This is by no means a complete list of what one needs to learn, but it does
indicate the topics that will receive more emphasis.
| Section | Objectives |
|---|---|
| 1.1 | know the graphs of power and root functions |
| understand what is meant by increasing and decreasing functions | |
| know the definition of the the absolute value function and its basic properties | |
| 1.2 | know the basic examples of functions: polynomials (includes constant, linear, and power functions), rational functions, algebraic functions |
| know the six trigonometric functions and their graphs | |
| know the statement and understand the meaning of the Fundamental Theorem of Algebra | |
| 1.3 | know the four basic operations on graphs: vertical and horizontal translation and vertical and horizontal stretching ( f(x) + c, f(x-c), cf(x), f(cx) ) |
| be able to manipulate graphs of functions using the four basic operations | |
| understand the definition of composition of functions, be able to calculate compositions and recognize when a function is a composition | |
| 2.1 | be able to calculate slopes of secant lines, and using them to determine a tangent line slope |
| understand position ( s ), average velocity, and instantaneous velocity ( v ) | |
| 2.2 | be able to interpret the notation lim_{x --> a} f(x) = L |
| have an intuitive understanding of the meaning of lim_{x --> a} f(x) = L | |
| have an intuitive understanding of infinite and one-sided limits | |
| 2.3 | be able to calculate basic limits involving the basic arithmetic operations, have an appreciation of the hazards of calculations involving infinite limits |
| understand how limits preserve inequalities, know the statement of the Squeeze Principle and its meaning | |
| 2.4 | know the precise definition of limit -- be able to reproduce its statement but also to apply it to other situations, e. g. lim_{t --> 5} k(t) = M |
| be able to use the definition of limit to verify specific limits, such as limits of linear functions, or other simple cases such as lim_{x --> 0} x^5 = 0 | |
| 2.5 | know the definition and intuitive meaning of continuity |
| know the statement of the Intermediate Value Theorem and its meaning, be able to carry out simple applications of the IVT | |
| 2.6 | be able to calculate tangent line slopes using limits |
| 3.1 | be able to calculate f '(a) using limits |
| be able to sketch graphs of functions satisfying certain conditions on their derivatives | |
| 3.2 | understand the definition of the derivative function f '(x) and the information it contains about f(x) |
| be able to sketch the derivative of f '(x) by examining the graph of f(x) | |
| be able to calculate derivatives of functions using the definition of derivative as a limit | |
| be able to use f '(x) to solve problems involving tangent lines to the graph of y = f(x) | |
| 3.3 | be able to calculate derivatives using the algebraic formulas for the derivative of a sum, difference, product, quotient, and the power rule |
| 3.4 | be familiar with some of the derivatives that arise in engineering and science applications |
| 3.5 | know how to calcualte the derivatives of sin(x) and cos(x) algebraically (given the sum formulas for sine or cosine) |
| know derivatives of all six trigonometric functions and how to use them in calculation of derivatives of more complicated functions | |
| 3.6 | know the Chain Rule in both standard notations, and be able to apply it to calculate derivatives of composite functions |
| know the geometric interpretation of the Chain Rule in terms of stretch factors | |
| 3.7 | understand what is meant by an implicit function, and know how to calculate dy/dx using implicit differentiation |
| 3.8 | understand the notations f^(n) (x) and d^n y / dx^n for higher derivatives, and understand how to calculate them |
| be able to calculate a general formula for f^(n) (x) in examples where it is simple enough to guess by inspection | |
| 3.9 | understand what is meant by a related rates problem |
| be able to solve related rates problems | |
| 4.1 | know the definitions of local maximum, local minimum, absolute maximum, and absolute minimum, and be able to identify them given the graph of the function |
| know the definition of critical point (also called critical number) | |
| understand the relationship between f'(a) = 0 and f(x) having a local extreme value at x = a | |
| know the statement of the Extreme Value Theorem | |
| 4.2 | know the statement of Rolle's Theorem and of the Mean Value Theorem, and understand their geometric content |
| know and be able to derive simple consequences of the Mean Value Theorem | |
| know examples of how the conclusion of the Mean Value Theorem can fail for functions which do not satisfy its hypotheses | |
| 4.3 | know how to detect local extrema using the first derivative |
| know how to detect local extrema using the second derivative | |
| be able to sketch examples satisfying various conditions on their first and second derivatives | |
| know the definition and geometric meaning of an inflection point | |
| 4.4 | know the definition and geometric meaning of an asymptote of a function (horizontal, vertical, and slant asymptotes) |
| be able to determine limits of functions as x --> \infty and x --> -\infty | |
| 4.5 | be able to sketch graphs of functions by making use of domain, range, extent, symmetry, translation, intercepts, periodicity, asymptotic behavior, first derivative (ranges where f is increasing or decreasing, local extrema), second derivative (concavity, inflection points), knowledge of specific functions, common sense, and ingenuity |
| 4.9 | understand the geometric idea of Newton's method, and how the iteration formula is derived |
| be able to apply Newton's method to find solutions of equations, possibly after initial investigation to get a rough idea of where the solutions are located | |
| be aware of some of the many ways in which Newton's method can fail to find a root, or fail to find the desired root | |
| 4.7 | know how to set up optimization problems as problems of finding maximum and minimum values of functions on a given domain |
| be able to find extreme values of functions on domains that are intervals | |
| 4.10 | know the definition of antiderivative, understand that all antiderivatives of a given function f can be obtained from any one particular antiderivative f by adding constants, which may be independently selected for each interval in the domain of f |
| be able to sketch an antiderivative of f, given the graph of f | |
| be able to calculate antiderivatives of simple functions such as sums of polynomials and sine and cosine functions, be able to solve for the unknown constants if particular values of the antiderivatives are known | |
| understand that the area function for the graph of f is one of the antiderivatives of f |