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!

Sections 3.1 and 3.2

Main Points: Sections 3.1 and 3.2 begin to introduce the derivative and the many rules for finding the derivative from a given equation. In the first section, the book states that the derivative of a constant, k, always equals zero. It then explains that the derivative of a linear function, mx + b, equals m since the derivative of a function is the slope. Also, if a function is multiplied by a certain number, say x^2 to 3x^2, then the derivative of the second function is simply the derivative of the first times the number it is multiplied by. Furthermore, derivatives of the sums or differences of two functions are simply the d/dx of function 1 +/- d/dx of function 2. Finally in section 3.1, the power rule is explained as d/dx [x^n]= nx^(n-1). In 3.2, the derivative of e is explained. Simply put, the derivative of e^x = e^x. Also, the exponential rule is stated as d/dx (a^x)= (ln a) * a^x. Finally, d/dx of ln x = 1/x. The graph of the function ln x is shown next to its derivative in order to show the differences and similarities between the two.

Challenges: In section 3.2, I found it very confusing the way the written explanation of a d/dx (a^x). However, once I studied the formula, I found it much less confusing.

Reflections: In calculus last year, we spent much time on derivatives. I found section 3.1 to be very easy, but section 3.2 was a much needed review for me. I always seemed to have trouble with the e^x and ln x derivative rules, but I now have a much better grasp on the situation.

Sections 2.2 and 2.3

Main Points: Section 2.2 defines the derivitive as f ' (x) = the instantaneous rate of change of f at x. It also states that a person can can tell if a function is increasing or decreasing by looking at the graph of the derivitive. For example, if the sign of the derivitive is greater than zero at a certain interval, it is increasing. The same goes for negative and decreasing, as well as zero and constant. Also, if f ' is large in magnitude, f is steep, and if f ' is small in magnitude, f is gently sloping. The book also gives examples on how to estimate a derivitive numerically. Section 2.3 gives alternate ways of stating the derivitive of a function, such as f ' (x)= delta y/ delta x, dy/dx. and d/dx (y). It is also important to remember that derivitives are much like slope in that they are composed in the manner of rise over run. The units of a derivitive are the units of the dependent variable over the units of the independent variable. Also, the derivitive of velocity equals acceleration, with the units being the length unit / (time unit ^2). The derivitive can also help estimate the values of a function. The book states that the local linear approximation of a function is: delta y is about equal to f ' (x)*delta x for delta x near zero.

Challenges: In section 2.2, I did not understand the ways to improve numerical estimating of a derivitive. The book explained it really bizzarrely and I couldn't comprehend it. Also, I didn't understand why the book was going in such a round about way to explain the derivitive, especially in the last example. Section 2.3 didn't really give me any problems but the different ways of how to state the derivitive could have confused a student who has never taken calculus or worked with derivitives before.

Reflections: This is still all kind of a big review for me, but I love derivitives so it's okay! My only problem is that the book takes somewhat simple concepts and explains them in super weird ways which totally throws me off! I really liked how the authors explained how to choose the units for derivitives though - I used to become confused on units when I took Calculus last year.

Sections 1.3 and 2.1

Main Points: Section 1.3 talks about the average rate of change. It defines it as the rate of change of y between t=a and t=b as delta y / delta t, or [f(b) - f(a)]/ b-a. The average rate of change of a linear function is the slope, and the function is linear if the rate of change is the same at all intervals. This section also states that a function is increasing if f(x) increases as x increases, and is decreasing if f(x) decreases as x decreases. Furthermore, it talks about concavity. A function is concave up if it bends up from left to right (like a smiley!) and concave down if it bends down from left to right (like a sad face!). Finally, section 1.3 says that average velocity is the change in distance over the change in time. Section 2.1 opens by defining instantaneous velocity of an object at time = t as the limit of the average velocity of the object over shorter and shorter time intervals containing t. The instantaneous rate of change of f at a equals the limit of the average rates of change of a f over shorter and shorter intervals of a. The book then defines a derivitive of a function f at a, f ' (a), as the instantaneous rate of change of f at a. The derivitive of a certain point equals the slope of the function at the point as well as the slope of the tangent line of that point. Section 2.1 then discusses how to numerically and graphically estimate derivitives.

Challenges: Section 1.3 didn't give me many challenges because it basically reviewed a lot of what I learned in Calc last year. However, concavity could have confused people because it's definitely difficult to remember which is CCU and CCD. Section 2.1 was also a big review for me, but I still got a little confused remembering the difference between instantaneous rate of change and instantaneous velocity because they are so similar.

Reflections: This reading made me super happy because I love derivitives and am very excited to do them in class! I also really liked how they explained everything in these sections: it was much easier to understand than my highschool calculus book!

Sections 9.1 and 9.2

Main Points: Section 9.1 talks about functions with two independent variables and states that the domain of a function f is a collection of all posssible inputs in (x,y). Also, the function is increasing if one variable increases while the other independent variable is held constant. This goes the same for decreasing functions. Contour functions can be represented numerically by a table of values, algebraicially by a formula, and pictorally by a contour diagram. A cross section of the function can be created by holding one variable fixed and letting the other variable change. Section 9.2 discusses contour diagrams more in depth. Types of contour diagrams include weather maps and topographical graphs. Contour graphs are made of curvy lines that show data, generally called contours. Weather maps have curves called isotherms which means "same temperature." Topographical graphs separate regions of lower elevation from regions of higher elevation. In this case, contours are also called level curves or level sets. The Cobb-Douglas Production model helps to figure out economic problems and has the standard form of: P=f(N,V)= cN^alpha * V^beta. In this formula, P is the total quantity produced, c, alpha, and beta are all positive constants, and both alpha and beta are greater than zero but less than one.

Challenges: The tables in section 9.1 really confused me. They were all really complex and I didn't really understand the examples at first. I understood the parts where the book gave equations, but looking at a picture got me really confused. In section 9.2, I wasn't really sure why they gave the Cobb-Douglas model and didn't do any sample problems with it. Also, the reasoning behind some of their explanations was worded oddly and I had to reread it to fully understand.

Reflections: These sections were totally new to me. I have never done these types of problems, but I like the fact that I can show increases and decreases in a picture. Sometime formulas are boring! Also, the section about topographical maps will help me understand those types of maps in classrooms better since before reading this, I wasn't really sure what they were for.

Section 1.10

Main Points: The main points of section 1.10 basically explain what a periodic function is, what characteristics make it periodic, and defines certain aspects of this type of function. A periodic function is one that has values repeating at regular intervals. The book also explains that in the function of y=Asin (Bt), A and B are parameters called amplitude and period. Amplitude is half the difference between the functions max and min values on the graph. The period is the time it takes the function to go through one complete cycle. Finally, a function in the form of y=Asin(Bt) + C or y=Acos(Bt) + C is periodic as long as Amplitude = A, period= 2pi/B, and vertical shift = C.

Challenges: My Calc teacher focused on periodic functions for a very long time, so I understood 99% of the reading. However, for someone who hasn't had as much emphasis on this type of function, he or she may have had difficulty understanding that when the period of a function (ie y=sinx) is increased (y=sin2x), the graph becomes narrower and when it is decreased (y=sin(1/2)x, the graph becomes wider. When I first learned this I was super confused.

Reflections: I really liked the sample problems the book gave for this unit because it actually showed how periodic functions relate to the world. In my past calculus class, we just did problems about finding the amplitude, etc., but the sample problem about high and low tide actually applied math to real life. Good deal!