Test 1 will be held in class on Friday September 17. It will cover sections 1.1, 1.2, 1.3, 1.5, 1.6, 2.1, 2.2, 2.3, 2.5, and 2.6. A more detailed description of each section, with points to pay attention to, is provided below:1.1 Four Ways to Represent a Function
- know the definition of a function, what it means graphically, and the vertical line test
- know what is meant by the domain and range of a function and how to find them in explicit examples
- know how to handle piecewise defined functions (i.e. graphing them, computing their values, finding domain and range, etc.)
- know what is meant by a function being even or odd, both in terms of equations and the appearance of the function's graph
- know how to set up equations of functions from word problems (i.e. review those in the text and the homework)
1.2 Mathematical Models
- know what is meant by a linear model and how to find one given data from a word problem
- know what is meant by a polynomial, and also know the standard graphs for y = x2, y = x3, y = x4, etc.
- also know the standard graphs for y = x1/2, y = x1/3, y = 1/x, y = 1/x2, etc.
- know the standard graphs for y = sinx, y = cosx, y = tanx
1.4 New Functions from Old Functions
- know how to translate, stretch, and reflect the graphs of functions to get the graphs of other functions
- know what is meant by the composition of functions
1.5 Exponential Functions
- know the standard graphs for y = ax and y = a-x and their domains, ranges, and special values
- make very sure you now your rules for exponents
- know how the graphs for y = 2x, y = ex, y = 3x, etc., compare to one another
- know how the graphs for y = (1/2)x = 2-x, y = (1/e)x = e-x, y = (1/3)x = 3-x etc., compare to one another
1.6 Inverse Functions and Logarithms
- know what is meant by one-to-one: interms of equations, in terms of graphs, and through the horizontal line test
- know what is meant by an inverse function, i.e. f and f -1 cancel each othe out when composed, and how to find it
- know how the domain and range of f -1 are related to those of f
- know how to get the graph of f -1 from that of f
- recall the definition of logax as the inverse of ax and how to use this in solving equations involving exponentials and logs
- know the standard graph for y = logax and its domain, range, and special values
2.1 The Tangent and Velocity Problems
- know how to compute the secant from P to Q to a graph and how the slope of this secant relates to the slope of the tangent at P
- know how to compute the average velocity from t to t + h and how it relates to the instantaneous velocity at t
2.2 Limit of a Function
- know the definition of limit in terms of f(x) "approaching" a number L as x "approaches" the number a
- be able to use this definition to pick off limits from a function's graph
- know what is meant by one-sided limits
- know what are meant by infinite limits and how they relate to vertical asymptotes
2.3 Calculating Limits Using the Limit Laws
- know the limit laws and how to apply them
- know your list of special known limits (see page 104, in particular)
- be sure you know how to compute all sorts of limits by first simplifying the expression before applying the rules
- know how to find one-sided limits and how they are related to true limits
2.5 Continuity
- know what is meant by continuity of a function and the ways in which it can break down
- know what continuity means graphically and what the three types of discontinuities are
- know the rules for forming new continuous functions from known ones, as well as a list of known continuous functions
- know the continuity rule for composite functions
- know the Intermediate Value Theorem and how to apply it
2.6 Limits at Infinity; Horizontal Asymptotes
- know the definition of limit at infinity in terms of f(x) "approaching" a number L as x "gets larger and larger"
- know how such limits relate graphically to horizontal asymptotes
- know how to find horizontal asymptotes by computing limits at infinity (there are techniques here!)
- know what infinite limits at infinity are, how to justify what they are, and how they relate to the graph of a function