Math 5: Precalculus. Homework Problems

Transcription

1 Math 5: Precalculus Homework Problems Note: Short answer questions are labeled by SQ and long answer questions are labeled by LQ. You will be graded on the clarity of your written work only with the long answer questions. For short answer questions, you need only submit your answers. Short Answer Questions: You will earn five points for completing the short answer questions by the due date. Long Answer Questions: There are two grading scales. (1) You have completely correct and typed solutions posted by the due date. For this earn six points (5 if not typed). The homework is graded out of only five points. (2) You earn 1 point for having a rough draft due by the due date. 1 point for completing the assignment by the due date (getting all problems correct). 2 points for getting all problems correct. 1 point for attending a TA office hours to seek help on that assignment. 1 point for typing the assignment. 1

2 1. Proportion and Basic Manipulation 1.1 The Algebra of Intervals Set and Interval Notation Unions and Intersections of Intervals Cases and Conditions 1.1 SQ 1 Graph on a number line the set ( 1, 7] (3, 10). Describe this set using interval notation. 1.1 SQ 2 Graph on a number line the set [1, 3] (5, 7]. 1.1 SQ 3 Graph on a number line the set ( 2, 7] (5, ). Describe this set using interval notation. 1.1 SQ 4 Graph on a number line the set (, 2) (5, ). Describe this set using interval notation. 1.1 SQ 5 Graph on a number line the set ( 2, 7] ( ([ 5, 0) [3, 10) ). Describe this set using interval notation. 1.1 LQ 1 Suppose x > 0. Simplify the expression expression defined? 1 x+h 1 x h. For what values of h is this 2

3 1.2 Manipulating Simple Equations to Solve for Variables Linear Equations Systems of Linear Equations 1.2 SQ 1 Solve for x in the equation 1.2 SQ 2 Solve for x in the equation 1.2 SQ 3 Solve for x in the equation 2x + 3 = 9. 2x + 3 = 7x x 3 5 = 2 3 x SQ 4 Is it possible that a can be chosen so that has exactly one solution? x + 2y = 9 3x + 6y = a 1.2 LQ 1 Find a point (x, y) in the plane that is a solution to the linear system 4x + 2y = 9 x + 3y = 2. Solve for this point in two ways, first using substitution and then by adding a multiple of one equation with the other. Finally, check that your solution is correct. 1.2 LQ 2 Find a real number a so that the linear system x 2y = 9 3x + ay = 2 has no solutions. How many choices of a are there so that the system has no solutions? 3

4 1.3 Linear Inequalities Graphing Inequalities on a Line Solving Linear Inequalities Restrictiveness of Conditions 1.3 SQ 1 Graph on the real line all points x such that x SQ 2 Graph on the real line all points x such that x < SQ 3 Graph on the real line all points x such that 2x 1 > SQ 4 Graph on the real line all points x such that 3x SQ 5 Graph on the real line all points x such that x 3 and x < SQ 6 Graph on the real line all points x such that 2x + 1 > 7 and x SQ 7 Graph on the real line all points x such that x + 3 < 5 or x LQ 1 Graph on the real line all points x such that (1) 4x 2 6 and 3 x 7, or (2) 3x > 18 and 2x

5 1.4 The Algebra of Physical Units Physical Quantities Manipulating Physical Quantities Commensurable Units Conversion Factors 1.4 SQ 1 Alice can run at 15 miles per hour, how many feet per second is this? 1.4 SQ 2 A container carries 3 cubic feet of fluid. How many cubic inches is this? 1.4 SQ 3 A drain drains water at 100 cubic centimeters per minute. How fast does it drain in units of cubic meters per hour? 1.4 SQ 4 The formula for the kinetic energy of an object of mass m traveling at a speed v is KE = 1 2 mv2. What are the fundamental units of kinetic energy? 1.4 SQ 5 What is the fundamental unit of the conversion factor that converts the quantity 200 square feet per second to the same quantity in units of square meters per hour? 1.4 LQ 1 A force, F, on a body of mass m causes the body to accelerate. Denote the acceleration of the body by a. The relationship between force, mass, and acceleration is F = ma. If we take the unit of mass to be a kilogram, the unit of distance to be a meter, and the unit of time to be a second, then what are the units of force? 1.4 LQ 2 The force, F, between two bodies of mass m 1 kilograms and m 2 kilograms whose centers of mass are a distance of r meters apart is given by the formula F = Gm 1m 2 r 2, where G is the gravitational constant of the universe. What are the units of the gravitational constant G? 1.4 LQ 3 The radius of the earth is approximately 4,000 miles. The acceleration due to gravity is approximately 32 feet per square second. The mass of the earth is approximately kilograms. Calculate the value of G in units of kilograms, meters, and seconds. 5

6 1.5 Proportionality Linear Relationships Power Laws Reciprocal Relationships Utility of Analyzing Proportionality 1.5 SQ 1 Five munchkins working together can lay 300 pounds of yellow brick in an hour. How many hours will it take seven munchkins to lay 400 pounds of brick? 1.5 SQ 2 The intensity of light is proportionate to the inverse square of the distance to the light source. Lightbulb A shines at 9 times the intensity as Lightbulb B. How many feet do you need to stand from Lightbulb B so it appears to have the same intensity as Lightbulb A at 12 feet away. 1.5 SQ 3 The force applied to a body is F. The mass of the body is m. The acceleration that the body experiences as a result of the force is a. We then have that F = ma. If you apply twice the force on a new body and it experiences half the acceleration of the original, what is the mass of the new body compared with m? 1.5 SQ 4 The weight of a solid steel ball is proportionate to its volume. If the weight of a particular steel ball is 100 pounds, how much will a solid steel ball with twice its radius weigh? 1.5 LQ 1 It takes three workers eight hours to shovel 200 cubic feet of sand. How many workers are required to shovel 100 cubic yards of sand in seven hours? 1.5 LQ 2 The weight of paint is proportionate to its volume. Paint is always applied at the same thickness. It takes 10 ounces of paint to paint a ball that has a volume of 3 cubic feet. How many ounces of paint does it take to paint a ball that has a volume of 24 cubic feet. 6

7 2.1 Relations Ordered Pairs The Cartesian Product Relations and their Domains 2. Functions and Transformations 2.1 SQ 1 Suppose that Write down all elements of the set X Y. X = {x, y, z} and Y = {a, b}. 2.1 SQ 2 Suppose that X is a set with 5 elements and Y is a set with 4 elements. How many elements does X Y have? 7

8 2.2 Graph of a Function Definition Functions and Formulas Piecewise Defined Functions 2.2 SQ 1 Suppose that Suppose that X = {a, b, c, d, e, f, g, h} and Y = {1, 2, 3, 4, 5}. f = {(a, 1), (b, 4), (c, 5), (a, 3)}. Is f a function from X to Y? If so, what is its domain and its range? 2.2 SQ 2 Suppose that Suppose that X = {a, b, c, d, e, f, g, h} and Y = {1, 2, 3, 4, 5}. f = {(a, 1), (b, 4), (c, 4), (d, 3)}. Is f a function from X to Y? If so, what is its domain and its range? 2.2 SQ 3 What is the domain and range of the function f given by f(x) = 2x 1 x + 5? 2.2 SQ 4 What is the domain and range of the function f given by f(x) = x + 3 2x 4? 2.2 SQ 5 What is the domain and range of the function f given by 2.2 SQ 6 Write the function f given by as a piecewise defined function. 2.2 LQ 1 Define f by Write f(x) as a piecewise defined function. 2 if x < 5 f(x) = x + 1 if 0 < x < 3. f(x) = x + 2 f(x) = 3x x

9 2.3 Obtaining Information from a Graph The Sign of a Function Monotone Intervals Extremal Values Periods 2.3 SQ 1 Consider the graph of the function, f, below. (0, 3) (3, 3) 3 (3, 5 3 ) (a) Find all monotone intervals, all strictly monotone intervals, and say respectively whether the function is increasing or decreasing, or strictly increasing or strictly decreasing on these intervals. (b) Find all local maximums and local minimums. (c) Find all with f(x) 0. (d) Find all with f(x) > 0. (e) Find all with f(x) 0. (f) Find all with f(x) < SQ 2 Let f be the function whose graph is given below

10 (a) Find all x such that f(x) > 0. (b) Find all x such that f(x) 0. (c) Find all x such that f(x) < 0. (d) Find all x such that f(x) SQ 3 Let f be the function whose graph is given below (a) Find all x such that f(x) > 0. (b) Find all x such that f(x) 0. (c) Find all x such that f(x) < 0. (d) Find all x such that f(x) 0. 10

11 2.4 The Algebra of Functions Sums and Products Quotients Compositions Suppose that f(x) = x and g(x) = 3x SQ 1 Write a formula for (f + g)(x). 2.4 SQ 2 Write a formula for (f g)(x). 2.4 SQ 3 Write a formula for (f g)(x). 2.4 SQ 4 Write a formula for ( f g ) (x). 2.4 SQ 5 Write a formula for (f g)(x). 2.4 SQ 6 Write a formula for (g f)(x). Suppose that f(x) = 2x + 3 and g(x) = x 1 x SQ 7 Write a formula for (f + g)(x). 2.4 SQ 8 Write a formula for (f g)(x). 2.4 SQ 9 Write a formula for (f g)(x). 2.4 SQ 10 Write a formula for ( f g ) (x). 2.4 SQ 11 Write a formula for (f g)(x). 2.4 SQ 12 Write a formula for (g f)(x). 11

12 2.5 Identifying Structural Components of Functions Decomposition into Structural Components Domain of a Compound Function Range of a Compound Function Period of a Compound Periodic Function 2.5 SQ 1 Suppose that f is given by What is the domain and range of f? 2.5 SQ 2 Suppose that f is given by What is the domain and range of f? f(x) = 2x 5. f(x) = 1 x LQ 1 For any real number x, the floor function, denoted floor, inputs x and outputs the first integer value larger than or equal to x. We write where for example, floor(x) = x, 0 = 0, 1 = 1, 1.2 = 1, 1.7 = 1, 2.3 = 2, 1.3 = 2,.9 = 1 and so on. What is the principle period of the function f where Sketch the graph of this function. Now, set What is the principle period of f g? 2.5 LQ 2 Write the function f(x) = x x? g(x) = 3x f(x) = x x x as sums, products, and compositions of functions. Try to make these functions as simple as possible. 12

13 2.6 Transformation of Functions Transformations and Composite Functions Vertices and x-intercepts of Parabolas Suppose that S a (x) = ax and T b (x) = x + b. 2.6 SQ 1 Suppose that f is given by f(x) = 1 x. Write g = T 3 S 5 f T 1 S 2 as a quotient of two functions. 2.6 SQ 2 Suppose that f is given by f(x) = 1 x. Write g = T d S c f T b S a as a quotient of two functions. Use this expressions to write in the form in which g is written. 2.6 SQ 3 Suppose that f is given by h(x) = 10x + 1 5x 3 f(x) = x 2. Suppose that g = T 1 S 3 f T 4 S 7. Write g(x) in as simplified a form as possible. 2.6 LQ 1 Let f be the quadratic polynomial given by f(x) = x 2 + 4x 10. Find the vertex f by using transformations of functions. Find all points where f is zero, again by only appealing to transformations of functions. Describe which transformations you have used and what your primitive function is. 13