Math 110.107, Calculus II (Biological and Social Sciences)

Fall 2010 Course Lecture Synopses

http://www.mathematics.jhu.edu/brown/courses/f10/107.htm

 

 

 

 

·       Monday, August 30:  Today I gave a brief description of how I want to organize the course, along with some discussion of who should or should not be in the course.  I talked about how your training is what we consider Calculus I (first semester calculus) may be different from the material we consider a pre-requisite for this course.  I placed the syllabus from 110.106 on the website to help you see whether your training is adequate, or needs to be supplemented by outside study.  I then started an example of a function you may not have seen in the AP system at the AB level, but which is a part of the 110.106 syllabus:  The function .  This function does not fit the mold of an exponential function nor a polynomial.  Even simple concepts like domain, and how one analyzes limits and its derivatives, can be tricky with functions like this one.  I analyzed this function’s behavior near 0 on the domain , as a way to generate discussion.  I will continue next time with its derivative, and some other concepts that may or may not be a part of your math past.

·       Wednesday, September 1: I started today’s lecture with some final comments about the function , including one would go about calculating the derivatives.  One would need to restructure  via the exponential identity  and then correctly manipulate the result to calculate.  Next, I spent some time discussing the derivatives and antiderivatives of a function like .  The latter can be solved by a substitution in a rather straightforward way.  I will ask the class to solve for the antiderivative via Integration by Parts in the homework.  I also spent some time on the behavior of the function  both near zero and as x goes to infinity (the horizontal asymptote).  Calculating these quantities allowed for a discussion on L’Hospital’s Rule for the limt at 0, and for a treatment of the Sandwich Theorem (this is how the book finds the limit at 0), and a modified version of the Sandwich Theorem appropriate for limits at infinity.  And lastly, I talked about some of the populations models that litter this book and how we will approach them.  The exercises in Section 4.6 are a good place to review some of the populations models we will or may use.  On Friday, I will start 7.4.

·       Friday, September 3:  Today, I focused on Section 7.4 on Improper Integrals.  I defined the improper Integral by playing with the example:  Let  on the interval .   is continuous, and I calculated its definite integral via the Fundamental Theorem of Calculus on a finite subinterval  of the domain;    Holding  fixed and pushing  out to infinity, we get the improper integral , defined as the limit of the proper integral as $b$ goes to infinity.  If the limit exists, the improper integral is said to converge.  I then formally defined the improper integral using infinity as one of the limits of the integral and used a few more examples to show how sometimes the integral converges and sometimes it does not.  I also related the convergence of an improper integral to the existence of the horizontal asymptote of the antiderivative of the integrand.  This latter statement is not stressed in the book.  Next class, I will discuss the second type of improper integral before moving into Chapter 8. 

·       Wednesday, September 8:  There is a second type of improper integral;  one where the edge of the interval of integration is not in the interval of integration.  This does not present a problem when the function simply has a hole at this point.  But it does when the function has a vertical asymptote or other abnormality there.  We defined in a very similar way how to rewrite an improper integral of this type as a limit of proper integrals, and the analysis is the same.  This works also for the integrals of functions where the interval of integration has a problem point inside the interval.  Simply break up the seemingly proper integral into two improper integrals at the problem point, and evaluate each improper integral separately.  The integral will converge only if the two improper integrals converge.  Be careful, as in examples, like , where one may think cancelation may lead to an integral converging when it does not.  This one does not.  However, the similar one  does (what is its value?).  After some examples, I moved into chapter 8 by introducing the idea of a differential equation as any equation involving an independent variable, an unknown function (the dependent variable), and some of its derivatives.  I gave some examples, and defined what the order of a differential equation is, as well as what it means to solve a differential equation (solving for the unknown function represented by the dependent variable).  I then defined the first type that we will analyze in this class;  that of a separable differential equation of the form , where the functions on the right hand side are continuous functions.  Some examples of differential equations which are and are not separable finished the discussion.

·       Friday, September 10:  Today, I continued the discussion on separable differential equations by realizing that this form of differential equations leads directly to a solution by integration:  Pull  to the left hand side to get .  This separates the variables  from .  Considering both sides as functions of , the equation says that these two functions of  are equal.  Hence their antiderivatives are equal also (at least up to a constant!).  Hence .  The left side is to be interpreted via the anti-Chain Rule (the Substitution method):  For , , so that .  I stressed that this is NOT the same thing as simply cancelling out the  ’s, although in this case, it amounts to the same thing.  Hence the solution to a separable differential equation is given by doing the integration .  Remember that once the integrations are completed, there will be an unknown constant lying around.  And once the integrations are finished, the resulting equation, involving  and  and the unknown constant, are collectively called the general solution.  Like in integration, one can solve for the unknown constant if one had knowledge of a simgle point of the curve that represents the antiderivative.  Here, many times, a differential equation comes with a data point, in the form .  Once you have the general solution, you can uses this “initial value” to solve for the unknown constant, and find the particular solution that fits your data.  I did a couple of examples to ensure that the class understands this in practice.  Then I started the discussion of the two special cases.  The first are called Pure-Time differential equations, where the function .  Then the entire problem amounts to finding the antiderivative of .  In general, with the initial value, I derived the formula in the book .  This comes directly from the Fundamental Theorem of Calculus.  I finished today’s discussion with yet another example, and concluded that for Pure-time separable differential equations of this type, all particular solutions look like vertical translates of each other (just like all antiderivatives look like vertical translates of each other.

·       Monday, September 13:  The second special case is called autonomous, or time-independent;  when .  Then the separable, first order differential equation looks like .  One consequence of this is that no matter where in time one starts a solution (using the initial value), the evolution of the resulting solution will appear the same.  This is important in application like population dynamics, where one does not really care when a population is at a certain value, but just how it is changing over time.  I use the example of the solutions to  , which are .  Different choices of initial value result in different values for the constant c, and for different values of the constant, the graphs of particular solutions look like horizontal translations of each other.  I then detailed three basic example of autonomous population models:  1) Exponential Growth , 2) the Bertalanffy Equation , and 3) the Logistic Equation .  We know how to solve and analyze the first.  I detailed the general and particular solutions to the third, and graphed it.  With the graph, one can see why it is a good basic model for populations.  Then I started a discussion on Allometric Growth, the study of the relative growth rate between two things which are growing exponentially at fixed rates , and .  Really the entire discussion revolves around the solutions to the differential equation , which are the curves , where  .

·       Wednesday, September 15:  Today, I finished the discussion of allometric growth by cleaning up the previous discussion, and generalizing to the model shown in the book.  I then discussion one application of this type of differential equation.  It is Example 9 in the text, and deals with homeostasis.  Then I went back to the Logistic Equation , and presented the solution without calculating it.  It is in the book, but I will have you guys calculate it explicitly in any case.  Using the solution , I graphed many solutions and discussed their properties.  We noticed that all solutions that started with  all satisfied .  This was especially true for the solution corresponding to the initial point .  Here, the solution is the horizontal line in the plane corresponding the solution .  This type of solution is called an equilibrium solution.  For autonomous differential equations , the equilibrium solutions are easy to find:  they are precisely the places where .  And without actually solving the differential equation, we can locate all of them by solving for the zeros of the function .  I ended with the example of the differential equation , for , and the Logistic Equation.  Notice that for the Logistic Equation, there are actually two equilibrium solutions;  one at K and one at 0.  The former is called the carrying capacity of the model, and represents the long-term stable population of the species.  The other represents the basic fact about populations:  one cannot grow a population if one starts with no members of that population.  There is one question I posed to the audience:  if you look at the general solution to the Logistic equation, , it is easy to “see” the equilibrium  (Indeed, stick K in for  ).  But trying to stick 0 in for  won’t work (why not?).  Why can we not see this other equilibrium?  The answer is that in solving the differential equation, one first separates the variables into  , and then integrates.  But to even do this step, one implicitly assumes that the denominator will not be zero.  Hence one is discounting that one solution in order to help find all of the others.  We call this lost solution an extraneous solution, but that is not important.  The important thing is that we can find the equilibrium solutions directly from the differential equation, without actually solving it.

·       Friday, September 17:  To be written….

·       Monday, September 20:  To be written….

·       Wednesday, September 22:  To be written….

·       Friday, September 24:  Here, I continued the discussion of how to use matrices to solve systems of linear equations by looking at the matrix version of the system , where ,  and .  For , these three quantities are numbers, and one can solve an equation like  simply by dividing each side of the equation by , thus isolating .  Of course, this only works when .  But really, what one is doing here is multiplying by the multiplicative inverse of , something also called the reciprocal.  Can we do something similar for the case ?  Yes, if we knew what the multiplicative inverse of a matrix was.  Then I defined the inverse of a (square) matrix, when it exists, and made mention of a general way to find it.  It exists only when the matrix has a special property;  that the determinant of the matrix is not 0.  I defined what the determinant is, and discussed its calculation in the case for  matrices.  I then spent time on two special matrix equation forms that show up in many applications.  The first was the equation .  Solving this would either require writing out the system of equations, and then simplifying, or realizing that one can try to combine like terms as matrices.  The trick was to understand that to bring all term to one side and factor out the matrix  needs a bit of care.  Specifically, the equation looks like .  The other special matrix equation was .  Notice that for , and , the solution ,  is a solution no matter what  looks like.  This is called the trivial solution.  The real question is, are there other solutions?  It turns out that there are when and ONLY when .    I ended the class with this last statement as a theorem.

·       Monday, September 27:  Today, I started Section 9.2, introducing a new notation for matrices which are vectors:  instead of using a capital letter for a vector of variables, I am using a small case letter with an arrow over it.  So call an n-vector .  I then introduced the assignment of a vector  to a new vector , where  is a square matrix as a function whose input is vectors and whose output are vectors of the same size;  a map of vectors .  The maps of vectors are called linear maps, where a map is called linear (in this case) is it satisfies  and  for  any real number.  For comparison, check that the function ,  is linear, for  a real number, while the two other functions of one real number  and  are not linear.)   In the case of 2-vectors, the set of all 2-vectors is precisely the set of points in the plane, and I discussed the visual representation of vectors as well as their sums, lengths, representations with polar coordinates and how they look when multiplied by a constant.  Using the notation , where  and , we can then say that a function ,  is linear.  I started looking at just how linear maps behave (in how they move around vectors in the plane), and talked about some particular ones like the identity map, a diagonal map, and a rotation.  Then I started the discussion about general linear maps and how their behavior on vectors can be studied by looking at the properties of the matrix .  We will continue this next class.

·       Wednesday, September 29:  For general linear maps, one can watch how they act on individual vectors (how the matrix changes the components of a vector).  Most vectors simply get moved around, but certain ones do not change direction.  They are simply magnified by a fixed amount.  Since any multiple of a vector of this type also only gets magnified, I explained that there are certain lines, passing through the origin, that remain invariant under a linear map.  These special vectors and the amount they are magnified are called the eigenvalues and eigenvectors of a matrix and give lots of information about the linear map.  I used as motivation the linear map given by the matrix  throughout this discussion, and defined the eigenvalue  and eigenvectors as the solutions to the matrix equation .  Detailing the example, I mentioned that the last equation can be written .  If we need to find solutions to this, recognize that we would need non-trivial solutions for the vectors .  This last equation will have them ONLY if .  This becomes our way of finding the eigenvalues, by solving this last equation for .  For a  matrix , we have , and the equation, called the eigenvalues equation or the characteristic equation, is .  I again used the example to solve for the eigenvalues.  To find the eigenvectors, simply go back to , and for each specific value for , solve for .  You will find, in this case, that the resulting system of equations will always have tons of solutions, since when  is an eigenvalues, there are tons of eigenvectors.  I finished with noting that the characteristic equation can be rewritten , where  is the trace of A and is the sum of the elements of the matrix on the main diagonal.

·       Friday, October 1:  In this lecture, I started with some special cases of eigenvalues finding that aid calculations as well as understanding.  First, it is perfectly acceptable that 0 be an eigenvalue of a matrix, and have eigenvectors associated to it.  Really, this means that in certain directions, the matrix simply takes all vectors in that direction to the zero-vector.  I showed that this must mean that the determinant of the matrix must also be zero here (and that the determinant of any matrix is a product of its eigenvalues in general.  I also showed that for a matrix, all of whose entries are either above or below the main diagonal, the eigenvalues ARE the main diagonal, and can simply be read off.  I backed this up with a couple of calculations.  Also, if the matrix is a diagonal matrix, then the eigenvectors can also be easily found without calculation.  Lastly, sometimes the eigenvalues are not even real, even if the matrix is (has real entries).  This happens when the characteristic equation has no real solutions (the quadratic formula used to solve for the eigenvalues of a  matrix has negative discriminant).  I spent some time defining and playing with complex numbers (a la Section 1.1.6).  Then I used the example of a rotation to calculate.  I also noted that all rotations (with the exception of no rotation and the half-way around rotation) have complex eigenvalues.  I then started the discussion of Section 9.4 on vectors in , discussing the notation, where they live, how to visualize them, how to calculate their length, what the unit vector in a certain direction is, and what the transpose of a vector is.  We will need this for the next lecture.

·       Monday, October 4:  Today, I started the class with a definition of one way that vectors can be multiplied together, the scalar or dot product.  Recall, that vectors are  matrices, and, carefully chosen, the transpose of one vector times the other is a scalar.  I then spent the class going over many applications of the dot product, as a way to develop properties of vectors in .  Re-writing the length calculation of a vector in terms of its dot product with itself is one.  After a bit of development of what it means to have a vector based at a point other than the origin, I introduced the geometric notion of the difference vector (a vector which is the difference of two other vectors.  The difference vector, for two vectors based at the origin, makes a nice triangle with the other two.  Then the Law of Cosines for triangles, relates the lengths of the three vectors (sides of the triangle) to the angle between the two based at the origin.  One can solve for the angle here, and write the calculation in terms of the dot product of the two vectors forming the angle.  A nice trick and the second application of the dot product.  The third follows directly with the knowledge that, for two vectors that form a right angle, the Law of Cosines reduces to the Pythagorean Theorem, and leads to the idea that the dot product of two vectors that form a right angle is 0.  The fourth and last application I will talk about is the way the dot product of two vectors can lead to a straightforward way to write the equation of a line in the plane (or of a plane in 3-space) using any vector on the line and a normal vector (one whose dot product with the vector in the line is 0.  I ended here.

·       Wednesday, October 6:  This lecture completes the discussion of Section 9.4.  Today, we defined some geometric objects in real space via the use of vectors.  To start, we completed the discussion we started last time using the dot product and a vector as a way to defines a unique line passing through a particular point.  The line actually consists of all vectors in the plane that are normal to the given vector, and I restated this process of finding the equation of the line.  This process generalizes to higher dimensions, but in a surprising way:  That the set of all vectors based at a point and perpendicular to a given vector actually form a  -dimensional space inside .  This means that the same procedure we used to define an equation describing a line in  can be used to write an equation to describe a plane in .  And so on.  I then talked about the notion that for a line in the plane, we can use the two coordinates of the plane to describe points on the line (via the equation that defines the line).  Alternatively, we can define a new coordinate directly on the line, in the same way that the real line has one coordinate.  This new coordinate is called a parameter, and the process is called a parameterization of the line.  Given any vector based at a point, there is a unique line that contains the entire vector.  But ANY multiple of that vector still lives on the line.  In a sense, the set of all multiples of that vector IS the line.  But the set of all multiples (call the parameter  ) is a copy of , and gives a position on the line sitting in the plane.  The  of this line is at the base point of the vector.  The  point on the line is at the head of the vector, and so on.  The vector becomes the measuring stick used to define the unit of measurement of the line.  The plane coordinates  and  are then functions of this parameter , and I wrote down how to do this in a similar way to the book, with examples from  and .  I then talked about how to create this parameter given either a vector, two points in space, or the equation of the line.  This ended the lecture and the section.

·       Friday, October 8:

·       Tuesday, October 12:  First exam day  (Tuesday is Monday…, day…  umm….)

·       Wednesday, October 13:  Today, I defined the idea of a contour line, or  -level set of a function of two or more independent variables.  It is a way to gain information about how a function behaves by looking at its values WITHIN the domain of the function.  For , the  -level set is a curve in the plane, and is defined as the solutions to the equation , where  is in the domain of the function.  We see this often in topographic maps, where the demarcations of constant height are noted by curves in a plan of an area.  This also allows us to view contour surfaces, or  -level surfaces in  for functions like .  I gave the example of , whose level surfaces are concentric spheres.  Then I showed many pictures of graphs of surfaces to the class via Mathematica on my computer (projected), along with samples of their contour lines in the planes.  One such function of interest was , .  The graph showed a hole at the origin, and a complicated surface graph whose contour lines all appeared to be straight lines that converged to the origin.  This motivated the notion of a limit of a function of more than one variable.  After a rehash of the formal and informal definitions of a limit of a function of one variable at a point, I showed via some graphics and schematics how limits look and work in two dimensions.  The definition in the book is best used only for showing a limit will not exist at a point (its evaluation along different paths to the limit point yield different values), it is still useful.  I went through some examples, and went back to the graph of the above function to show that the limit at 0 from different directions will yield different numbers.  I calculated the limit along paths through each axis direction and from the line .  I finished with looking again at the graph of  to “see” how the calculations match the graph.

·       Friday, October 15:  In this lecture, I continued the discussion of limits of functions of more than one independent variable, noting that all of the limit laws one uses in single variable calculus also work here in this setting, like the Limit of a sum of functions is a sum of the limits (at least when the individual limits exist, that is), etc.  I went over the notion of continuity explicitly, and noted through a few examples just how the three elements of continuity can individually fail.  I also spent some time on the notion of composition of functions, making sure it is understood that the range of the inside function must agree with at least part of the domain of the outside function.  Hence, for example, one cannot compose  and .  I then moved into Section 10.3.  With the example of the function , I asked general questions about how this function behaves as we vary one or both of the variables.  With a review of single variable calculus and the definition of a derivative, I showed that the corresponding notion for functions of more than one variable is more complicated yet relies on the same principles.  One can get an idea of how a function of more than one variable varies as one changes only one of the variables by pretending that the function has only one variable and has fixed the other variables like parameters.  Then single variable calculus allows us to “see” how this function is changing via a derivative of  with respect to only one variable (  for example).  Visually (geometrically), by holding the other variable fixed (  in this case), we are looking only at the part of the graph of   that intersects the  constant plane (this plane is an  -plane).  The intersection of the graph of the function with this plane is a curve whose graph is given by , where  is a function of  only.  Then the tangent line is well-defined and its slope is a derivative of   with respect to .  It is called the partial derivative of  with respect to , and written .  There is a corresponding derivative with respect to  also. 

·       Monday, October 18:  Here, I expounded on this notion of a partial derivative, by stating that this construction certainly works for function of more than 2 independent variables, that one can write out the equation of the tangent line in the  plane in the same way one does in single variable calculus:  , and in the  plane by , that really, calculating partial derivatives is no more difficult and similar to calculating single variable derivatives:  one simply hold all other variable fixed like parameters, and use Calc I techniques.  I gave a few examples.  And lastly, via the example of , I talked about how partial derivatives of functions of more than one variable are still functions of more than one variable, and hence we can take derivatives of derivatives, like , , and so on.  I then showed that the mixed partials existed in this case, and that they were equal.  This turned out not to be a coincidence, and I gave the Theorem in the book which established the criterion necessary for the mixed partials to be equal.   This lead into Section 10.4 and the notion of a tangent plane to a surface.  I then turned back on the computer and projected some examples of computed tangent lines and planes to surfaces.  I ended here.

·       Wednesday, October 20:  Here, …

·       Friday, October 22:  Today, I continued the discussion of the tangent plane approximation to the graph of a surface at a point.  I reiterated the structure of this tangent plane (see last class), and rewrote it as a matrix equation.  For a function  which is differentiable at a point , it looks like

.

This way, the matrix that contains all of the derivative information is in one place.  This is called the derivative matrix (or the Jacobi Matrix) of  and is denoted .  In this case,  is a  matrix containing the two partial derivatives of .  Keep this in mind.  I then generalized the discussion to functions of two or more independent variables that are vector-valued (the output now has more than one component.  These are written like a vector output (our first example was the linear maps we defined via a  matrix  (from September 27, above).  In our case, we started with a function , , noting that there are two outputs here, called either the coordinate functions or the component functions.  Each one of these component functions is a real-valued function of  and .  Hence each has its own linearization.  In general, write  , .  Then the linearization of  is given by

.

Each coordinate function has its own linearization, and the linearization of   is the 2-vector combination of these.  I cautioned that visualization is rather difficult (the graph of  is a surface in  ), but in calculations, this approximation can be quite valuable.  I did an example explicitly for the first function I mentioned above.  One can rewrite this approximation again as a matrix equation as

 

Now the matrix of derivative information, again denoted  is a  matrix (2 components, each with 2variables).  This is what is meant by the derivative of a function of more than one variable that may be vector-valued.  I then did an example using  and the initial point .  Generalizing even further, suppose , where .  Then we can say .  Suppose  is differentiable at a point  (we need this notation here, since we are using the subscript slot for the variables.  We cannot also use it to denote the initial point like in  ).  Then the derivative matrix is an  matrix and looks like

.

And not incidentally, we can write the linear approximation of  at the point  as

.

Finally, Notice that all of these matrix versions of the linear approximation follow the same pattern you already know really well:  For a Calculus I function of one variable , the tangent line approximation has the equation , or .  Thus the line as a function is simply the initial value  plus the derivative (the slope) times the independent variable shifted to .  But the vector form of this is EXACTLY in this same format.  It is the initial value  plus the derivative   time the shifted independent variables .  Really, vector calculus and the Calculus I stuff you already know are really the same.  We will see this again next week.  But really, there is simply more bookkeeping in .