Things to Know Before A



THINGS TO KNOW BEFORE A.P. CALCULUS AB

CALCULATOR SKILLS

You should have your own graphing calculator, preferably a TI-83 or TI-84. You should be able to do all of the following:

1. Evaluate expressions including the four basic arithmetic operations, parentheses, exponents and radicals, trig functions and their inverses, and exponential and log functions (especially base e)

2. Graph a function in an appropriate window. This includes being able to find a window that shows the relevant features of the graph.

3. Make a table of values for a function.

4. Solve an equation, either on the graph screen using Calc Zero or Calc Intersect or from the home screen using the Solver.

BASIC ALGEBRA SKILLS

Weak algebra skills do not make success in calculus impossible but they do make it much more difficult. Many students learn their calculus concepts really well but have trouble actually solving problems because of poor algebra skills. What follows is surely not a complete list of everything you need to know but hopefully includes the more important stuff.

Laws of exponents

You must know your exponent rules cold. This includes

- Multiplying and dividing with exponents

- Powers with exponents (powers of powers)

- Negative exponents

- Rational exponents (very important that you be able to easily convert back and forth between exponent form and radical form)

Note: related to the above, it is really helpful if you can do arithmetic with simple fractions (things like [pic], [pic], [pic], [pic]and so on) easily in your head.

Polynomial operations

You must be good at adding, subtracting and multiplying polynomials and dividing polynomials by a monomial. It is helpful (though not critical) that you be familiar with polynomial division. It is critical to have some strong factoring skills: be able to factor out common terms, including binomial terms, with rational exponents, and factor quadratic binomials and trinomials. (Factoring sums and differences of cubes is nice but not essential. Ditto for factoring by grouping.)

Rational Expressions

You should be comfortable working with rational expressions. In particular, you should be able to simplify complex rational expressions.

Solving Equations

Any really ugly equations you encounter in AP Calc AB you can (and should) solve with your calculator. However, you should be able to solve simple equations algebraically. These include linear, pure and/or easily factored quadratics, and simple exponential, log and trig equations. You should also be able to solve both linear and nonlinear inequalities. Being able to construct and also interpret “sign charts” is very helpful. Finally, you should be able to solve systems of two equations (not necessarily both linear) in two unknowns.

GEOMETRY FORMULAS

You should know the following geometry formulas:

Areas of triangles, rectangles, trapezoids, and circles.

Perimeter and circumference.

Volumes of rectangular prisms and cylinders.

It is not necessary to memorize other volume formulas (spheres or cones); if you need them, they will be given.

COORDINATE GEOMETRY

You must be familiar with the idea of change of a quantity, (x = x2 – x1, and understand that if (x > 0,

x increased (moved right) and if (x < 0, x decreased (moved left).

You should thoroughly understand the idea of slope of a line; especially

1. The formula: [pic] and

2. The interpretation of slope as a rate of change of y with respect to x (i.e., a slope of m means that each time x changes by 1 unit, y changes by m units). In an application problem, the slope has “units:” units of y per units of x.

You should be confident with the idea of distance between two points in one and two dimensions:

1. Distance in one dimension: d = |(x| (horizontal) or d = |(y| (vertical).

2. Distance in two dimensions: [pic] (Pythagorean Theorem)

The midpoint formula is not very important; however, you should be able to find the middle value of an interval. For example, you should have no trouble telling what number is in the middle of the interval [6, 12].

You should know the various forms of the equations of lines:

Vertical line: x = a

Horizontal line: y = b

General form of equation of a line: ax + by = c (a, b not both 0)

Slope-intercept form: y = mx + b

Point-slope form: y – y1 = m(x – x1) (See note below.)

Note: The point-slope equation of a line is by far the most important one. You must know this formula and how to use it to write equations of lines.

TRIGONOMETRY

All calculus involving trig functions is done in radians. So you must be comfortable using radian measure for angles. You should know (without needing your calculator) the trig function values for certain angles:

0, [pic], [pic], [pic], [pic], (, [pic] and 2(. Conversely, you should know (or be able to easily figure out without your calculator) special inverse trig function values such as (but not limited to) sin-1(0), cos-1(0.5), sin-1(1),

tan-1(1), etc. You should know the signs of the trig functions in the various quadrants. You should be able to solve simple trig equations both with and without a calculator.

FUNCTIONS

Basic Ideas and Notation

It is absolutely critical that you be comfortable with functions and function notation. For example, given the functions f(x) = 2x2 – x and g(x) = e-x, you should be able to do the following without suffering a heart attack:

1. Evaluate f(3), f(3 + h), f(x + h), f(g(x)) and g(f(x))

2. Solve f(x) = 10, f(x) = 0, f(x) > 0, f (x) < 0 and f(x) = g(x).

You must know what roots (or zeros) of a function are and be able to find them algebraically for simple functions and with your calculator for more complicated ones.

You must know what domain and range mean and be comfortable with the “natural domains” of commonly used functions. The natural domain of a function (aka “the largest possible domain in the reals”) includes all values of x for which the function is defined and real. In particular, we need

1. Denominator ( 0 (f should be defined)

2. Radicand ( 0 for even roots (f should be real)

3. Arguments of logs > 0 (again, f should be defined and real)

You should be able to evaluate and graph “piece-wise functions” such as [pic]

Graphs of functions

You should be able to find the y-intercept and root(s) of a function both graphically and analytically. You should recognize the two main types of symmetry – y-axis symmetry (aka an “even” function) and origin symmetry (aka an “odd” function) – and know how to test for them analytically. If would be helpful if you understood the concepts of increasing and decreasing functions, relative maxima and minima (extrema) and absolute (global) maxima and minima (but we will go over all these in class).

Average Rate of Change

You should know the definition and formula for the average

rate of change of a function over a closed interval:

The average rate of change of f on [a, b] is the slope

of the secant line connecting (a, f(a)) to (b, f(b)).

ARC = [pic]

Compositions

You should be comfortable with compositions of functions, f(g(x)) and g(f(x)). In particular, you should never confuse a composition with a product: f(g(x)) ( f(x)g(x). It is helpful if you can mentally decompose a composition into simpler functions. For example, if [pic], what are the functions f, g and h?

Inverses

You should understand the idea of the inverse of a function and be able to find the inverses of basic functions algebraically and graphically. You should know what it means for a function to be one-to-one. You should know and understand the result of compositions of functions and their inverses; i.e. [pic] and [pic].

Transformations

You should be familiar with basic transformations of functions:

1. Reflections: f(-x), -f(x) and -f(-x)

2. Translations: f(x) + k, f(x – h) and f(x – h) + k

3. Dilations: af(x), f(bx) and af(bx)

Knowledge of rotations of functions is not necessary.

Library of Basic Functions

You should have a mental picture of the basic functions listed below (in other words, you should be able to visualize and sketch them without needing to resort to your graphing calculator). You should know their important features including domain and range, intercepts, asymptotes and “end behavior.”

Linear functions: y = mx + b

Quadratic functions: y = ax2 and the more general y = ax2 + bx + c or y = a(x – h)2 + k

Power functions: y = xn for n ( 1

Radical functions: [pic] and [pic]

Simple rational functions: [pic] and [pic] and transformations of them

Exponential functions: y = ex and y = e-x and transformations of them.

Log function: y = ln x and transformations of it.

Trig functions: y = sin x and y = cos x and transformations of them (y = tan x is nice, but not as

important)

Direct variation: y = kx

Inverse variation xy = k or [pic]

Special Functions

Polynomials

You should know (at a minimum):

The Factor Theorem: (x – r) is a factor of a polynomial P(x) if and only if x – r is a root of P(x).

The Leading Coefficient Test: The “end behavior” (aka the “tails”) of a polynomial is determined by the degree, n, of the polynomial and the sign of the leading coefficient, an:

n even, an > 0 n even, an < 0 n odd, an > 0 n odd, an < 0

Multiplicity of roots: The graph of a polynomial will cross the x-axis at a root of odd multiplicity but will be tangent to the axis at a root of even multiplicity:

Rational functions

You should be able to identify the domain of a rational function, find its roots, find its vertical asymptotes and tell whether they are even or odd (and know what that means for the graph), and describe the end behavior of the function (especially a horizontal asymptote if there is one).

You should be able to tell your asymptote from a hole in the graph.

Trig functions

You should be comfortable with the graphs of y = asinbx + d and y = acosbx + d: you should be able to tell the amplitude, the period and the vertical shift and know how they affect the graph. Phase (horizontal) shift is nice to know but not really critical for AP Calc AB. You should be familiar with the graph of y = tan x to the extent of knowing where its roots and asymptotes are but you don’t need to know much else about it. You should know the definitions of sec x, csc x and cot x but their graphs are not important. You should be familiar with the inverse sine and inverse tangent functions; you should know their domains and ranges and what their graphs look like. The other inverse functions will not be used.

Conic sections

You should know the equation of a circle, especially one centered at the origin. You should recognize the equations of ellipses and hyperbolas and be able to find their intercepts but it is not important to know about foci or eccentricity.

Exponential and log functions

Exponential functions are used frequently in AP Calc AB, almost exclusively base e. (If you have never heard of base e, run to your favorite math teacher and demand to be enlightened.) You should instantly know the graphs of y = ex and y = e-x and transformations of them and you should be at least somewhat comfortable dealing with combinations and compositions of them with other functions: for example y = x2e-x and [pic].

Almost as important as the exponential function is its inverse, the log function. Again, the only one you really need to know is the natural log function (base e, y = ln x and transformations of it) but you should know it well. You should know the value of ln(1) and you should know what happens to the value of ln x as ( 0. You should know for what values of x is ln x > 0 and for what values of x is ln < 0.

It would be helpful (not essential) if you were familiar with exponential growth and decay models.

INTERVAL NOTATION

You should be familiar with interval notation.

Open interval: (a, b) means a < x < b

Closed interval: [a, b] means a ( x ( b

Half-open intervals: (a, b] means a < x ( b

[a, b) means a ( x < b

Infinite intervals: (-(, a) means x < a

[a, () means x ( a

Unions of intervals: (-(, a] ( [b, () means x ( a ( x ( b

General notes:

1. Pay attention to context: (-1, 3) could be either an ordered pair or an open interval.

2. a is always less than b. Correct: [-1, 3]. Wrong: [3, -1]

3. ( and -( are always “open:” Correct: (-(, (). Wrong: [-(, (].

SUMMATION NOTATION: (

You should be comfortable with sigma notation for sums, e.g., [pic].

THINGS YOU DON’T NEED TO KNOW

The following topics are all good to know and may be important if you take more than just first-semester calculus in college but are not necessary for AP Calculus AB.

Vectors in the plane: Wonderful stuff for physics and Calc II but we will not need it for our class.

Matrices: Helpful for some college statistics; not needed for this course.

Sequences and series: Very important for future calc courses, minimally important for this one.

Complex Numbers: We should be able to get through the whole year without ever needing i.

Polar coordinates: Fun stuff, useful in future calc courses, but not needed for our class.

Probability and statistics: Knowing calculus may be helpful if you take statistics classes in college but you don’t need to know any probability or statistics to learn calculus.

Regression: You should never need to do a regression in this course and if you are tempted to do so anyway, resist.

Calculus: That’s right. This course is taught on the assumption students have solid precalc skills but start with no actual calculus knowledge. Please don’t bother spending your summer taking derivatives and doing integrals.

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