Review of week eight and preview of week nine (homework in separate post)

Hi all,

Last week we had our second test and talked about the coordinate system, next week we will talk about lines in the coordinate system and quadratic equations.

The coordinate system is a powerful tool to combine algebra with geometry: pairs of points, distances and equations are all presented on the coordinate plane with explicit pictures. We have plotted some pictures on Thursday, and you have seen the general way of plotting graphs of functions: take some arbitrary values of x and compute the corresponding y values from the function, plot those dots in your plane and connect them. The examples we did in class on Thursday are all linear functions, next week we will see some quadratic functions.

Tests will be handed out to you next Tuesday, make sure to do corrections! If you want to talk about any of the problems, come for help! The corrections are due Nov 17.

Homework problems are posted in a separate post, don’t forget to do them!

Alright, have a good weekend, see you next week!!

Review of week eight and preview of week nine (homeowk in separate post)

Hi all,

Last week we covered trig integrals, trig substitutions and rational integrals. Next week, we will finish with rational integrals and introduce improper integrals. Just to remind you, a test is coming up, so make sure to review!

The major goal for trig substitution is to handle nonlinear functions such as fraction exponents, and typical examples we see are summed up in the table on page 467. By doing these substitutions and using the trig identities, we are left with one whole piece under the square root. Therefore, if you see strange functions under the square roots, try to complete the square and leave only constant outside– an example is Example 7 on page 471.  Start making your note sheet and make sure to write down the trig identities on your sheet for your test!

After your trig substitution, you should get an integral shaped like the ones in section 7.2: product of powers of sine and cosine, or product of powers of tangent and secant. Use the general methods we studies in section 7.2 and classify the resulting trig integrals by the powers of the trig functions. The results are summed up in tables on page 462 and 463.

Rational integrals have integrands that are rational functions. Any rational functions can always be simplified to the sum of a polynomial and a proper rational function. By proper, we mean the numerator has degree strictly lower than the degree of denominator. If the rational function is already proper, we do partial fractions and decompose  the integrand into things we know the antiderivative of; if not, first simplify by long division.

We covered the case where the results of partial fractions are distinct linear factors, next week we will study the cases where the results of the partial fraction contain repeated factors or irreducible quadratic factors.

The third true and false problem in our test 1 contains an integral that has a undefined spot in the integration domain, such integrals are one kind of improper integrals. Improper integrals are definite integrals that contain either undefined points in the integration interval, or has infinity as one integration limit. For these integrals, we need to be careful and check if it makes sense to compute them first.

Homework problems are in a separate post, make sure to check them out!

Alright, hope you have a good weekend, and don’t forget to do quiz corrections, they are due Nov 12/13!

Review of week seven and preview of week eight (test on Tuesday 20th)

Hi all,

Last week, we talked about systems of linear equations. There are two ways of solving systems: substitution and elimination. Either way, the goal is to convert the new problem (a system of two equations with two unknowns) into an old problem (one single linear equation with one unknown variable).

Next week, we will have our second test on linear equations, linear inequalities and linear systems. Please practice by reviewing your homework and doing the practice test. Email me if you have any questions, or meet up with me on Monday!

On Thursday, we will make the connection between algebra and geometry by introducing the coordinate plane. By then, we can talk about what linear equations, inequalities and systems mean: they are all related to the geometric object called line (therefore linear).

There is no homework due next week, but review to study for the test!

Alright, that’s it for now. Hope you have a good long weekend, let me know if you have any questions!

Review of week seven and preview of week eight (Homework due Thursday 29th/Friday 30th)

Hi all,

Last week we finished with the two applications of integration: for definite integrals, we applied the idea of Riemann sum to compute the volumes of solids; for indefinite integrals, we studied the separable differential equations.

In computation of volumes, we use the same idea as computation of areas under curve: cut the big solid into small pieces that we can approximate with regular shapes, add up the small pieces and take the limit as the cut gets more and more fine. This process leads to a definite integral that we can compute. Areas (from chapter 5) were approximated by long thin strips of rectangles, and solids are approximated by thin slices or thin shells.

We mostly focus on solids of revolutions, as they are easy to compute. With slicing, the thickness of your slice determines the variable in your integral. Find outer radius and inner radius in that variable, and compute the area of cross section by \pi R^2-\pi r^2. The volume is (\pi R^2-\pi r^2) thickness. With shell method, the thickness of your shell determines the variable in your integral. Find the radius and height of your shell, and compute the volume by 2\pi radius height thickness. These quantities are normally related to the x or y coordinates of the points on the bounding curves, depends on the rotating axis.

Differential equations contains a large class of different types of equations, as long as they are equations with derivatives. We look at separable equations, the kind that you can separate independent variable and unknown function onto two sides of the equation. This way, on each side of the equation you see a differential form explicit in only the independent variable or the unknown function. To solve the differential equation, integrate both sides respectively. Remember to add the arbitrary constant C and check for zeros when dividing by the unknown function. If you are given an initial conditions, use it to determine the constant C.

Next week, we will talk about the last technique in integration: trig substitution. Trig functions come with a lot of identities that can help simplify integrands, therefore are good to try if u-sub or IBP don’t work. Please review the identities in our textbook for next week!

There is a quiz on Monday/Tuesday, on volumes (both set up and compute).

Finally, here is the homework:

Reading: 9.3,7.2,7.3

9.3: 2,3,5,9,10,12,13,16,19,35

7.2: 7,11,12,19,51(no need to graph)

7.3: Read 1-3, do 6,7,9,10,18

Alright, that is it for now. See you next week and have a good long weekend!

Review of week six and preview of week seven, homework (due 22, Thursday)

Hi all,

This week we covered linear equations and linear inequalities and applications of them.

To solve linear equations or linear inequalities, your goal is to isolate the variable (say x, for example). An approach I normally take is to put all constants on one side of the equality/inequality, and all terms with variable on the other side. Finally divide out by the coefficient multiplied by the variable. In equality, you don’t need to worry as much, but for inequalities, make sure to change the direction of your inequality if you are dividing by a negative number.

Next week, we will look at two linear equations with two variables and solve them. The idea is to transfer the two equations into one, then the problem becomes something we did in section 1.1.

On Thursday, we will review for the test on Oct 27. I will put a practice test online next week, and we will work on that.

Here is the homework due Oct 22 (Thursday):

1.2: 10,14,19

1.5: 3,4,16,21,45,84,87,88,115

9.1: 15,16,19,20,37,38,41,42

Alright, have a nice weekend and let me know if you have any questions! Don’t forget to do test correction!

Review of week six and preview of week seven, Homework (due 21 Wednesday/22 Thursday)

Hi all,

Last week we covered two methods of computing the volume: the slicing method and the shell method. Next week we will finish talking about the two methods and introduce my favorite topic in math: differential equations!

The two methods of slicing and shell are mostly used in computing the volume of solids of revolution, but you can also use them to compute the volume of general solids (for example, exercises 51 to 71 in section 6.2). The difference between the two methods is the small piece of volume you use for adding up: in slicing, small pieces of volume are thin slices with a base area (like sliced bread); in shell, small pieces of volume are thin shells with height and radius (like one layer in the bottom of a chuck of leek or green onion). Therefore in slicing, focus on finding the area of the cross section; and in shell method, focus on finding the height and the radius. To find out which variable is used in your integral, look at the thickness of one slice in slicing method, and the thickness of the shell in shell method.

Next week, we will see more exercises on computing volumes, and you will choose whether to choose slicing or shell method. The principle is always use whichever is easier to set up and compute.

The differential equations we will see are called separable equations. The idea is basically to translate your problems into two separate integrals. You will see then why it is important to add the arbitrary constant for indefinite integrals: the constants are related to the initial conditions.

There is a quiz on Thursday/the Monday of 26th. I will put up a practice quiz sometime next week.

Finally, here is the homework:

6.2: 25-30, 33,34,41,43,63

6.3: 1,2,3,7,8,11,23,24,30,38,40

Make sure to do quiz/test correction, the deadline is next Friday!!

Alright, have a good weekend and let me know if you have any questions!!