LA 4.3

Main Points: When two values, x and y, that are proportional to each other but do not have equal constant values, the points are not found on the same line. However, a multiple of x can reach the closest to the y value, and this is called the projection of y on the line spanned by x. The step from x to y is caleld the residual vector, r. Because the residual vector and the quantity mx (the multiple of x) are perpendicular, the vectors equal 0. To solve for y, the dot product is necessary. The line y=mx is called the least squares fit to the data. Curve-fitting with two parameters leads to projections onto 2-dimensional subspaces. Scatter plots are used to collect data in order to find the values of m,a, and b in the equation s=ma +b. Finding the linear combination that is closest to s allows us to approximate the answer. Least-squares curve-fitting with J parameters leads to projection onto subspaces spanned by J vectors. The residual vector is perpendicular to all of the vectors in the set.

Challenges: I'm very confused about the J area of the LA. I really don't understand what J actually is or what it stands for. Also, I was confused on why the LA solved the equation in 4.3.1 the way that it did.

Reflections: This reading seemed to add onto what we were doing in class on Thursday. Even though I didn't fully understand the material, I think it will be more helpful in continuing what we were learning from the lecture on Thursday.

Reflections:

LA 4.1 and 4.2

Main Points: Some equations do not have solutions due to the number of equations being larger than the number of variables. For example, three equations with two variables will sometimes not have a solution. When this happens, an approximate solution is found. Dot products help to find approximate solutions to these equations.

Challenges: I understand the information about having to find approximate solutions, but I'm a little fuzzy as to where dot products come in to help.

Reflections: I still can't really comprehend why this is important in real life. The parts about angles and lenghts make sense, but when would you use something like this in the real world? Maybe more of what I'm asking is what kind of question/problem would lead you to use dot products to help find an approximate solution?

LA 1.3, 2.0-2.3, 3.0-3.3

Main Points: Linear combinations involve adding multiples of two vectors, or summing multiples. Problems involving this idea includes ones that ask you to find what multiples of vectors u and v can make vector w. Using simultaneous equations, a solution can be found and plotted, where the the x and y values are the intersection of the two lines on the graph. Two fundamental but different geometric interpretations can be seen when looking at the equations ax + by = e and cx + dy = f. They are: 1. what point is on the intersection of lines? and 2. What multiples are added to give vector w? This entire idea of combinations works for dimensions higher than 2 as well. Matrices are lists of vectors, and one list of vectors is in one column. An m x n vector is a rectangular array of numbers arranged in m rows and n columns. Three main iddeas accompany vectors. First, the span of a set of vectors is the set of all linear combinations that can be made with those vectors. Second, if a vector on the listt is not a linear combination of the other vectors on that list, the vector is said to have linear independence. Finally, subspace is the set of vectors that is a span of some list of vectors. The dimension of subspace is the minimum number of vectors required to span the subspace. Subspace is a very special collection of vectors.

Challenges: I'm a bit confused about the matrices. Specifically, if you have 2 sets of vectors, such as

(a b) and (e f)
(c d) (g h)

can you multiply a and g and c and f? Also, can you still multiply c and d and g and h?

Reflections: I know I've stated this in previous blog entries, but I'm not quite sure how any of this actually fits into the real world. I don't really understand exactly what a matrix is either and how it is helpful in anything that we are doing.

Section 10.7

Main Points: Differential equations can predict an outbreak of a disease epidemic as well as the level of vaccination necessary to stop the disease. The standard model used for this is the S-I-R model. S stands for number of susceptibles, or the people who are not yet sick but can become ill. I stands for the number of infecteds, or those that are currently sick. R represents the number of recovered people, or people who cannot be reinfected or continue to infect others. The number of susceptibles decreases with time as more people become infected. The sum of S, I, and R is unchanged, however, the numbers pertaining to each group can change within that certain number. Trajectories and graphs are also used in this type of problem to show how a disease is spreading. Where the rate of change of infecteds vs. susceptibles equals 0 (dI/dS = 0), a threshold population occurs. If the number of susceptibles is larger than the threshold population, an epidemic is present.

Challenges: I was a little confused wiht the constants used, and I don't really understand how the threshold population equals the proportion b/a.

Reflections: This section was pretty interesting because I always hear on the news about epidemics, but was never sure how people at the CDC and NIH decided that something was an epidemic. Now that I pretty much understand the S-I-R method, I can see how the scientists come to these conclusions.

Section 10.6

Main Points: Differential equations can be used to consider growth of two interacting populations, such as species competing for food, predator/prey, and symbiosis. This action requires a system of differential equations. This system is found by using the Lotka-Volterra equations. It states that the number of "encounters" between the two species is proportional to the product of the populations. Phase planes are used to graph the two populations against time. The point of the two populations versus eachother moves, and this path is called the phase trajectory. Slope fields are also used to see what the solution will look like. Setting the equations equal to zero helps find the equilibrium points. The shape of the trajectory tells how the populations vary with time, and a closed curve means that both populations oscillate periodically.

Challenges: I was confused on how the book came to the equations for the worm/robin example. I don't quite follow the part about the different constants either.

Reflections: The Lotka-Volterra equation is really helpful in finding how different populations depend on eachother. I'm sure that ecologists use this alot to see how different species are changing in number and how the food chain works.

10.4 and 10.5

Main Points: The general solution to to dy/dt=ky is y=Ce^kt for any constant C. For populations, the rate of grouth is usually proportional to the size of the population. In the equation dy/dt= k(y-A), k and A are constants. Also, y= A + Ce^kt. However, C is not the initial value of the variable, but the initial value of y-A. An equilibrium solution is constant for all values of the independent variable. The graph is a horizontal line. Equilibrium solutions can be identified by setting the derivative of the function to zero. It is stable if a small change in the initial conditions gives a solution which tends towards the equilibrium as the independent varibale tends to positive infinity. It is unstable if th curve veers away from the equilibrium as the independent variable tends to positive infinity. An example of equilibrium problems could involve Newton's Law of Heating and Cooling.

Challenges: The only part I had trouble understanding was the section involving dy/dt=k(y-A). I am unsure on how to use this equation.

Reflections: Differential equations can be very helpful in numerous ways, such as predicting population growth/decay, how fast a a drug can dissolve in a body, or the amount of pollutant in a lake or sky. I've already done differential equations, but I definitely need a refresher on them.

Sections 10.1 and 10.2

Main Points: Differential equations are made from information we know, such as the derivative or the rate of change and are used to find an unknown value. This is useful for finding the net worth of a company or the rate of change in a population. Differential equations involve the derivative of an unknown function. A solution to a differential equation is any function that satisfies the differential equation. These sections also talk about finding an unknown constant in a differential equation. The general solution is a family of functions while a particular solution is one for a specified value. The differential equation and the initial condition together are called an initial-value problem.

Challenges: Some students may have found the idea of finding the unknown constant challenging since it involves more information. Also, creating a differential equation is a bit difficult because you have to know exactly which information goes where in the differential equation.

Reflections: Differential equations are super helpful in the real world. Finding the net worth of a company or increases or decreases in population growth are important values to know, especially for other companies or projects, etc.

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!

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!

First post!

My name is Sam DelSerra. I'm from Scranton, PA and I'm a first year at Mac. I plan to major in biology with an emphasis on biochemistry. Applied Calculus is my first math course at Mac, but in high school i took Geometry, Algebra II/Trig, Pre-Calc, and AP Calculus AB. The weakest part of my math background is probably geometry (too many thereoms!) and certain areas of calculus that I can't really remember because I tend to forget things I don't like/don't understand easily. My strongest math skills are deriving and integrating and I'm also pretty good at linear functions. I decided to take Applied Calculus because I need a math course for the Bio major and I felt that I would be able to remember how to do most of the problems since I just finished AP Calc in June. Also, I wanted to see how calculus actually works in the real world outside of the classroom. I hope to be able to actually use what I learn in the course in real life rather than just forgetting everything once I take the final.

My interests include singing, dancing, sleeping, eating, and science. I love almost all types of music, but my favorites include John Mayer, Jack Johnson, and rap (mostly Pitbull and Lil' Wayne). I also enjoy hanging out with friends and family and aimlessly driving around in my car (not so much now since gas is ridiculous!).

The worst math teacher I've ever had was my Geometry teacher in freshman year. She moved too fast and didn't make sure everyone in the class understood the material. She usually just asked the kids who had a knack for math and ignore the rest of us. Also, if you answered a question wrong, she would scold and embarrass you. I'll admit it - I cried a few times in class. The best math teacher I've ever had was my Pre-Calc and Calc teacher. She was strict, but she always made sure everyone understood everything. She made herself super available to get help from, and since she had me for two years in a row, she could tell when I was lost in class and would make sure to re-explain the problem. Also, she had a lot of faith in me and would call me out when I would psyche myself out on a problem (which I often tend to do). That faith really helped me become more comfortable with math in general.

I would love to hear Jack Johnson played before class because his music wakes me up! I recommend Waiting for Summer or Sitting, Waiting, Wishing!

SECTION 1.2

Main Points: This section explains what slope is, and it focuses on linear equations. Slope is the rate of change vertically divided by the rate of change horizontally, or in simpler terms, rise over run. Also, this section gives the standard point-slope form for linear equations as well as the standard form for a linear equation. Point slope is y -y1 = m(x - x1) and standard form for a linear equation is y= mx +b, where m is the slope and b is the y intercept.

Challenges: I was challenged at finding some of the patterns for an equation from the tables. After working on them for a bit, I did eventually come to the correct answer. Also, some of the word problems were oddly worded and confused me at first. I had to read and reread them in order to figure out what the question was actually asking.

Reflection: I've done this type of problem many times throughout my highschool career, and I actually enjoy linear equations. I think this is one of the easier areas of Calculus and it's nice to be able to review what I learned in highschool.

SECTIONS 1.5 and 1.7

Main Points: In Section 1.5, the book discusses exponential functions and growth. Exponential functions help show phenomena, such as population growth, in natural and social sciences. These functions can increase and decrease and the standard form is P=Poa^t. If a > 1, the function is growing, but if a<1, e =" 2." p=" Poe^(kt)" interest =" P="Po" interest =" P="Poe^(rt).">

Challenges: In 1.5, I found problem number 21 difficult. I can't understand why the final answer is P=670(1.096)^(t/1000). When I did the problem, I got P=670(1.096)^t. Also, I had trouble with number 27. In 1.7, I had difficulty with problem number 29. I wasn't sure how to go about the problem because I was confused as to what it was asking.

Reflection: Towards the end of my senior year, we focused a lot on exponential functions. My teacher used slightly different equations, so it took me a little bit to get used to the P's and Po. (She taught us y=Ce^kt) I really like doing these problems because I think its pretty cool how you can figure out how much money you will make in an account or how many people will be living in a city in 200 years. I like when math actually applies to the real world!!