Course:MATH110/Archive/2010-2011/003/Learning Goals

From UBC Wiki

Here are the learning goals for this course. You should refer to them while you study to make sure you can attain these goals. These also provide guidelines to understand what will be asked on midterms and exams.

Learning goals related to the theoretical content

Students will learn the basic ideas, tools and techniques of differential calculus which prepares them for solving application problems.

Specifically, students will learn:

  • B1. the idea of limit and to evaluate limits involving basic functions using the limit laws, the squeeze theorem and/or the l’Hospital’s rule;
  • B2. the relationship between limits and asymptotes and to find asymptotes using limits;
  • B3. the idea of continuity and to construct/determine functions that are continuous or discontinuous using the definition of continuity or theorems involving continuity;
  • B4. the idea of derivative in terms of the slope of the tangent line to a curve, the rate of change of a quantity with respect to another quantity, and the limit definition of derivative;
  • B5. the idea of differentiability and to construct/determine functions that are differentiable or non-differentiable;
  • B6. the graphical/numerical relationship between a function and its derivative;
  • B7. to differentiate basic functions using the definition of derivative or the differentiation rules (derivative formulas, product rule, quotient rule, chain rule, logarithmic differentiation);
  • B8. to implicitly differentiate an equation involving two variables and to find tangent lines to the graph of an implicit function;
  • B9. to find the critical points and the local/absolute maxima/minima of a function defined on any open/closed interval;
  • B10. to find the intervals of increase/decrease of a function using derivative tests;
  • B11. to find the intervals of concavity, and to find the inflection points of a function defined on any interval using derivative tests;
  • B12. to graph a function by analyzing the behaviour of the function using limits and derivatives;

Learning goals related to the applications of the course's content

Students will apply the above skills and knowledge (including the learning goals related to basic skills) to translate a problem involving higher-level abstractions or real-life applications into mathematical problems and solve it. In general, when solving a problem students should be able to:

  • after reading a problem, correctly state in their own words what the problem is asking and what information is given that is needed in order to solve the problem;
  • after restating the problem, identify which mathematical techniques and concepts are needed to find the solution;
  • apply those techniques and concepts and correctly perform the necessary steps to obtain a solution;
  • interpret results within the problem context and determine if they are reasonable.

Specifically, students will learn:

  • C1. the idea of linear approximation and to find linear approximations to functions;
  • C2. the idea of approximation error and to estimate the error bound of a linear approxi- mation;
  • C3. to solve application problems involving velocity and acceleration of moving objects, rate of change, economics, natural growth/decay, related rates, linear approximation and optimization by using basic mathematics, limits, and derivatives.

Students will also learn how to construct simple proofs. They will show that a given mathematical statement is either true or false by constructing a logical argument using appropriate explanations, theorems and properties of functions.

Specifically, students will learn:

  • C4. to properly read a theorem or an implication and construct the contrapositive statement;
  • C5. to prove (or disprove) a statement or a mathematical formula by logical arguments without using the same statement or formula being proved;
  • C6. to apply a theorem by satisfying its hypotheses and drawing logical conclusion, or by negating the conclusion and concluding that not all the hypotheses are satisfied;
  • C7. the intermediate value theorem, Rolle’s theorem and mean value theorem, and to use these theorem to prove mathematical statements.