DOC PREVIEW
Stanford MATH 19 - Background Material

This preview shows page 1-2-3 out of 8 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 8 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 8 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 8 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 8 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Math 19 Calculus Winter 2008 Instructor: Jennifer Novak KlokeBackground MaterialThis handout summarizes all the background material I expect you to know when you start Math19. You will be quizzed on this material on Monday, January 14. Many of the formulas in thishandout are also available on the reference page at the very beginning of your textbook.1 Simplifying Algebraic Expressions1. You should know how to handle fractions (e.g., adding, multiplying, dividing, using commondenominators, cancelling).2. You should know how to factor a perfect square out of a square root. For example:√18 = 3√2,px3− x2= x√1 − x.3. Throughout the course, we will have equations that have constants in them. For example,I might say: let y = cx, where c is a constant. This means that c is some fixed number, butwe do not know what it is. In this case, we know that the graph of y = cx is a straight linegoing through the point (0, 0) with slope c (we just don’t know what that slope c is).We often state rules using constants. For example, the Quadratic Formula says that if0 = ax2+ bx + c, where a, b, and c are constants, and a 6= 0, thenx =−b ±√b2− 4ac2a.For example, the Quadratic Formula is still true if a, b, and c are replaced by any numbers(assuming a 6= 0).However, the Quadratic Formula is not true if a is replaced by x, because x is not a constant.That is to say, if 0 = x·x2+bx+c, the Quadratic Formula is not the correct way to solve for x.2 Solving Equations and Inequalities1. Avoid dividing by zero. For example, to solvex3− x = x,you cannot divide everything by x and say thatx2− 1 = 1,1and so x = ±√2. The reason you cannot do this is that x might equal 0, and in fact, x = 0is a solution to the original equation. So instead of dividing by x, you collect everything toone side of the equation and then factor out an x:x3− x = xx3− 2x = 0x(x2− 2) = 0x(x +√2)(x −√2) = 0So we see x = 0 or x = ±√2.2. You should know how to solve equalities and inequalities with absolute values. For example:|x − 3| = 4x − 3 = 4 or x − 3 = −4x = 7 or x = −1|x − 3| ≤ 4−4 ≤ x − 3 ≤ 4−1 ≤ x ≤ 7|x − 3| ≥ 4x − 3 ≤ −4 or x − 3 ≥ 4x ≤ −1 or x ≥ 73. When solving equalities and inequalities, you have to be careful with things like square roots.For example:x2= 4 has two solutions, x = ±2,x2≤ 4 has the solution − 2 ≤ x ≤ 2, not x ≤ 2,x2≥ 4 has the solution x ≤ −2 or x ≥ 2, not x ≥ 2.4. Remember that when solving inequalities, if you divide by a negative number, you need toflip the inequality sign around:−2x ≤ 4x ≥ −223 Functions3.1 Polynomials1. A linear polynomial has the form y = ax + b and is the equation of a straight line. Aquadratic polynomial has the form y = ax2+ bx + c and is the equation of a parabola. Acubic polynomi al has the form y = ax3+ bx2+ cx + d and is the equation of a cubic curve.2. Given two points in the plane, you should feel comfortable finding the slope of the line passingthrough these two points. If the points are (x1, y1) and (x2, y2), then the slope is defined asslope = m =y2−y1x2−x1.It is easy to remember this formula as “rise over run.”3. The equation of a straight line with slope m and y-intercept b is y = mx + b. The equationof a straight line that goes through the points (x1, y1) and (x2, y2) isy − y1= (y2−y1x2−x1)(x − x1).4. The y-intercept(s) of a curve is/are the y-value(s) at which the curve intersects the y-axis.This can be found by letting x = 0 and finding the value(s) of y. The x-intercept(s) of acurve is/are the x-value(s) at which the curve intersects the x-axis. This can be found byletting y = 0 and finding the value(s) of x.3.2 Trigonometric Functions1. In this class (and in future math classes) you should always use radians, never degrees.2. Given the right triangle below, we have the formulassin a =yzcos a =xztan a =yx33. Trig functions are those functions which involve the functions sin x or cos x. There are otherstandard trigonometric functions (tangent, cotangent, secant, and cosecant), but they all ariseby taking products and quotients of these original two.tan x =sin xcos xcot x =1tan x=cos xsin xcsc x =1sin xsec x =1cos x4. Note: Sine and cosine are functions of a variable. That variable must appear somewherewhenever you write a trigonometric function.Right Wrongsin x sinsin2x + cos2x sin2+ cos2Also, when the input for the trig function is more than just x, write the input inside paren-theses. For instance, ‘sin(x + 3π)’ means “the value of sine at x + 3π”.5. In the first quadrant, the sine, cosine, and tangent functions have the following values:x sin x cos x tan x0 0 1 0π/6 1/2√3/2√3/3π/4√2/2√2/2 1π/3√3/2 1/2√3π/2 1 0 —You should be able to figure out the corresponding values of these functions in the other threequadrants. If you have difficulty with this, please come see me for some help.6. The expression (sin x)2is usually written sin2x out of laziness. These two expressions are notequal to sin(x2).7. There are several key equations that will allow you to simplify expressions involving trigono-metric functions. Essentially these identities come from either the periodicity of sin x andcos x, or the equationsin2x + cos2x = 1.Here are a few that you should keep in mind:sin(x + 2π) = sin x sin2x + cos2x = 1 sin(−x) = −sin xcos(x + 2π) = cos x 1 + tan2x = sec2x cos(−x) = cos x1 + cos2x = csc2x tan(−x) = −tan x48. The half-angle formulas for sine and cosine let you rewrite sin2x and cos2x. The double-angle formulas for sine and cosine let you rewrite sin(2x) and cos(2x). The formulas canbe found at the beginning of your textbook.Although you do not need to have these formulas memorized, you do need to know that theyexist. If one of the above expressions appears in a homework question, you should immediatelythink of these formulas. If one of these formulas is required on the exam, it will be given toyou.9. Similiarly, you should know that the sine and cosine addition formulas exist, and they allowyou to rewrite sin(x + y) and cos(x + y).3.3 Exponential functions1. When b is a positive integer and a is a real number, the symbol abdenotes the product of awith itself b times:ab= a . . . a| {z }b times.When b is a positive fraction, say b = c/d, we defineab= ac/d=d√ac.When b < 0, we defineab=1a|b|.2. An exponential function has the form y = axfor some positive number a. While a is oftena positive integer, the most


View Full Document

Stanford MATH 19 - Background Material

Download Background Material
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Background Material and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Background Material 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?