a (b + c) = ab + ac.  \n~ ~  \n~a · b = b · ~a = axbx + ay by + . . . = ab cos ϕ  \n~ ~ ~ ~  \n| ~~a × b| = ab sin ϕ; ~a × b = −b × ~a ⊥ ~a, b~a × b = (ay bz ~− by az)e~~x + (az bx − bz ~ax)~ey + . . .  \n~a × [b × ~c] = b (~a · ~c) − ~c(~a · b) . Mixed prod. (volume [of parallelep. def. by](of parallelep. def. by) 3 vec.):  \n~ ~ ~ ~  \n(~a, b, ~c) ≡ (~a · [b × ~c]) = ([~a × b] · ~c) = (b, ~c, ~a) .  \n6. Cosine and sine laws:  \nc2 = a2 + b2 − 2ab cos ϕ a/ sin α = b/ sin β = 2R  \n7. sin(α ± β) = sin α cos β ± cos α sin β cos(α ± β) = cos α cos β ∓ sin α sin β tan(α ± β) = (tan α + tan β)/(1 ∓ tan α tan β) cos2 α = 1+co2s~~ ~~2~~α~~ , sin2 α = 1~~ ~~−~~ ~~co2s~~ ~~2~~α~~  \ncos α cos β = cos(α+β)+2cos(α−β) ,    \ncos α + cos β = 2(cos α+2β + cos α~~ ~~−2β ), . . . .  \n8. An angle inscribed in a circle is half of the central angle that subtends the same arc on the circle. Conclusions: hypotenuse of a right triangle is the diameter of its circumcircle; if the angles of a quadrilateral are supplementary, it is a cyclic quadrilateral.  \n9. Surface area of a triangle = 12aha = pr = pp(p − a)(p − b)(p − c) = abc/4R.  \n10. Triangle’s centroid: intersection point of medians, divides medians to 2:1 .  \n11∗ . Vector approach to geometry problems.  \n12. Taking derivatives:  \n(fg)0 = fg0 + f0g, f [g(x)]0 = f 0 [g(x)]g0 (sin x)0 = cos x, (cos x)0 = − sin x  \n(ex )0 = ex , (ln x)0 = 1/x, (xn )0 = nxn−1 (arctan x)0 = 1/(1 + x2 )    \n(arcsin x)0 = −(arccos x)0 = 1/ p1 − x2 13. Integration: the formulas are the same as for derivatives, but with swapped left-hand-side and rhs. (inverse operation!), e.g.  \nZ xndx = xn+1/ (n + 1) . Special case of the substitution method:  \nZ f (ax + b)dx = F (ax + b)/a.  \n14. Conic sections: a 11 x2 + 2a12 xy + a22 y 2 + a 1 x + a2 y + a0 = 0 with a 11 = a22 — circle; with a11 ·(a11 a22 − a212) > 0—ellipse,   \u003C 0—hyperbola, with a 11 a22 = a212 —parabola. Ellipse: l 1 + l2 = 2a, α 1 = α2 , A = πab; hyperbola: l 1 − l2 = 2a, α 1 + α2 = 0; parabola: l + h = const, α 1 = α2 .  \n15. Numerical methods. Newton’s iterative method for finding roots f (x) = 0:  \nxn+1 = xn − f(xn)/f 0 (xn) .  \nTrapezoidal rule for approximate integration: Zab f (x)dx ≈ b~~ ~~−2na [f(x0 ) + 2f(x1 ) + . . .+2f(xn−1) + f(xn )]  \n16. Derivatives and integrals of vectors: differentiate/integrate each component; alternatively differentiate by applying the triangle rule for the difference of two infinitesimally close vectors.  \nII General recommendations  \n1. Check all formulas for veracity: a) examine dimensions; b) test simple special cases (two parameters are equal, one param. tends to 0 or ∞ ); c) verify the plausibility of solution’s qualitative behaviour.  \n2. If there is an extraordinary coincidence in the","cbCaiopDsLkqiBqD","https://ap.wps.com/l/cbCaiopDsLkqiBqD","pdf",302764,"English","# I Mathematics\n## General recommendations\n# II General recommendations\n# III Kinematics","[{\"question\":\"What approximation tools are included under Taylor series?\",\"answer\":\"The document provides Taylor-series truncation for approximations and lists special linear approximations such as sin x ≈ x, cos x ≈ 1 − x^2/2, and (1 + x)^n ≈ 1 + nx.\"},{\"question\":\"Which vector and geometry relations are provided?\",\"answer\":\"It includes distributive dot/cross product identities, mixed product (volume) identities, sine/cosine laws, the inscribed angle theorem, triangle surface area and centroid properties.\"},{\"question\":\"What kinematics formulas for different types of motion are covered?\",\"answer\":\"It gives translational relations using integrals/areas under graphs, rotational analogs using ω and ε, curvilinear motion guidance, rigid-body motion constraints, and notes on non-inertial reference frames.\"}]","Formulas for IPhO - Quick Reference - July 4, 2018 | PDF",1791928745]