Mathematics 133 – Intensive Calculus for Science 1 Things to Know – Exam 1 September 25, 2000

Recall that the first full-period exam of the semester will be given in class on Friday, September 29. There will be seven or eight questions similar to problems quizzes so far. You should know the following material. We will review for the exam in class on Thursday, September 28.

1) Linear functions and their properties (Section 1.2)

a) The slope-intercept (y = mx + b) and point-slope (y − y_{0} = m(x − x_{0})) forms for linear functions

b) The meaning of the slope and how to determine it from either a formula for the function, or from a table of values

2) Exponential functions and their properties (Section 1.3)

a) The general formula for exponential functions f(x) = ba^{x} (or using different letters, P (t) = P_{0}a^{t}, as on page 17 of the text).

b) Exponential growth versus exponential decay (which values of a give which case)

c) Be able to determine an equation for an exponential function, given a graph or a table of values.

d) How to tell exponential functions apart from linear functions

3) Power functions and their properties (Section 1.4)

a) Algebraic rules for exponents

b) The general form of power functions: y = kx^{p}

c) The general shapes of their graphs.

d) Which functions grow faster as x grows: powers or exponentials?

4) Inverse functions. Know:

a) How to tell whether or not a function is invertible from its graph,

b) How to derive a formula for the inverse function f^{−1} from a formula for f,

c) How to sketch the graph of the inverse function from the graph of f.

5) The logarithm functions f(x) = log_{a}(x) and their properties a) g(x) = log_{a}(x) is the inverse function of the exponential function f(x) = a^{x}.

b) Formulas for logs of products, quotients, powers and how to apply them

c) The shapes of the graphs y = log_{a}(x)

d) Using logarithms to solve equations involving exponentials

e) The natural logarithm function f(x) = ln(x) (the logarithm function with base a = e = 2.71828 ⋯)

6) Trigonometric functions. Know: a) Radian measure for angles and how to determine the signs of sin θ, cos θ, tan θ

for an angle θ in radians

b) How to sketch graphs for sinusoidal oscillations

y = A sin(Bx) + C y = A cos(Bx) + C

and the meanings of A, B, C

c) How to find a formula for a sinusoidal oscillation, given the graph. (Note: we have not talked about shifting the period to start at points other than 0, so that will not be included)

7) Polynomials and rational functions. Know:

a) How to determine the shape of the graph of a polynomial for big x from the equation.

b) How to determine equations for polynomials given information from a graph such as x- and y-axis intercepts.

c) How to determine where rational functions have vertical and horizontal asymptotes

Good Review Problems:

From the Review problems at the end of Chapter 1: 1, 2, 6, 11, 22, 23, 24, 25, 27,29, 34, 35, 36,3 8, 39, 41, 44, 45

From Section 1.10: 12 (sketch the graphs), 18 - 21