Main topics for the First Midterm

Transcription

1 Main topics for the First Midterm Midterm 2 will cover Sections , , , This is roughly the material from the first five homeworks and and three quizzes. In particular, I will expect you to know how to solve each problem from the homeworks and the quizzes. In addition, you will certainly need to know the material from Math Make sure you are comfortable with the following: Understand the material from Math In particular: Understand what an integral is, both intuitively (as a sum of infinitely small quantitites), and the actual definition (involving upper and lower sums). Make sure you understand the notations b f(x)dx and f(x)dx. In particular make sure a you understand why the first one is a number, and the second one is a function of x. Pay attention to which one you are asked to compute. If I ask you for a definite integral, your final answer should be a number, If I aks you for an indefinite integral, your final answer should be a function of x (or whatever the dummy variable is.) Know how to use the fundamental theorem of calculus to compute an integral by finding an antiderivative. Know how to use u-substitution to compute integrals. Make sure you are familiar with the forms for definite integrals and indefinite integrals. For definite integtals, remember that bounds change? (This can save you a lot of work by avoiding going back to the original variable.) For indefinite integrals, remember that you final answer should not be a function of u. Know the common trircks for u-substition (letting some complicated expression be u, letting u = x + a, finding a u that will make du appear in the integral, etc.) Don t forget that you also do simple things like u = ax + b, u doesn t have to be a complicated function of x in order to be useful. Have a look at section 8.1. Everything in there (except the hyperbolic functions) should seem pretty familiar to you by now. Know how to compute (f 1 ) (x) if f is one-to-one function. Remember that you can sometimes compute this (at least at specific points) without first finding f 1 (x) Know how to define ln x and exp(x), and know their basic properties. Know how to compute f (x) f(x) dx Trig Identities: A number of things we have covered this term involve trig functions, so you will need to know basic trig identities. These will be much more essential this term than they were the last term. 1

2 Know the simple identities like sin( x) = sin(x), cos x = sin( π 2 x), sin(π x) = sin x, etc. If you understand know the unit circle works, you won t need to memorize all of these. You should be able to come up with them on your own by just lookint at the picture. Know how to find the values of trig functions at certain special values (e.g. those you can get from or right triangles in terms of radians!) Know the Pythagorean identities: sin 2 x + cos 2 x = 1 and 1 + tan 2 x = sec 2 x. They can help you integrare expressions of the form sin n (x) cos m (x) where at least one of n, m is odd.they are also usefull to expressions of the form tan n x sec m x where n is even or m is odd. Know how to take the derivatives of the basic trig functions. In particular, make sure know that (sin x) = cos x, (cos x) = sin x (remember these signs!) and (tan x) = sec 2 x = 1. Also, remember that you can ger the others with the quotient rule if you cos 2 x have forgotten them. Know how to use derivatives of trig functions to help you integrare expressions of the form sin n (x) cos m (x) where at least one of n, m is odd. They are also usefull to expressions of the form tan n x sec m x where n is even or m is odd. Know the double angle identities: sin(2x) = 2 sin x cos x and cos(2x) = 2 cos 2 x 1 = 1 2 sin 2 x and the related half-angle formulas. Focus on how you can use these to integrate expressions of the form sin n x cos m x for n, m both even. It may also help to be somewhat familiar with the sum identities: sin(a+b) = sin a cos b+ cos a sin b, cos(a + b) = cos a cos b sin a sin b and the related product identities for sin a sin b, cos a cos b, sin a cos b. They are helpful in integrating experessions of the form: sin(nx) cos(mx), sin(nx) sin(mx), cos(nx) cos(mx). Inverse Trig Functions: Know how to define the basic inverse trig functions: arcsin and arctan (in particular, what are their domains and ranges?) Know how to define other inverse trig functions (e.g. arccos, arcsec etc.) in terms of these. Know how to compute certain values of these functions (if you know some value of a trig function, then you also know a value of the inverse trig function.) Know how to compute things like tan(arcsin x) of arcsin(sin x) (note that this is not always x) etc. If you can t immediately see how to do this with a trig identity, try drawing a picture. Most problems like this can be solved by drawing the init circle or an appropriate rignt triangle. Know how to take the derivatives of inverse trig functions (especially arcsin and arctan) Know how to use this to compute dx a 2 x 2 and dx x 2 +a 2 2

3 More generally, make sure you know how to compute things like dx or dx ax 2 +bx+c ax 2 +bx+c by completing the square (keep in mind that how you handle these will depend on whether the polynomial ax 2 + bx + c has real roots or not. Integration by Parts: Understand how the fundamental theorem of calculus and the product rule give the identity u(x)v (x)dx = u(x)v(x) u (x)v(x)dx. a Also know the definite version of this: u(x)v (x)dx = u(x)v(x) a b a u (x)v(x)dx. Remember that these are the limits for x, not for u or v. Know how to use this to sometimes compute f(x)g(x)dx - integrate one factor, and differentiate the other. If you are luck, the new integral will be easier to compute. If it is harder to compute, try the switching the derivative and the function. Remember that this often won t be as simple as memorizing a formula and plugging something in to get your answer. Integration by parts in useless if you can t figure out how to use if effectively. b There will sually be many different ways to break up your function into a product of two functions. Not all of them will be useful. Don t be afraid to try several things of your first guess doesn t work. Remember that there is always at least one way to break up a function as a product: f(x) = (1)f(x). Consider using this if f (x) is a much simple function than f(x), (e.g. f(x) = ln x, arcsin x, arctan x etc.) If using integration by parts doesn t immediately get you the answer, that doesn t mean you should give up on it. Did you end up with an integral which looks simpler? Consider using integration by parts again (e.g.. to compute x 2 e x dx). Maybe the integral you ended up with doesn t look any simpler. Can you find another way to relate if back to the original integral? Think about how we computed e x sin xdx: We used integration by parts twice and ended up with an equation we could solve for e x sin xdx. Don t forger about other techniques (like u-substitution) Is it possible that you could use u-substitution either before or after you use integration by parts? Products and Powers of Trig Functions: Know how to compute the integral sin n x cos m xdx for any m, n, Remember that if either m or n is odd, this is easy, simply use u substitution, with u = sin x, cos x and use the identity sin 2 x + cos 2 x = 1 to write the integral in the form f(u)du for some polynomial f. If m and n are both even, you can use the half-angle identities to decrease the exponents. Know how to hande something like sin(mx) cos(nx)dx (you can use the product identities) Also know how to deal with integrals involving other trig functions e.g. tan 3 xdx, sec 3 xdx. Remember the various tricks you can use, like u-substition, trig identities, and integration by parts. 3 b

4 Trig Substitution: Be familiar with the substitutions x = a sin u, x = a tan u, x = a sec u. How do they work? What is dx in each situation? When/why are they useful? (They are useful for simplifying expressions like a 2 x 2, a 2 + x 2, x 2 a 2 respectively.) What interval whould you pick u to lie in? For example, if you let x = a tan u, then you may take u ( π 2, π 2 ) in which case you will always have sec u > 0 and so x 2 + a 2 = a sec u. If you do this to compute an indefinite integral, then you will need to put the answer back in terms of x Make sure you know how to do this, Often simplifying your answer will involve computing things like sin(2 arctan x). Know how to do this (in this case you would want to first use a double angle identity and then a suitable right triangle.) Know what to do with other quadrativ functions. Any quadratic can be written in the form u 2 a 2, u 2 + a 2, a 2 u 2 by completing the square, Make sure you know how to do this and that you can use this to compute an integral involcing some quadratic function. Also keep in mind that an expression like a 2 x 2, a 2 + x 2, x 2 a 2 doesn t mean that you have to use trig substituion, Keep an eye for simpler approachs. You can compute x 4 x dx by setting x = 2 sin u, but is there something much 2 simpler you could do? Integrating Partial Fractions: Remeber that to use the partial fraction decomposition method, you first need a proper rational function. If the degree of the numerator is higher than the degree of the denominator, you will need to first divide the numerator by the denominator beore using the method. Also make sure that quadratics you end up with are really irreducible (for x 2 + bx + c you need b 2 4x < 0) Make absolutely sure that you know how to compute A (x a) dx and Bx+C dx for n (x 2 +bx+c) n x 2 + bx + c. If you don t know how to compute these integrals, then you won t be able to use this method. Remember, the first one is easy to compute. Just let u = x a, then you are left with du u Don t try to over complicate this. You don t need to complete the suare, or use n trig substitution of anything like that. This is the sort of things we were computing last term. The second one is harder. Your first step should be to complete the square. Then you are left with u du and 1 du. The first one of these can be done with a (u 2 +a 2 ) n (u 2 +a 2 ) n simple substition (let v = u 2 + a 2 ). The second one requires trig substitution at least of n > 1. In the case of Ax+B dx it will be useful the break the numerator into two parts x 2 +bx+c Ax + B = const.(x 2 + bx + c) + const. Why is this helpful? 4

5 Partial Fraction Decompostion: Know how to integrate a rational function (write it a sum of partial fractions, then integrate each partial fraction.) Remember that only proper rational fractions can be written as a sum of partial fractions. Know how to write a rational function P (x) r(x) r(x) Q(x) in the form p(x)+ Q(x) where Q(x) is proper. (This is just polynomial long division.) Know how to tell which partial fractions can appear in the decomposition of a given rational function. You can do this by factoring the denominator. A factor of (x a) n corresponds to A factor of (x 2 + bx + c) n corresponds to A 1 (x a) + A 2 (x a) A n (x a) n B 1 x + C 1 (x 2 + bx + c) + B 2x + C 2 (x 2 + bx + c) B nx + C n (x 2 + bx + c) n Remember, the numerator is always constant in the first case (and linear in the second case), regardless of the exponent n. If you have something like Bx+C, then (x a) 2 you have done something wrong. As a check: the number constants you introduce should always equal the degree of the denominator. If you have too many constants, you will not be able to solve for all of them. If you have too few, you will (probably) get a contradiction. If you set up things incorrectly, you will most likely get stuck. Be careful! One you have set things up, you will need to find the values of the constants. Your first step will (probably) be to clear up the denominatos by multiplying by the original denominator. There are two main methods for finding the constants: plugging in different values of x (think about whihc ones to use! What whill make your work easier?) or expanding and comparing the coefficients. Know how to use both of these. Also learn how to recognize which one will be less work. Also, be aware that you don t have to restrict yourselfto only using these Sometimes it must be useful to combine them. Maybe you can figure out a few of the coefficients by plugging in values of x, and then find the rest by expanding (before you expand, can you maybe factor out something like (x a) to make your life easier?). Don t be afriad to try other things, if they look like they might make things easier. You can use any method you want, as long as it works. Differential Equations: 5

6 Know what a differential equation is, and what it means to solve one. Know the difference between a geneal solution (i.e. any function that satisfies the equation) and a particular solution (i.e. one satisfying y(a) = b. Know how to find a particular solution if you know the general solution. Be sure to distinguish between the two types of diffential equations we studied. You won t be able to solve one type by using the method for the other type. Know how to solve a first order linear differential equation: y + p(x)y = q(x). (Multiply by an integrating factor e H(x), to turn the LHS into (e H(x) y). How do you figure out the appropriate H(x) to use? Really check that derivative of e H(x) y is exactly the LHS. If it is not, then you have done something incorectly. Note the plus sign in front of p(x). Know how to solve a separable differential equation (p(x) + q(y)y = 0). Sequences: Know how to tell when your equations is separable, often you will need to rewrite it. Remeber that you need to integrate both sides of an equation with respect to the same variable. Only because dy = y dx one of the integrals turn into integral for y. So pay attention to the integrating variable (differential.) Be careful with things like 1 xdx = ln x + C. Remember that g(x) = x is not differential at x = 0. Be aware that a separable differential equation has as solutions integral curves. They may or may not represent the graph of a function y = y(x). Thing about the curves y 2 + x 2 = C. Know what a sequence is both intuitively and the formal definition as a function. Know how to find the general term of a sequence by noticing the patern of the first few terms or using a recursive formula. Know how to prove your formula using a mathematical induction. Know what it means for a sequence to be increasing/decreasing or to be bounded above/below/both. Know how to show that a sequence is increasing by either checking the ratio a n+1 a n or calculating the derivative f (x) for a n = f(n). (In particular, know in which case which method to use). Know the order of increasing of the following expressions log n, n c, c n, n!. The Least Upper Bound Axiom: Know what it means for the number b to be an upper bound (or a least upper bound) for the set S. 6

7 Know (and understand) the statement of the least upper bound axiom. In particular, remember it applies to real numbers, not sets of rational numbers. Understand why an axiom is different from a theorem - its not something we can prove, its simply a basic property of the real numbers (like x+y = y +x or x(y +z) = xy +xz). It it wasn t true, then we we simply wouldn t be talking about the real numbers anymore. Understand what it is useful forl it gives us a way to show tat some real number exists without being able to explicitely find that number. We have been implicitly using it like this last quarter. If we didn t, we wouldn t even be able to talk about basic things like integrals. 7