Section 9.6

Main Points: Constrained optimization is used when certain limits are placed on a problem, such as a budget. To maximize something subject to contraints using a graph, the point on the contour line completely exhausts the budget, point below the line gives possible values of x and y that will not use all of the budget, and a point about the contour line cannot be afforded. The maximum occurs at a point P where the budget constraint is tangent to the production contour. Lagrange multipliers solves three equations: partial derivative of f with respect to x (x,y) = lambda g(sub x) of (x,y). The second equation is the same except with respect to y. Finally g(x,y)= c (or some constant). lambda is the Lagrange multiplier. If f has a constrained max or min, then it occurs at one of the Lagrange solutions or at the endpoint of the constraint. Lambda is approximately equal to the change in optimum value of f when the value of the constraint is increased by one unit. Also, lambda represents the rate of change of optimum values of f as the constraint increases. Finally, the Lagrangian function allows a person solving a constraint optimization problem the opportunity to use 2 steps: first, write the Lagrangian function and then find the critical points of that function.

Challenges: I'm unsure about how to use the Lagrangian function. I understand that the answers from the equation are the critical points, but I don't understand how to get an answer from it, if that makes sense?

Reflections: This is really helpful in getting the most for your money. Because companies and businesses want to be able to make more money than they spend, finding the optimization under constraints is a really important concept.

Section 4.3

Main Points: Section 4.3 discusses global maximums and global minimums. A global max or min occurs where the function is greater or less than all other points on the function. The function, f, has a global maximum at p if f(p) is greater than or equal to all values of f. It has a global minimum at p if f(p) iis less than or equal to all values of f. To find a global max or min of f on an interval including endpoints, compare values of f at all critical points on the interval and also the endpoints. However, a function defined as domain is all real numbers or on an interval exluding endpoints may or may not have a global max or min. To see if the function has a global max or min in this case, simply find values of f at all the critical points and sketch the graph.

Challenges: I've already done this section, except my teacher called them "absolute" max or min. However, some students may get confused if they forget to include the endpoints when checking for global maxes or mins since this could affect their final answer.

Reflections: I understand that global maxes or mins give the highest or lowest value of the function, but how can you apply that into real life situations? The book gave an example about gas, but other than that, what would you use the second derivative and global maxes/mins for?

Sections 1.3, 2.4, 4.1, and 4.2

Main Points: Section 1.3 was already read a few weeks ago, but it was a review of the first derivative to prepare us for the second derivative. Section 2.4 introduces the second derivative. Its notation is either f " (x) or d^2y/dx^2. It is the rate of a change of a rate of change. The second derivative gives information about the first derivative, such as if it is increasing or decreasing. It also tells whether the original funciton is concave up or down. Section 4.1 talks about local maxes and mins. If f(p) is less than or equal to valuesf for points near p, then the function has a local minimum at p. Also, f has a local maximum at p if f(p) is greater than or equal to values of f for points near p. Also, a critical point is a point, p, where f ' (p) = o or undefined. A critical value is the value of f(p) where the critical point is. If a function that is continuous on an interval has a local max or min at p, then p is a critical point or enpoint on the interval. Two ways to test for the maxes and mins of a function are the first and second derivative tests. The first derivative test says that if f changes from negative to positive at a point, then f has a local minimum at p. Also, if f changes from positive to negative, it has a local maximum at p. The second derivative test states that if p is a critical point of f and f ' (p)=0, and if f is concave up at p, then f has a local min at p. Also, if f is concave down at p, then f has a local max at p. However, not ever critical point is a max or min. In section 4.2, the text introduces inflection points. Inflection points are points where the graph of a function change concavity. This happens at points where f '' (x) = 0 or undefined. However, not all points where f '' (x) = 0 is an inflection point.

Challenges: The theory behind maxes and mins can be a bit confusing, especially the difference critical values and critical points.

Reflections: My teacher in high school calc spent a lot of time on maxes and mins, along with inflection points. Although I am pretty familiar with these concepts, I still don't understand their importance in solving problems in real life or how they are actually applied to real world situations.

Supplementary Notes pp 4-8

Main Points: The idea of a gradient is introduced. A gradient is a vector that consists of partial derivatives and is denoted as grad(f). A directional derivative tells the rate of change in any direction on the xy plane where as a partial derivative tells the rate of change in either the x direction or the y direction. Directional derivative can be justified in the exactly the same way a regular derivative can be justified. The properties of a gradient are as follows. It always points in the direction of the greatest increase. Similarly, the direction opposite the direction of the gradient shows the greatest decrease. The length of the gradient directly corresponds to the steepness of the slope. Thus, larger slopes mean larger gradients. A gradient is always perpendicular to the level curve at which it is rooted.

Challenges: The idea of gradients is a bit hard to grasp. I'm a bit confused as to their importance and how exactly they are calculated. I don't have a specific question to ask about gradients - I'm just confused about everything!

Reflections: Gradients are totally new to me and I'm still a little puzzled as to how to go about solving problems in this area. However, I think that once you explain it in class I will be able to grasp the concept a lot better!

LA Sections 1.0-1.2; 4.2.0

Main Points: In the LA sections 1.0 through 1.2, the idea of matrices is introduced. Points are thought of as vectors and can be be written horizontally in (x,y) form or horizontally, in matrix form. Two types of actions are introduced: scalar multiplication and vector addition. Scalar multiplication involves multiplying each term by the constant outside of the parantheses. This does not change direction of vector, but rather makes it longer or shorter. Vector addition, also known as coordinate-wise addition, involves adding the top terms together and the bottom terms together, creating new coordinates and a new vector. In section 4.2.0, dot products are discussed. Also called scalar products, this operation produces a real number by multiplying corresponding coordinates of vectors and adding the products together to reach a new, single number. Furthermore, the dot product of two vectors is zero if the vectors are perpendicular, such as (2, -3) and (3, 2).

Challenges: I was not confused by this reading because I am somewhat familiar with matrices. However, someone may be confused with dot multiplication if one does not add the products together.

Reflections: Although I have done matrices before and understand the operations, I am still not sure why they are important. However, they are pretty easy to understand and I enjoy doing them!

Sections 3.5, 9.3, and 9.4

Main Points: In Section 3.5, it is stated that a periodic function has a periodic derivative. Also, the proof that dy/dx (sinx)= cosx and dy/dx (cosx)=-sin(x). Also, if a constant, k, is present in the the periodic function, the derivatives are as follows: [d/dt(sinkt) = kcoskt] and [d/dt(coskt)=-ksinkt]. In section 9.3, the book discusses partial derivatives with respect to both x and y (or a and b). The formulas for the partial derivative are, with respect to x, lim (h-> 0) [f(a+h,b) - f(a,b)]/h, and with respect to y, lim(h-> 0) [f(a,b+h) - f(a,b)]/h. Furthermore, it is proven that the partial derivative can be estimated through both tables and contour diagrams. Local linearity is introduced ( delta f is about equal to f with respect to x times delta x + f with respect to y times delta y). In 9.4, the book states that partial derivatives are able to be computed algebraically by fixing one of the variables. The second order derivative formulas are all listed (if I knew how to get the special characters, I would list them all!). Also, if fxy and fyx are continuous at point (a,b), then fxy(a,b)= fyx(a,b). This is called mixed partial derivatives.

Challenges: In section 9.4, I was a little confused on the Second Order Partial Derivatives. I'm not so sure about specifics, but I think I just need to hear them explained in real life rather than in a text.

Reflections: This information is very useful when trying to find an answer to a problem with more than one variable. Also, since we have already studied contour diagrams, getting partial derivatives using these graphs is the next and more complex step in our math education. Furthermore, partial derivatives encompass all of the knowledge we have gained from the techniques for differentiation for single-variable function and then add on another step.

Sections 3.3 and 3.4

Main Points: Section 3.3 discusses the chain rule when taking the derivative of a more comlex function The chain rule is: dy/dt=(dy/dt)(dz/dt). The book also states that the derivative of (f(g(t))) = f ' (g(t)) * g ' (t). The derivative of e^kt = ke^kt. In section 3.4, the product rule and quotient rule are given. The product rule, in simpler terms than the book, is u'v + uv' and the quotient rule is (u'v - uv')/ v^2. These rules help to solve for derivatives of more than one function.

Challenges: Possible challenges in these sections could include confusion on the introduction of the z term in section 3.3. At first glance, it is a bit different to substitute a one variable term for an entire function, however, it just takes getting used to. The quotient rule could also cause a problem because if not used in the correct order of u'v - uv' in the numerator, the answer found will be wrong.

Reflections: The e formula in section 3.3 is very helpful in solving problems concerning interest and money matters. Also, the chain rule makes solving more complicated derivatives much simpler. My calculus teacher focused a lot on the product and quotient rule last year, so this reading assignment was pretty easy for me! She even made a song for the rules!