10.1 and 10.2: Differential eqns

Main Points

Differential equations are those which express information about the deriv. or rate of change of the original eqn. Instead of starting with the eqn and computing the deriv, you can use the deriv and work backwards. This method can be applied to many different type of real world problems sucha s marine harvesting, the net worth of a company, and pollution in a lake. The rate of change of a population can be expressed by the constant of proportionality times the current population times the difference btwn. the carrying capacity and the current population. The solution to a differential eqn is in fact another equation, it is the function with that deriv. The general solution is the family of functions that satisfies it, while the particular solution is the function with variables set at specific variables.

Challenges

How can an equation be the answer to another equation?

Reflection

Because we start with a deriv and work off that, isn't using differential eqns like finding the indefinite integral? 

9.6. constrained optimization

Main points

When the function has a limit, to express the "constrained" max or min you set it equal to the upper or lower level, respectively. Graphically, the limit is represented by a line that intersects with the contours of the function line. The max lies on this line because anything bellow is not reaching full potential by not hitting the limit, and everything above is disallowed because it exceeds the limit. Lagrange multipliers can also be applied to a constrained eqn to optimize it. Local max.s occur at a particular point that are greater than all immediate points and meets the requirements of the constrainment, while the global max this is true for all points of the function, vice versa for min.s. The optimization is found by plugging the function into three specific eqns. The Legrange multiplier used in these eqns is defined by the optim value of a funtion as the constrained value is increased by one unit. The Legrange multiplier is the rate of change of the increase in a function per unit. Furthermore a constrained function can be optimized by plugging it into the Legrange function and determining the critical points.

Challenges

Why do you set the limit equal to the limit instead of it having be less than or equal to? If there is a limit, like the 378,000 used in the book, how can that possibly represent a line as opposed to a point? so the legrange multiplier is not like e right, in that it is a special number? is it a special ratio? i dont understand it's significance. 

Reflections

Because the Legrange multiplier represents a rate of change, is it closely connected with the derivative function? Is it the deriv.?

4.3: global max and min

Main Points

While local max and min are the function values lowest or highest on a specific curve of the graph, the global max and mins are the highest and lowest points of the entire function, respectively. Because those points are the lowest and highest of the entire function, they are more extreme than any points on the same curve as well. Therefore they are also local max and mins,  and the lowest local min and highest local max are the global mins and maxs, respectively.

To find the global max and mins you compute all critical points and find the most extreme ones. 

Challenges

I understand that the global min would also be the local min. Because all animal are not cats while all cats are animals, then why does the local min have to be a global min. I just realized that its not, but one has to be. The contrapostive is that if its not a local min it's not a global min. So i guess i no longer have a challenge but the whole point of the blog is to teach yourself so i hope that's okay.

Reflections. 

It makes sense that the global max and min are important because this can show you how to obtain the greatest yield or minimize damage for a real world problem. Now I don't understand why the local mins and max's are important. Is it just to find to global ones? Perhaps i should switch my challenges and reflections paragraph.

Main Points

Because f'(x) is function, evident by its graph, it too has a deriv. f"(x), like any deriv, shows certain properties for f'(x), which in turn indicates certain things for f(x).  When f" is positive, the function f' is increasing, therefore f is concave up. The opposite is true for when f" is negative. The local maximum is the highest value on the part or the curve, while the local minimum is the lowest. A critical value is a point where the deriv is undefined or =0. It is also the maximum or minimum of a function. For a function f, the min occurs when a function changes at p when the function changes from decreasing to increasing. When the change is form increasing to decreasing, it is a max. Because concavity is related to a function increasing or decreasing, when x is concave down there is local max, when it concaves up there is a local min.

Changes in convavity result in inflection points.

Challenges. 

How can p be a point. Isn't a point an x and y value, there for how can there be f(p)?

Reflection.

I have gone over maximum and minimums and inflection points in my previous calc course.  I think the better understanding of derivs i now have allows me to understand the implications of concavity and therefore max and mins, because they occur at points of concavity.

4-8

Main Points

The gradient is the vector that included the derivative of both the x and y coordinates of a point.

While partial derivatives state the rate of change in either a straight vertical or horizontal point, a directional derivative shows the rate of change in any direction. Gradients have certain defined traits. They are always directed toward that of the largest increase and point away from greatest decrease. Length and slope steepness are directly proportional. For contour diagrams, the gradient vector is perpendicular to the curve it starts at. All gradient vectors are directly related to and there fore correspond to curves,

Challenges

Don't we normally find the deriv at a certain point. Why is this now a partial deriv? I don't understand the notation for the directional deriv. What does it mean that the gradient vector points in the direction of greatest increase? Greatest increase of what?What are the curves in the gradient contour diagrams?

Reflection

The idea of gradients and directional derivs builds nicely on my knowledge of derivs and partial derivs. However I cannot think of them as contour diagrams.

3.5, 9.3, 9.4.

Main Points

The derivatives of sine and cosine are closely related to the functions themselves. The derivative of sine is cosine, and the deriv. of cosine is sine. This is true regardless of if you are finding the deriv. of x in radians or an actual function. To find the deriv of a function times a constant, the constant is taken out and multiplied by the sin/cosin of the function. 

To find the deriv of a multi-variable function, the same rules apply for when finding the deriv.s of regular functions. To determine the partial deriv of f(x,y) either x or y is held constant. They can be determined through tables, equations, and contour diagrams. The deriv. of a partial deriv is called a second-order partial deriv. Any function with two variables will have 4 of these type of derivs.

Challenges

 Why is the deriv of sin=cos, but the deriv of cos=-sin, not sin? What is a mixed partial deriv.?

Reflection

I have never heard of second order partial deriviatives, but the concept makes sense. if each partial derivative has 2 derivs, it would make sense that they each have 2 derivs, resulting in 4 second order partial derivs. I don't really understand them, but I think that you can leave one variable constant each time, resulting in the 4 posibilities. it is interesting that leaving y constant for the partial deriv and x constant for the second-order is equal to if you made x constant and then y. I hope to learn more about why this is true.