Setting Up Differential Equations

Essentially a differential equation can be written from the information provided. All of the examples in section 10.1 were from verbal description. The most difficult part was the logistic model, because I personally have difficulty imaging a graph from a verbal description, for example when dP/dt =kP(L-P) the graph showed delta p small's Im not sure how that was known to satisfy the curve.

My favorite example was about the dug in the body, I like bio-pharmaceutical problems, they are easier to visualize... much more favorable than economics problems... I know its a personally preference, but hey if it helps the math skills all the better. Also, since I work in a pharmacy it was funny to actually recognize the drug used in the example, Vancomycin.

Constrained Optimization

This was sort of a strange concept but page 381 broke down how each step was obtained was amazing! I believe the constraint on a function is actually a second function that limits the first. The hardest part was the Lagrange multiplier.It was interesting that it is a rate of the change in the optimum value of f over the change in g. but i don't understand the purpose of the Lagrange multiplier, what does it have to do with constrained optimization?

On a random note... are constraints represented graphically as contour maps?

What was neat was how useful constrained optimization is, it really applies to real world situations. Like businesses and budgets. Seriously if you do not put constraints on functions than you can see all possible values, but not the ones that are important to a situation.

Seriously we are in college we need to budget just about everything from time to money to life!

Global Maxima and Minima

It was really interesting to see that global maximum and minimum differ if the endpoints of a function or included or not. But it makes sense that if there are no endpoints there can not be a global min or max because the function is continuing, you may not be able to see the real max or min. I had some difficulty with example #2; the question asks for the time when photosynthesis is fastest, but the rate at that time so is time the first derivative and the rate of at the time ,t, the second rate? Its the units I am trying to get to that's confusing.

What I found interesting was the graphical example, about minimizing gas consumption because I commute to my job about 45 minutes away every weekend and that driving speeds effect gas consumption and that is helpful to know since I unfortunately drive and SUV... yeah 14 miles per gal....

The Second Derivative, Local Maxima and Minima, and Inflection Points

Its really cool to think that f",f', and f are related. f" lets us know depending on whether it is less than or greater than zero, which describes if f' is concave up or down! I enjoy clear connections between ideas. However I did not understand the government defense budget example, perhaps that was worded funny?

Local Maxima and Minimum were not too hard of an idea to grasp, but one clarification is the critical point f'(p)=0 reflected in the original function as crossing an axis? I want to make sure I am drawing the correct conclusions from the reading. I had the most difficulty with the idea that critical points could divide the domain of f into intervals. If there are critical points, must we look at the function f in sections defined between two critical points or can it still be looked at as a whole?

The idea of an inflection point was pretty easy, its just when the concavity changes, but its neat to realize inflections occur when f" changes signs.

I think the class will have the most difficulty with keeping all the relationships together. Also perhaps the units of the second derivative. Understanding f" as a rate of change, I'm not quite sure if it is as simple if the second derivative is positive then the rate of change is increasing and vice versa. It seems to simple to be right... Or am I over thinking it?

The mid-term exhausted mind is a funny thing!

Partial Derivatives and Computing Them Algebraically

I have actually never been exposed to the concept of partial derivatives before and was confused when the books definition of a partial derivative contained limits. I thought this class wasn't planning on using limits??? Also i don't think the purpose of a partial derivative is explained either? Is it just a fraction of a rate of change?

Perhaps the class will have the most difficulty with second-order partial derivatives, especially if the concept of partial derivatives was difficult to begin with. But what I did like about these sections were familiar examples used to try and explain this type of function.

Derivatives of Periodic Functions

This section was not too terribly difficult, and rather short too. But its good to know that the derivative of sine is cosine. But what I found the most difficult is that the reverse is not true. The derivative of cosine is not sine, it is -sine. Why is that?

I think the most difficult thing the class will work with is looking at a function and having to figure out which of all the available rules we were introduced to in this chapter to use!

The Product and Quotient Rules

After Monday's class the chain rule was much clearer and section 3.4 introduces to new rules to help with derivation; the product and quotient rules. I didn't find this section to terribly difficult, but I wished a couple more q's were assigned for homework for more practice. Honestly the most difficult part was remembering that for the quotient rule the denominator is squared. By the way why is it that the derivative is squared. Can you explain that in class?

I believe that the most difficult idea for the entire class would be the order of which derivative should be taken for each piece. For example the product rule, (fg)'= f'g+fg'
It looks simple but sometimes when you forget, it can mess it all up. So the class has to remember the derivative of f multiplied by g, then add that to f multiplied by the derivative of g.

The Chain Rule

The eight different examples used to explain the chain rule were very helpful. I found the car situation the best at explaining this rule. To find the rate gas is being consumed is rationalized by seeing how units cancel eachother out. What I found most difficult was example 2, I felt like the chain rule was trying to use composite functions... Maybe it was just fuzzy, because I understand the idea of inside and outside functions but just reading on my own i don't feel confident in it.


The class may have the most difficulty with using the chain rule without units, such as the bank account example (#8) where you cannot double check you answer by seeing if the units are correct. On the bright side there is familiarity in the idea its a question about rates.

Short Cuts to Derivatives and Exponential and Logarithmic Functions

Finally a direct rule! The power rule is officially my favorite. Section 3.1 wasn't too terribly difficult. It was more exciting to have more rules to use so that taking derivatives is more efficient. But what i did have some difficulty with was trying to understand how derivative of ln(x) = 1/x. Is there a proof to show how this is true or came about? I just don't like to use rules if i don't understand how it works.

As a whole the class may have trouble with the behaviors of the original function and the derivative. for example Monday's NYT about what f'(x) can tell about f(x) not everyone could understand what happens from one range of values to another. (increase,decrease, no change) So perhaps before we step forward into analyzing more complex functions.

until next time