{"id":348,"date":"2023-07-16T18:15:46","date_gmt":"2023-07-16T22:15:46","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/abehandbook\/chapter\/mathematics-provincial-level-calculus\/"},"modified":"2023-07-16T18:15:46","modified_gmt":"2023-07-16T22:15:46","slug":"mathematics-provincial-level-calculus","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/abehandbook\/chapter\/mathematics-provincial-level-calculus\/","title":{"raw":"Mathematics: Provincial Level\u2014Calculus","rendered":"Mathematics: Provincial Level\u2014Calculus"},"content":{"raw":"\n<h3>Mathematics: Provincial Level\u2014Calculus<\/h3>\n\n<hr>\n\n<h5>Goal Statement<\/h5>\nABE Provincial Level Calculus is designed to (1) provide students with the mathematical knowledge and skills needed for post-secondary academic and career programs and (2) ease the transition from Provincial level Mathematics to first year calculus at college\/ university.\n<h6>1. Prelude to Calculus<\/h6>\nIt is expected that learners will be able to:\n<ul>\n \t<li>demonstrate an understanding of the concept of the limit and notation used in expressing the limit of a function<\/li>\n \t<li>evaluate the limit of a function analytically, graphically and numerically<\/li>\n \t<li>distinguish between the limit of a function as x approaches a and the value of the function at <em>x = a<\/em><\/li>\n \t<li>demonstrate an understanding of the concept of one and two-sided limits<\/li>\n \t<li>evaluate limits at infinity<\/li>\n \t<li>determine vertical and horizontal asymptotes using limits<\/li>\n \t<li>determine continuity of functions at a point <em>x = <\/em><em>a<\/em><\/li>\n \t<li>determine discontinuities and removable discontinuities<\/li>\n \t<li>determine continuity of polynomial, rational, and composite functions<\/li>\n<\/ul>\n<em>Optional <\/em><em>Outcomes:<\/em>\n<ul>\n \t<li>determine continuity of trigonometric functions<\/li>\n \t<li>determine limits of trigonometric functions<\/li>\n<\/ul>\n<h6>2. The Derivative<\/h6>\nIt is expected that learners will be able to:\n<ul>\n \t<li>define and evaluate the derivative at <em>x = a<\/em> as: [latex]\\displaystyle f'(x) = \\lim_{x \\to a} \\frac{f(x)-f(a)}{x-a}[\/latex]<\/li>\n \t<li>distinguish between continuity and differentiability of a function<\/li>\n \t<li>determine the slope of a tangent line to a curve at a given point<\/li>\n \t<li>calculate derivatives of elementary, rational and algebraic functions<\/li>\n \t<li>distinguish between rate of change and instantaneous rate of change<\/li>\n \t<li>apply differentiation rules to applied problems<\/li>\n \t<li>use Chain Rule to compute derivatives of composite functions<\/li>\n \t<li>solve rate of change application problems<\/li>\n \t<li>determine local and global extreme values of a function<\/li>\n \t<li>solve applied optimization (max\/min) problems<\/li>\n<\/ul>\n<em>Optional <\/em><em>Outcomes:<\/em>\n<ul>\n \t<li>calculate derivatives of trigonometric functions and their inverses<\/li>\n \t<li>calculate derivatives of exponential and logarithmic functions<\/li>\n \t<li>use logarithmic differentiation<\/li>\n \t<li>calculate derivatives of functions defined implicitly<\/li>\n \t<li>solve related rates problems<\/li>\n \t<li>use Newton\u2019s Method<\/li>\n<\/ul>\n<h6>3. Applications of the Derivative<\/h6>\nIt is expected that learners will be able to:\n<ul>\n \t<li>determine critical numbers and inflection points of a function<\/li>\n \t<li>compute differentials<\/li>\n \t<li>use the First and Second Derivative Tests to sketch graphs of functions<\/li>\n \t<li>use concavity and asymptotes to sketch graphs of functions<\/li>\n<\/ul>\n<em>Optional <\/em><em>Outcomes:<\/em>\n<ul>\n \t<li>differentiate implicitly<\/li>\n \t<li>understand and use the Mean Value Theorem<\/li>\n \t<li>apply L\u2019Hopital\u2019s Rule to study the behaviour of functions<\/li>\n<\/ul>\n<h6>4. Antiderivatives<\/h6>\nIt is expected that learners will be able to:\n<ul>\n \t<li>compute antiderivatives of linear combinations of functions<\/li>\n \t<li>use antidifferentiation to solve rectilinear motion problems<\/li>\n \t<li>use antidifferentiation to find the area under a curve<\/li>\n \t<li>evaluate integrals using integral tables and substitutions<\/li>\n<\/ul>\n<em>Optional <\/em><em>Outcomes:<\/em>\n<ul>\n \t<li>use antidifferentiation to find the area between two curves<\/li>\n \t<li>compute Riemann sums<\/li>\n \t<li>apply the Trapezoidal Rule<\/li>\n \t<li>solve initial value problems<\/li>\n<\/ul>\n<h6>5. Differential Equations<\/h6>\nIt is expected that learners will be able to:\n<ul>\n \t<li>derive a general solution of differential equations and find a particular solution satisfying initial conditions<\/li>\n \t<li>derive differential equations that explain mathematical models in the applied sciences<\/li>\n<\/ul>\n","rendered":"<h3>Mathematics: Provincial Level\u2014Calculus<\/h3>\n<hr \/>\n<h5>Goal Statement<\/h5>\n<p>ABE Provincial Level Calculus is designed to (1) provide students with the mathematical knowledge and skills needed for post-secondary academic and career programs and (2) ease the transition from Provincial level Mathematics to first year calculus at college\/ university.<\/p>\n<h6>1. Prelude to Calculus<\/h6>\n<p>It is expected that learners will be able to:<\/p>\n<ul>\n<li>demonstrate an understanding of the concept of the limit and notation used in expressing the limit of a function<\/li>\n<li>evaluate the limit of a function analytically, graphically and numerically<\/li>\n<li>distinguish between the limit of a function as x approaches a and the value of the function at <em>x = a<\/em><\/li>\n<li>demonstrate an understanding of the concept of one and two-sided limits<\/li>\n<li>evaluate limits at infinity<\/li>\n<li>determine vertical and horizontal asymptotes using limits<\/li>\n<li>determine continuity of functions at a point <em>x = <\/em><em>a<\/em><\/li>\n<li>determine discontinuities and removable discontinuities<\/li>\n<li>determine continuity of polynomial, rational, and composite functions<\/li>\n<\/ul>\n<p><em>Optional <\/em><em>Outcomes:<\/em><\/p>\n<ul>\n<li>determine continuity of trigonometric functions<\/li>\n<li>determine limits of trigonometric functions<\/li>\n<\/ul>\n<h6>2. The Derivative<\/h6>\n<p>It is expected that learners will be able to:<\/p>\n<ul>\n<li>define and evaluate the derivative at <em>x = a<\/em> as: [latex]\\displaystyle f'(x) = \\lim_{x \\to a} \\frac{f(x)-f(a)}{x-a}[\/latex]<\/li>\n<li>distinguish between continuity and differentiability of a function<\/li>\n<li>determine the slope of a tangent line to a curve at a given point<\/li>\n<li>calculate derivatives of elementary, rational and algebraic functions<\/li>\n<li>distinguish between rate of change and instantaneous rate of change<\/li>\n<li>apply differentiation rules to applied problems<\/li>\n<li>use Chain Rule to compute derivatives of composite functions<\/li>\n<li>solve rate of change application problems<\/li>\n<li>determine local and global extreme values of a function<\/li>\n<li>solve applied optimization (max\/min) problems<\/li>\n<\/ul>\n<p><em>Optional <\/em><em>Outcomes:<\/em><\/p>\n<ul>\n<li>calculate derivatives of trigonometric functions and their inverses<\/li>\n<li>calculate derivatives of exponential and logarithmic functions<\/li>\n<li>use logarithmic differentiation<\/li>\n<li>calculate derivatives of functions defined implicitly<\/li>\n<li>solve related rates problems<\/li>\n<li>use Newton\u2019s Method<\/li>\n<\/ul>\n<h6>3. Applications of the Derivative<\/h6>\n<p>It is expected that learners will be able to:<\/p>\n<ul>\n<li>determine critical numbers and inflection points of a function<\/li>\n<li>compute differentials<\/li>\n<li>use the First and Second Derivative Tests to sketch graphs of functions<\/li>\n<li>use concavity and asymptotes to sketch graphs of functions<\/li>\n<\/ul>\n<p><em>Optional <\/em><em>Outcomes:<\/em><\/p>\n<ul>\n<li>differentiate implicitly<\/li>\n<li>understand and use the Mean Value Theorem<\/li>\n<li>apply L\u2019Hopital\u2019s Rule to study the behaviour of functions<\/li>\n<\/ul>\n<h6>4. Antiderivatives<\/h6>\n<p>It is expected that learners will be able to:<\/p>\n<ul>\n<li>compute antiderivatives of linear combinations of functions<\/li>\n<li>use antidifferentiation to solve rectilinear motion problems<\/li>\n<li>use antidifferentiation to find the area under a curve<\/li>\n<li>evaluate integrals using integral tables and substitutions<\/li>\n<\/ul>\n<p><em>Optional <\/em><em>Outcomes:<\/em><\/p>\n<ul>\n<li>use antidifferentiation to find the area between two curves<\/li>\n<li>compute Riemann sums<\/li>\n<li>apply the Trapezoidal Rule<\/li>\n<li>solve initial value problems<\/li>\n<\/ul>\n<h6>5. Differential Equations<\/h6>\n<p>It is expected that learners will be able to:<\/p>\n<ul>\n<li>derive a general solution of differential equations and find a particular solution satisfying initial conditions<\/li>\n<li>derive differential equations that explain mathematical models in the applied sciences<\/li>\n<\/ul>\n","protected":false},"author":1935,"menu_order":8,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-348","chapter","type-chapter","status-publish","hentry"],"part":340,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/abehandbook\/wp-json\/pressbooks\/v2\/chapters\/348","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/abehandbook\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/abehandbook\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/abehandbook\/wp-json\/wp\/v2\/users\/1935"}],"version-history":[{"count":0,"href":"https:\/\/pressbooks.bccampus.ca\/abehandbook\/wp-json\/pressbooks\/v2\/chapters\/348\/revisions"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/abehandbook\/wp-json\/pressbooks\/v2\/parts\/340"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/abehandbook\/wp-json\/pressbooks\/v2\/chapters\/348\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/abehandbook\/wp-json\/wp\/v2\/media?parent=348"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/abehandbook\/wp-json\/pressbooks\/v2\/chapter-type?post=348"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/abehandbook\/wp-json\/wp\/v2\/contributor?post=348"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/abehandbook\/wp-json\/wp\/v2\/license?post=348"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}