Review Sheet for Exam 1 - Precalculus (Physics and Mathematics) | MATH 1113, Exams of Pre-Calculus

Download Review Sheet for Exam 1 - Precalculus (Physics and Mathematics) | MATH 1113 and more Exams Pre-Calculus in PDF only on Docsity! MATH 1113 Review Sheet for Test #1 (Chapter 2, Chapter 3) Section 2.1 Functions  Defining a function in words (we discussed three ways to do this)  Different ways to specify a function: Geometrically, Analytically, Numerically, Verbally  Relation, Domain, Codomain, Range, Dependent Variable, Independent Variable  Function Notation , i.e. f(a) = b  Given that f is a function, what does f(a) represent? If a is in the domain of the function f, f(a) is the value that function f assigns to the element a. If a is not in the domain of f, f(a) is not defined.  Operations on Functions (sum, product, difference, and quotient of functions)  The implicit domain for a function from real numbers to real numbers is the largest subset of the real numbers for which the formula has meaning. Section 2.2 The Graph of a Function  Defining the graph of a function  Determining when a graph represents a function; the vertical line test  Given the graph of function f, solving f(x) = k for x given k and f(x) =k for k given x using the graph.  Reading the domain and range of a function from the graph Section 2.3 Properties of a Function  Even and Odd functions  Local Extremum (local maximum, local minimum) Note the plural forms: extrema, maxima, minima  Secant Line  Average Rate of Change of a function over an interval, connection to the slope of a secant line  Increasing/Decreasing/Constant on an interval  Intercepts (horizontal intercepts, vertical intercepts) o A function may have any nonnegative integer number of horizontal intercepts  (i.e. have 0, 1, 2, 3, …..horizontal intercepts) o A function has at most one vertical intercept Section 2.4 Library of Functions  The library of functions include the following: Linear, Constant, Identity, Square, Cube, Square Root, Cube Root, Reciprocal, Absolute Value, and Greatest Integer functions  For each function in the library, be familiar with the following: its graph, intervals on which the function is increasing/decreasing/constant, its horizontal and vertical intercepts, whether it is even or odd or both or neither, its domain and range, its local maxima and local minima (i.e. local extrema)  piecewise-defined functions Section 2.5 Graphing Techniques: Transformations  Scaling (Shrinking and Stretching)  Reflection in the horizontal axis  Translations (Horizontal Shifts and Vertical Shifts)  Rigid vs. Non-rigid transformations; Determining the coordinates of a transformed point Section 2.6 Mathematical Models: Constructing Functions  What is Mathematical Modeling? (Formulate, Solve, Interpret, Test)  Particular Examples of the Modeling (examples and problems from the section)  Connections between an application/context and a mathematical model, Meaningful or Practical Domain Section 3.1 Linear Functions  Definition: A Linear Function is a function with formula description of the form L(x) = mx + b where m and b are real numbers. The slope of a non-vertical line is the signed vertical displacement corresponding to a one unit increase in the horizontal coordinate.  Connection between slope positive, negative, and zero and increasing, decreasing, and constant  Linear Functions are characterized by a constant average rate of change, i.e. Given L is a linear function, there is a real number k so that the slope of the secant line between any two points on L is always k.  Using this characterization to determine whether a set of points are collinear  Applications of linear functions Section 3.2  Given a set of points, displaying the scatter plot (or scatter diagram) of these points (i.e. the graph)  Given a scatter plot, having an impression of the strength of the linear relationship between the quantities  Using technology (graphing calculator) to perform linear regression on a set of points  Intuition for linear regression: What criteria are used to find the line of best fit with linear regression?  Interpreting slope as well as coordinates of intercepts and other points in contextual application problems Section 3.3 Quadratic Functions  Definition: A Quadratic Function is a function with formula description of the form q(x) = ax2 + bx +c where a, b, and c are real numbers.  Solving quadratic equations using factoring, completing the square, and the quadratic formula  Features of Quadratic Functions: Domain, Range, Vertex, Axis of Symmetry, Intercepts, Local Maximum or Local Minimum, Intervals on which the quadratic function is increasing or decreasing  Connection between the forms q(x) = ax2 + bx +c and q(x) = a(x-h)2 +k o h = b/(2a) k = (b2-4ac)/(4a)  Using transformation theory (Section 2.5 ) to obtain the graph of q(x) = a(x-h)2 +k Section 3.4 Quadratic Models  Finding coordinates of a point including the vertex and the intercepts  Interpreting the coordinates of a point on the graph (serving as a model) in context  Quadratic Regression Section 3.5 Inequalities Involving a Quadratic Function  First note that all such inequalities are equivalent to one of the form ax2 + bx +c < 0 or ax2 + bx +c ≤ 0  Two standard approaches both begin by finding the horizontal intercepts (solutions to ax2 + bx +c = 0), where these intercepts are the finite boundaries of the interval(s) comprising the solution. o Use the graph to determine the intervals for which the desired relationship holds o Test the intercepts, use a test point from each interval, where these intervals are determined by the horizontal intercepts. As there can be zero, one, or two horizontal intercepts, there can be one, two or three intervals to test respectively i.e. the number of intervals to test exceeds the number of horizontal intercepts by exactly one. ___________________________________________________ Suggested Problem List 2.1 in text: 15, 19, 33, 39, 51, 61, 73, 87 others: 17, 21, 23, 25, 43, 45, 47, 49, 55, 57, 65, 71, 77, 79, 85, 91, 95, 99 2.2 in text: 9, 13, 15, 25, 31 others: 11, 17, 19, 21, 27, 37, 46 2.3 in text: 11, 13, 15, 17, 19, 21, 33, 45, 53, 59 others: 22, 23, 25, 29, 31, 35, 37, 39, 55, 57, 61, 63, 81, 85 2.4 in text: 9-16, 29 others: 19, 23, 25, 31, 33, 37, 41, 43, 45, 47, 49, 55, 67 2.5 in text: 27, 35, 39, 41, 43, 57, 65 others: 7-18(all), 19, 21, 23, 25, 29, 31, 33, 37, 49, 53, 55, 59, 67 2.6 in text: 1, 7 others: 3, 5, 9, 11, 15, 19, 23 --------------------------------------------------------------------------------------------------- 3.1 in text: 13, 21, 41, 47 others: 15, 17, 19, 23, 24, 27, 31, 33, 37, 43, 49, 51, 53 3.2 in text: 3, 9 others: 5, 7, 15, 17, 19, 21 3.3 in text: 27, 35, 43, 47, 53 others: 21, 29, 31, 33, 37, 41, 45, 51, 55, 57, 59, 65, 85 3.4 in text: 3, 7, 11, 13, 27 others: 5, 9, 15, 17, 19, 29 3.5 in text: 5, 9, 13 others: 3, 7, 15, 17, 23, 27, 31, 33, 35