Formation Damage & Productivity Enhancement

Resources forÌýFormation Damage & Productivity EnhancementÌý- for more information about the course, please seeÌýcourse outlines.

MLC drop-in

Please note thatÌýwe give priority to students meeting maths in their degree for the first time.ÌýStudents from Formation Damage will have to wait a little longer to see a staff member if there are many first-year students in the room at the same time. Note alsoÌýthat some of the assumed knowledge is higher level maths that the tutors at the MLC are not necessarily experts in, but we'll give it a go.

Assumed knowledge

Below are resources that you can use to help you revise some of the assumed knowledge for Formation Damage.

Graphs and functions

An excellent , where you can enter several different functions either in "y = formula in terms of x" or "formula involving both x and y = number" format. You can change the settings to have them in different colours or with dots or dashes, and you can change the zoom to get a good picture.

It's a good idea to take a moment to notice how the graphs for specific functions look, and how changing the formula for a function changes its graph. Desmos can help you develop that intuition.

Also, note some alternative notations for familiar functions:

  • Ìý"exp(x)" is a very common way to write the function "ex".
  • "tg(x)" is a common way in Europe and South AmericaÌýto write the function "tan(x)".
  • "ctg(x)" is a common way in Europe and South AmericaÌýto write the function "cot(x)".

    Derivatives and partial derivatives

    In Semester 1 2022, David gave a revision seminar to Mathematical Economics I about differentiation and partial differentiation (and also and constrained and unconstrained optimisation of two-variable functions, which is not required for Formation Damage).

    The above seminar does not cover trig functions, so you might be interested in this seminar to Mathematics IM inÌýSemester 2 2019 David with a short section on the calculus of trig functions (starting at 1h39m18s).Ìý

    (Don't forget that in your course assignments, the notation "tg(x)" appears instead of "tan(x)" and "ctg(x)" appears instead of "cot(x)".)

    This PDF handout is useful for doing derivatives (and integrals):

    Techniques of integration

    This revision seminar for Maths IM from Semester 1 2020 had a section on integration techniques (starting at 53m26s).

    This lecture from the old MLC bridging course MathTrackX is about techniques of integration, and the content is similar to that in Maths IM.

      Multivariable and vector calculus

      In this seminar to Maths IB students from Summer Semester 2019,ÌýDavid spent a lot of time talking about different ways to imagine multivariable functions and their derivatives.

      In 2017, David did a fully worked example for Formation Damage students about calculating the integrals involved to verify Green's Theorem in the plane.

        This seminar to Maths IB students from Semester 1 2021 covered multivariable derivatives, including directional derivatives, partial derivatives and critical points.

        In Semester 1 2019, David didÌýa revision seminar on integrals for students in Multivariable and Complex Calculus II. He talked about each of the nine kinds of integrals in the course, comparing their features and finding where the various integral theorems connect them together. He finished off with a couple of examples of using the theorems to make some integrals easier. (David thinks this is one of his best revision seminars for any course ever.) He also turned his paper-and-play-dough diagram from the seminar into a handout, which is linked below.

          Differential Equations

          This seminar for Maths IB from 2014 covered various topics on differential equations, starting with first order separable equations, which are most important for this course. (Your lecturer calls it "separation of variables", but most textbooks mean an entirely different thing by that.)

          Note that it is possible to solve some second-order DEs by turning them into first-order DEs.
          For example, consider the equation
          d2y/dx2Ìý- x2Ìýdy/dx = 0.
          Let z = dy/dx, so that dz/dx =Ìýd2y/dx2, and so you can replace the derivatives in the original equation to get
          dz/dx - x2Ìýz = 0.
          And now this is a first order separable equation that gets the answer
          z = some function of x.
          This means
          dy/dx = some function of x.
          And now you can integrate to find y.

          Matrix inverses and solving systems of equations

          This lecture from the old MLC bridging course MathsTrack discusses matrix inverses and how to find them by hand using row operations.

          • Ìý

          Note that if you set up an equation like Ax=b, for a known column vectorÌýbÌýand an unknown column vector x,Ìýand if you know what the inverse of A is, then you can solve it by multiplying both parts on their left by the inverse to produceÌýxÌý= A-1b.Ìý

          Alternatively, you can do row operations as described in the following revision seminar from 2014 covers matrix operations and also using matrices to solve linear equations. The links will take you to the linear equations section, which begins at about 58 mins.Ìý

          • Ìý

          Numerical derivatives

          The courses Numerical Methods II and Statistics and Numerical Methods II contain methods of approximating derivatives using only a list of function values at equally-spaced points along the x-axis. You have a list of points xi-2, xi-1, xi, xi+1, xi+2, separated in steps of Δx (sometimes called h) and their matching function values fi-2, fi-1,fi, fi+1, fi+2, and you use some or all of these to calculate approximations to the derivative at xi.

          Unfortunately we do not have resources specifically about these concepts, especially for numerical partial derivatives, which are not covered in maths courses until the end of third year. We may be able to help you interpret information you find in a textbook or website.