Honors Precalculus Notes Packet 015-016 Academic Magnet High School
Contents Unit 1: Inequalities, Equations, and Graphs... 1 Unit : Functions and Graphs... 16 Unit 3: New Functions from Old... 37 Unit 4: Polynomial and Rational Functions... 50 Unit 5: Graphs of Functions Revisited... 7 Unit 6: Exponential and Logarithmic Functions... 81 Unit 7: Trigonometry Part 1... 95 Unit 8: Trigonometry Part... 109 Unit 9: Inverse Trigonometric Functions... 119 Unit 10: Sequences and Series... 17 Unit 11: Vectors in the Plane... 136
AMHS Precalculus - Unit 1 1 Interval Notation Unit 1: Inequalities, Equations, and Graphs Interval notation is a convenient and compact way to express a set of numbers on the real number line. Graphic Representation Inequality Notation x 3 Interval notation 1x 4 1 x x x 1 Inequality Properties 1. If a b, then a c b c. If a b and c 0, then ac bc 3. If a b and c 0, then ac bc Ex. 1 Solve each inequality (note that the degree is 1) and write the solution using interval notation: a) 3x 5 1 b) 9 x 10 5 c) 7 x 3 4 3
AMHS Precalculus - Unit 1 Ex. Solve each inequality and write the solution using inequality notation. a) 0 x b) 0 x c) 0 x 3 Polynomial Inequalities with degree two or more and Rational Inequalities Solve x 4x 7 4 by making a sign chart. Write your answer using interval notation. 1. Set one side of the inequality equal to zero.. Temporarily convert the inequality to an equation. 3. Solve the equation for x. If the equation is a rational inequality, also determine the values of x where the expression is undefined (where the denominator equals zero). These are the partition values. 4. Plot these points on a number line, dividing the number line into intervals. 5. Choose a convenient test point in each interval. Only one test point per interval is needed. 6. Evaluate the polynomial at these test points and note whether they are positive or negative. 7. If the inequality in step 1 reads 0, select the intervals where the test points are positive. If the inequality in step 1 reads 0, select the intervals where the test points are negative.
AMHS Precalculus - Unit 1 3 Ex. 3 Solve each inequality. Show the sign chart. Draw the solution on the number line and express the answer using interval notation. a) x( x 4)( x 3) 0 b) x 3x 4 0 x 3 c) 0 x 4 d) 3 x4 x1
AMHS Precalculus - Unit 1 4 Absolute Value x x x if x 0 if x 0 The absolute value of a real number x is the distance on the number line that x is from 0. Absolute value equations Ex. 4 Solve the equation (check your answers for extraneous solutions): a) x 1 x 3 4 b) x 3 1 Absolute value inequalities 1. if x a, then a x a. if x a 0, then x a or x a Ex. 5 Solve the inequality. Express your answers in interval notation and graph the solution: a) 4x 1.01 b) x 1 5
AMHS Precalculus - Unit 1 5 c) x + 3x 4 < 6 Equations and Graphs Lines The equation y mx b is a linear equation where m and b are constants. This is called Slope- Intercept form where m is the slope and b is the y-intercept. In general, m 0 m 0 m 0 m is undefined
AMHS Precalculus - Unit 1 6 The slope of a Line Point-Slope equation of a line: Ex. 1 Find the point-slope equation of a line passing through the points (-1, -) and (,5). Ex. Write the equation of a line passing through the points (4,7) and (0,3).
AMHS Precalculus - Unit 1 7 Parallel and Perpendicular Lines Two non-vertical lines are parallel iff they have the same slope. Two lines with non-zero slopes m1 and m are perpendicular iff m1 m 1. Ex. 3 Find the equation of the line passing through the point (-3,) that is parallel to 5xy 3. Ex. 4 Find the equation of the line passing through (-4,3) which is perpendicular to the line passing through (-3,) and (1,4). Ex. 5 A new car costs $9,000. Its useful lifetime is approximately 1 years, at which time it will be worth an estimated $000.00. a) Find the linear equation that expresses the value of the car in terms of time. b) How much will the car be worth after 6.5 years?
AMHS Precalculus - Unit 1 8 Ex. 6 The manager of a furniture factory finds that it costs $0 to manufacture 100 chairs and $4800 to manufacture 300 chairs. a) Assuming that the relationship between cost and the number of chairs produced is linear, find an equation that expresses the cost of the chairs in terms of the number of chairs produced. b) Using this equation, find the factory s fixed cost (i.e. the cost incurred when the number of chairs produced is 0). Ex. 7 Find the slope-intercept equation of the line that has an x-intercept of 3 and a y-intercept of 4.
AMHS Precalculus - Unit 1 9 Circles Recall the distance formula d ( x x ) ( y y ) 1 1 The Standard form for the equation of a circle is: Ex.1 Write the equation of a circle with center (-1,) and radius 3. Sketch this circle. Ex. Write the equation of a circle with center at the origin and radius 1. Ex.3 Find the equation of the circle with center (-4,1) that is tangent to the line x = -1.
AMHS Precalculus - Unit 1 10 Ex. 4 Find the equation of the circle with center (4,3) and passing through the point (1,4). Ex. 5 Express the following equations of a circle in standard form. Identify the center and radius: a) x y x y 4 6 3 b) x x y y 4 4
AMHS Precalculus - Unit 1 11 The intercepts of a graph The x -coordinates of the x - intercepts of the graph of an equation can be found by setting y 0and solving for x. The y -coordinates of the y - intercepts of the graph of an equation can be found by setting x 0 and solving for y. Ex. 1 Find the x and y intercepts of the line and sketch its graph: x y 1 Ex. Find the x and y intercepts of the circle and sketch its graph: x y 9 Ex. 3 Find the intercepts of the graphs of the equations. a) x y 9 b) y x x 5 1
AMHS Precalculus - Unit 1 1 Symmetry In general : A graph is symmetric with respect to the y axis if whenever ( xy, ) is on a graph ( xy, ) is also a point on the graph. A graph is symmetric with respect to the x axis if whenever ( xy, ) is on a graph ( x, y) is also a point on the graph. A graph is symmetric with respect to the origin if whenever ( xy, ) is on a graph ( x, y) is also a point on the graph. Tests for Symmetry: The graph of an equation is symmetric with respect to: a) the y axis if replacing x by x results in an equivalent equation. b) the x axis if replacing y by y results in an equivalent equation. c) the origin if replacing x and y by x and y results in an equivalent equation. Ex. 1 Show that the equation y x 3 has y axis symmetry.
AMHS Precalculus - Unit 1 13 Ex. Show that the equation x y 10 has x axis symmetry. Ex. 3 Show that the equation x y 9has symmetry with respect to the origin. Ex. 4 Find any intercepts of the graph of the given equation. Determine whether the graph of the equation possesses symmetry with respect to the x axis, y axis, or origin. a) x y b) y x 4 c) y x x d) y x 9
AMHS Precalculus - Unit 1 Difference of two squares: Algebra and Limits a b a b a b ( )( ) 14 Difference of two cubes: 3 3 a b ( a b)( a ab b ) Sum of two cubes: Binomial Expansion Binomial Expansion 3 3 a b ( a b)( a ab b ) n : ( a b) a ab b n 3: ( a b) a 3a b 3ab b 3 3 3 Limits x Ex. 1 Estimate lim x x 4 numerically by completing the following chart: x y x y 1.9.1 1.99.01 1.999.001 x Conclusion: lim x x 4 = Properties of Limits If a and c are real numbers, then lim c c,lim x a,lim x n a n xa xa xa Ex. Find the limit: a) 3 lim( x x 4) x b) lim(x 7) x1
AMHS Precalculus - Unit 1 15 Ex. 3 Find the given limit by simplifying the expression a) x x6 lim x x 5x6 b) lim x x x 3 4 8 c) x 3 lim x1 x 1 d) x lim x 7x10 x e) lim x x x 53 f) lim x 0 1 1 x 8 8 x
AMHS Precalculus - Unit 16 Unit : Functions and Graphs Functions A function is a rule that assigns each element in the domain to exactly one element in the range. The domain is the set of all possible inputs for the function. On a graph these are the values of the independent variable (most commonly known as the x values). The range is the set of all possible outputs for the function. On a graph these are the values of the dependent variable (most commonly known as the y values). We use the notation f( x) to represent the value (again, in most cases, a y - value) of a function at the given independent value of x. For any value of x, ( x, f ( x)) is a point on the graph of the function f( x ). Ex. 1 Given f ( x) to express the domain and range. x, graph the function and determine the domain and range. Use interval notation
AMHS Precalculus - Unit 17 Ex. Given f ( x) to express the domain and range. x, graph the function and determine the domain and range. Use interval notation Ex. 3 For the function f ( x) x x 4, find and simplify: a) f ( 3) b) f ( x h) Ex. 4 For x, x 0 f( x) x1, x0 find: a) f (1) b) f ( 1) c) f ( ) d) f (3)
AMHS Precalculus - Unit 18 Ex. 5 The graph of the function f is given: a) Determine the values: f ( ) f (0) b) Determine the domain: c) Determine the range: f () f (4) Ex. 6 The graph of the function f is given: a) f ( 3) f (0) f (4) b) For what numbers x is f( x) 0? c) What is the domain of f? d) What is the range of f? e) What is (are) the x -intercept(s)? f) What is the y - intercept? g) For what numbers x is f( x) 0? h) For what numbers x is f( x) 0?
AMHS Precalculus - Unit 19 Vertical Line Test for a Function: An equation is a function iff every vertical line intersects the graph of the equation at most once. Ex. 7 Determine which of the curves are graphs of functions: a) b) c) Domain (revisited) Rule for functions containing even roots (square roots, 4 th roots, etc): Ex. 1 Determine the domain and range of f ( x) 4 x 3 Ex. Determine the domain of f t t t ( ) 15
AMHS Precalculus - Unit 0 Rule for functions containing fractional expressions: 5x Ex. 3 Determine the domain of hx ( ) x 3x4 Ex. 4 Determine the domain of gx ( ) x 1 x15 Ex. 5 Determine the domain of hx ( ) 3 x x
AMHS Precalculus - Unit 1 Intercepts (revisited) The y -intercept of the graph of a function is (0, f (0)). The x - intercept(s) of the graph of a function f( x) is/are the solution(s) to the equation f( x) 0. These x - values are called the zeros of the function f( x ). Ex. 1 Find the zeros of f ( x) x(3x 1)( x 9) Ex. Find the zeros of f x x x ( ) 5 6 Ex. 3 Find the zeros of f x 4 ( ) x 1 Ex. 4 Find the x - and y - intercepts (if any) of the graph of the function 1 f ( x) x 4
AMHS Precalculus - Unit Ex. 5 Find the x - and y - intercepts (if any) of the graph of the function f x ( ) 4( x ) 1 Ex. 6 Find the x - and y - intercepts (if any) of the graph of the function f( x) x 4 x 16 Ex. 7 Find the x - and y - intercepts (if any) of the graph of the function 3 f ( x) 4 x
AMHS Precalculus - Unit 3 Transformations Horizontal and Vertical shifts Suppose y f ( x) is a function and c is a positive constant. Then the graph of 1. y f ( x) c is the graph of f shifted vertically up c units.. y f ( x) c is the graph of f shifted vertically down c units. 3. y f ( x c) is the graph of f shifted horizontally to the left c units. 4. y f ( x c) is the graph of f shifted horizontally to the right c units. Ex. 1 Consider the graph of a function y f ( x) shown on the coordinates. Perform the following transformations. y f ( x) 3 y f ( x) y f ( x 1) y f ( x 3)
AMHS Precalculus - Unit 4 Suppose y f ( x) is a function. Then the graph of 1. y f( x) is the graph of f reflected over the x -axis.. y f( x) is the graph of f reflected over the y -axis. Ex. Consider the graph of a function y f ( x). Sketch y f ( x ) 3 Common (Parent) Functions f ( x) x f ( x) x
AMHS Precalculus - Unit 5 f ( x) x f ( x) 3 x f ( x) 3 x 1 f( x) x f ( x) x or x
AMHS Precalculus - Unit 6 Combining common functions with transformations Sketch the graphs of the following functions. Determine the domain and range and any intercepts. Ex. 1 f ( x) x 1 Ex. f ( x) 1 x Ex. 3 f x 3 ( ) ( x ) 1 Ex. 4 f ( x) x 1 3
AMHS Precalculus - Unit 7 Symmetry (revisited) Tests for Symmetry The graph of a function f is symmetric with respect to: 1. the y -axis if f ( x) f ( x) for every x in the domain of the f( x ).. The origin if f ( x) f ( x) for every x in the domain of the f( x ). If the graph of a function is symmetric with respect to the y -axis, we say that f is an even function. If the graph of a function is symmetric with respect to the origin, we say that f is an odd function. In examples 1-3, determine whether the given function y f ( x) is even, odd or neither. Do not graph. Ex. 1 5 3 f ( x) x x x Ex. f ( x) x 3 Ex. 3 f ( x) x x
AMHS Precalculus - Unit 8 Transformations Vertical Stretches and Compressions Suppose y f ( x) is a function and c a positive constant. The graph of y cf ( x) is the graph of f 1. Vertically stretched by a factor of c if c 1. Vertically compressed by a factor of c if 0c 1 Ex.1 Given the graph of y f ( x) a) Sketch y f ( x) b) 1 y f ( x ) Ex. Sketch the graph of the following functions. Include any intercepts. f ( x) x 1 f ( x) 3( x 1)
AMHS Precalculus - Unit 9 Quadratic Functions A quadratic function y f ( x) is a function of the form constants. f ( x) ax bx c where a 0, b and c are The graph of any quadratic function is called a parabola. The graph opens upward if a 0 and downward if a 0. The domain of a quadratic function is the set of real numbers (, ). A quadratic function has a vertex (which serves as the minimum or maximum of the function depending on the value of a ), a line of symmetry, and may have zero, one or two x - intercepts. Ex. 1 Sketch the graph of f ( x) ( x 1) 3. Determine any intercepts.
AMHS Precalculus - Unit 30 The standard form of a quadratic function is parabola and x his the line of symmetry. f ( x) a( x h) k where ( hk, ) is the vertex of the Ex. Rewrite the quadratic function f ( x) x x 3 in standard form by completing the square. Determine any intercepts, the vertex, the line of symmetry and sketch the graph. Ex. 3 Rewrite the quadratic function f ( x) 4x 1x 9 in standard form by completing the square. Determine any intercepts, the vertex, the line of symmetry and sketch the graph.
AMHS Precalculus - Unit 31 Ex. 4 Complete the square to find all the solutions to the equation ax bx c 0 The vertex of any parabola of the form is ( b b, f ( )) a a. f ( x) ax bx c Ex. 5 Find the vertex of the quadratics from examples and 3 directly by using ( b b, f ( )) a a. Ex. 6 Find the vertex from example by using the x - intercepts and the line of symmetry.
AMHS Precalculus - Unit 3 Ex.7 Find the intercepts and vertex of the function 1 f x x x ( ) 1 Ex. 8 Find the maximum or the minimum of the function. 1. f x x x ( ) 3 8 1. f x x x ( ) 6 3 Ex.9 Determine the quadratic function whose graph is given.
AMHS Precalculus - Unit 33 Freely Falling Object - Suppose an object, such as a ball, is either thrown straight upward or downward with an initial velocity v 0 or simply dropped ( v0 0 ) from an initial height s 0. Its height, st () as a 1 function of time t can be described by the quadratic function s() t gt v t s 0 0 Gravity on earth is 3 ft / sec or 9.8 m / sec. Also, the velocity of the object while it is in the air is v() t gt v0 Ex. 10 An arrow is shot vertically upward with an initial velocity of 64 ft / sec from a point 6 feet above the ground. 1. Find the height st () and the velocity vt () of the arrow at time t 0.. What is the maximum height attained by the arrow? What is the velocity of the arrow at the time it attains its maximum height? 3. At what time does the arrow fall back to the 6 foot level? What is its velocity at this time? Ex. 11 The height above the ground of a toy rocket launched upward from the top of a building is given by s( t) 16t 96t 56. 1. What is the height of the building?. What is the maximum height attained by the rocket? 3. Find the time when the rocket strikes the ground. What is the velocity at this time?
AMHS Precalculus - Unit 34 Horizontal Stretches and Compressions Suppose y f ( x) is a function and c a positive constant. The graph of y f ( cx) is the graph of f 1. Horizontally compressed by a factor of 1 c if c 1. Horizontally stretched by a factor of 1 c if 0c 1 Ex.1 Given the graph of y f ( x) c) Sketch y f ( x) d) 1 y f ( x) Ex. Consider the function f x ( ) x 4 a) On the same axis, sketch f ( x), f ( x) and 1 f( x ). Identify any intercepts of each function.
AMHS Precalculus - Unit 35 b) On the same axis, sketch ( ), f x f x and 1 f( x ). Identify any intercepts of each function. List the transformations on f ( x) x required to sketch f ( x) x 1
AMHS Precalculus - Unit 36 Silly String Activity Objective: The use a quadratic function to model the path of silly string. Materials: Can of silly string, tape measure, stopwatch, clear overhead transparency, TI84 Personnel: Timekeeper, Silly-String operator, assistant Calculate the initial velocity v0 of the silly string as it exits the can. 1. Hold the can of silly string 1 foot above the ground. Have the timekeeper start the stopwatch and say go. At this time, shoot a short burst of silly string towards the ceiling. Have the class keep a casual eye on the maximum height the silly string achieves. When the silly string hits the floor, have the timekeeper stop the stopwatch and record the elapsed time.. Measure the maximum height of the silly string observed by the class. Use the position equation 1 s() t gt v0t s0 with g = 3 ft / sec to calculate v 0. ( s 0 = 1, get t from the timekeeper. This represents the time it took for the silly string to reach the ground, i.e. st () =0) Now that we know gvand, 0 s0 we can set up a position equation to model the height of the silly string as a function of time. Use this equation to determine the maximum height (the vertex holds this info) of the silly string. How does this compare to the actual height observed by the class. What factors might have caused it to be different? Now we are going to get the assistant to lean over the can of silly string (with the clear overhead transparency protecting the face) in its original position 1 foot above the ground and see if the assistant can move fast enough to avoid getting silly string in the face. Calculate the time it would take for the silly string to reach the assistant s face (set st () = the height of the assistant s face and solve fort ) Once the reaction time for the assistant has been calculated and discussed, see if the assistant can actually react that quickly, i.e. avoid silly string in the face. To date, it has never been done. Enjoy!
AMHS Precalculus - Unit 3 37 Unit 3: New Functions from Old Piecewise Defined Functions A function f may involve two or more functions, with each function defined on different parts of the domain of f. A function defined in this manner is called a piecewise-defined function. Ex.1 Sketch the graph of the given function and find the following: x if x 0 f ( x) x x if x 0 a) f ( 1) b) f () c) Domain: d) Range: Ex. 1b Express f ( x) x 3 as a piecewise function: Ex. Sketch the graph of the given function and find the following: 3x 1 if x 1 f( x) x 1 if x 0 a) f ( 1) b) f () c) Domain: d) Range:
AMHS Precalculus - Unit 3 38 Ex.3 Graph the following a) 1 if x 0 f ( x) 0 if x 0 x 1 if x 0 b) f( x) x 1 x 1 Hint: write this as a piecewise function Domain: Range: Domain: Range:
AMHS Precalculus - Unit 3 39 Graphing the Absolute Value of a Function Sketch the graph of the given functions. Include any intercepts. Ex.1 x if x 0 f ( x) x x if x 0 Ex. f ( x) ( x ) 4 Ex.3 Ex.4 f ( x) x 4x 3 f ( x) x 3 1
AMHS Precalculus - Unit 3 40 Compositions of Functions The composition of the function f with the function g, denoted f g is defined by ( f g)( x) f ( g( x)). The domain of f g consists of those x values in the domain of g for which gx ( ) is in the domain of f. Ex. 1 f ( x) x and g x ( ) x 1. Find the following: a) ( f g)( x ) b) Find the domain of ( f g)( x ) c) ( g f )( x ) d) Find the domain of ( g f )( x ) e) ( f g )() f) ( g f )(4) g) ( g f )(1) Ex. Write the function f x ( ) x 3 as the composition of two functions Ex.3 Write the function f( x) x 3 as the composition of three functions. 4x1
AMHS Precalculus - Unit 3 41 Ex. 4 Given F( x) ( x 4) x 4 find functions f and g such that F( x) ( f g)( x). Ex. 5 A metal sphere is heated so that t seconds after the heat had been applied, the radius rt () is given by r( t) 3.001t cm. Express the Volume of the sphere as a function of t. Ex. 6 f ( x) x and g( x) x, ( x 0). Find the following: a) ( f g)( x ) b) ( g f )( x) c) g (3) d) g (4) e) f (9) f) f (16)
AMHS Precalculus - Unit 3 4 Inverse Functions Suppose that f is a one-to-one function with domain X and range Y. The inverse function for the function f is the function denoted f 1 ( f ( x)) x and 1 f ( f ( x)) x. 1 f with domain Y and range X and defined for all values x X by Ex. 1 Prove that f ( x) x and g x ( ) x ( x 0) are inverse functions using composition. Steps for Finding the Inverse of a Function: 1. Set y f ( x). Change x yand y x 3. Solve for y 4. Set y f 1 ( x) Ex. Find the inverse of f( x) and f 1 ( x). 3x f( x) and check using composition. Find the domain and range of ( x 4)
AMHS Precalculus - Unit 3 43 The graph of f 1 ( x) is a reflection of the graph of f( x) about the line y x. One-to-One Functions A function is one-to-one iff each number in the range of f is associated with exactly one number in its domain. In other words, f ( x1) f ( x) implies x 1 x. Horizontal Line Test for One-to-One Functions A function is one-to-one precisely when every horizontal line intersects its graph at most once. Ex. 3 Determine whether the given function is one-to-one a) f x 3 ( ) x b) f ( x) x x
AMHS Precalculus - Unit 3 44 Ex.4 Given f ( x) x 3 Domain of f( x ): Domain of f 1 ( x) : Range of f( x ): Range of f 1 ( x) : Find f 1 ( x) and check using composition. Sketch the graph of f 1 ( x) and f( x) on the same axis.
AMHS Precalculus - Unit 3 45 Translating Words into Functions In calculus there will be several instances where you will be expected to translate the words that describe a problem into mathematical symbols and then set up or construct an equation or a function. In this section, we will focus on problems that involve functions. We begin with a verbal description about the product of two numbers. Ex.1 The sum of two nonnegative numbers is 15. Express the product of one and the square of the other as a function of one of the numbers. Ex. A rectangle has an area of 400 length of one of its sides. in. Express the perimeter of the rectangle as a function of the Ex.3 Express the area of a circle as a function of its diameter d.
AMHS Precalculus - Unit 3 46 Ex. 4 An open box is made from a rectangular piece of cardboard that measures 30cm by 40cm by cutting a square of length x from each corner and bending up the sides. Express the volume of the box as a function of x. Ex. 5 Express the area of the rectangle as a function of x. The equation of the line is xy 4.The lower left-hand corner is on the origin and upper right-hand corner of the rectangle with coordinate ( xy, ) is on the line. Ex. 6 Express the area of an equilateral triangle as a function of the length s of one of its sides.
AMHS Precalculus - Unit 3 47 The Tangent Line Problem Find a tangent line to the graph of a function f. m tan f a x f a lim x 0 ( ) ( ) x Ex.1 Find the slope of the tangent line to the graph of f x at x 1. ( ) x Ex. Find the slope of the tangent line to the graph of f x at x 3. ( ) x
AMHS Precalculus - Unit 3 48 Ex.3 Find the slope of the tangent line to the graph of f x at x. ( ) x The DERIVATIVE of a function y f ( x) is the function f ' defined by: f '( x) f x x f x lim x 0 ( ) ( ) x Ex.4 Find the derivative of f x ( ) x. Ex.5 Find the derivative of the tangent line at x. f ( x) x 6x 3 and use it to find the slope and then the equation of
AMHS Precalculus - Unit 3 49 Ex. 6 Find the slope of the tangent line to the graph of f( x) at x 1. x Ex.7 Find the derivative of line at x. f( x) and use it to find the slope and then the equation of the tangent x Ex. 8 Find the derivative of f ( x) x.
AMHS Precalculus - Unit 4 50 Unit 4: Polynomial and Rational Functions Polynomial Functions A polynomial function y p( x) is a function of the form p( x) a x a x a x... a x a x a n n 1 n n n1 n 1 0 where an, an 1,..., a, a1, a0 are real constants and are called the coefficients of px ( ). n is the degree of px ( ) and is a positive integer. an is called the leading coefficient and a0 is the constant term of the polynomial. The domain of any polynomial is all real numbers. Ex. 1 Determine the degree, the leading coefficient and the constant term of the polynomial. a) 4 3 f ( x) 5x 7x 3x 7 b) 3 g( x) 13x 5x 4x End Behavior of a Polynomial There are four scenarios: 1) Sketch p( x) x, p( x) x ( n is even, an 0) 4 ) Sketch p( x) x, p( x) x ( n is even, an 0 ) 4 As x, p( x) As x, p( x) As x, p( x) As x, p( x)
AMHS Precalculus - Unit 4 51 3) Sketch p( x) x, p( x) x ( n is odd, an 0) 3 5 4) Sketch p( x) x, p( x) x ( n is odd, an 0 ) 3 5 As x, p( x) x, p( x) As x, p( x) x, p( x) As x and x, the graph of the polynomial p( x) a x a x a x... a x a x a resembles the graph of n n 1 n n n1 n 1 0 y n anx. Ex. Use the zeros and the end behavior of the polynomial to sketch an approximation of the graph of the function. a) 3 f ( x) x 9x b) g x x x 4 ( ) 5 4
AMHS Precalculus - Unit 4 5 c) 5 f ( x) x x Repeated Zeros If a polynomial f( x) has a factor of the form ( x c) k, where k 1, then x cis a repeated zero of multiplicity k. If k is even, the graph of f( x) flattens and just touches the x -axis at x c. If k is odd, the graph of f( x) flattens and crosses the x -axis at x c. Ex. 4: Sketch the given graphs f ( x) x 3x x 4 3 g x x x x 3 ( ) ( 1) ( )( 3)
AMHS Precalculus - Unit 4 53 Ex. 5: The cubic polynomial px ( ) has a zero of multiplicity two at x 1, a zero of multiplicity one at x, and p( 1). Determine px ( ) and sketch the graph. Ex. 6: An open box is to be made from a rectangular piece of cardboard that is 1 by 6 feet by cutting out squares of side length x feet from each corner and folding up the sides. a) Express the volume of the box vx ( ) as a function of the size x cut out at each corner. b) Use your calculator to approximate the value of x which will maximize the volume of the box. Ex. 7: The difference of two non-negative numbers is 10. What is the maximum of the product of the square of the first number and the other?
AMHS Precalculus - Unit 4 54 The Intermediate Value Theorem Suppose that f is continuous on the closed interval [ aband, ] let N be any number between f( a) and f() b, where f ( a) f ( b). Then there exists a number c in ( ab, ) such that f () c N. Ex. 1: Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem. f ( x) x x 1, [0,5], f( c) 11 Ex. : Show that there is a root of the equation 3 x x1 0 in the interval (0,1).
AMHS Precalculus - Unit 4 55 The Division Algorithm Let f( x) and dx ( ) 0 be polynomials where the degree of f( x) is greater than or equal to the degree of dx ( ). Then there exists unique polynomials qx ( ) and rx ( ) such that f ( x) r( x) qx ( ) or f ( x) d( x) q( x) r( x). d( x) d( x) where rx ( ) has a degree less than the degree of dx ( ). Ex. 1: Divide the given polynomials. a) 3 6x 19x 16x 4 x b) 3 x 1 x 1 c) 3 3x x x 6 x 1
AMHS Precalculus - Unit 4 56 Remainder Theorem If a polynomial f( x) is divided by a linear polynomial x c, then the remainder r is the value of f( x) at x c. In other words, f () c r Ex. : Use the Remainder Theorem to find r when 3 f ( x) 4x x 4 is divided by x. Ex. 3: Use the Remainder Theorem to find f() c for 4 1 f ( x) 3x 5x 7 when c Synthetic Division Synthetic division is a shorthand method of dividing a polynomial px ( ) by a linear polynomial x c. It uses only the coefficients of px ( ) and must include all 0 coefficients of px ( ) as well. Ex. 4: Use synthetic division to find the quotient and remainder when a) f x 3 ( ) x 1 is divided by x 1 b) 4 f ( x) x 14x 5x 9is divided by x 4 c) 4 3 8x 30x 3x 8x 3 is divided by 1 x 4
AMHS Precalculus - Unit 4 57 Ex. 5: Use synthetic division and the Remainder Theorem to find f() c for 6 5 4 3 f ( x) 3x 4x x 8x 6x 9 when c. Ex. 6: Use synthetic division and the Remainder Theorem to find f() c for 3 f ( x) x 7x 13x 15 when c 5. The Factor Theorem A number c is a zero of a polynomial px ( ) ( pc ( ) 0 ) if and only if ( x c) is a factor of px ( ). Examples: Determine whether a) x 1is a factor of f x x x x 4 ( ) 5 6 1 b) x is a factor of 3 x 3x 4
AMHS Precalculus - Unit 4 58 Fundamental Theorem of Algebra A polynomial function px ( ) of degree n 0 has at least one zero. In fact, every polynomial function px ( ) of degree n 0 has at exactly n zeros. Complete Factorization Theorem Let c1, c,... c n be the n (not necessary distinct) zeros of the polynomial function of degree n 0 : p( x) a x a x a x... a x a x a. n n 1 n n n1 n 1 0 Then px ( ) can be written as the product of n linear factors p( x) a ( x c )( x c ) ( x c ). n 1 n Ex. 1: Give the complete factorization of the given polynomial px ( ) with given information: a) 3 p( x) x 9x 6x 1; 1 x is a zero. b) 4 3 p( x) 4x 8x 61x x 15 ; x 3, x 5 are both zeros.
AMHS Precalculus - Unit 4 59 c) 3 p( x) x 6x 16x 48 ; ( x ) is a factor. d) ; x(3x1) is a factor. 4 3 p( x) 3x 7x 5x x Ex. : Find a polynomial function f( x) of degree three, with zeros 1,-4, 5 such that the graph possesses the y - intercept (0,5).
AMHS Precalculus - Unit 4 60 The Rational Zero Test Suppose p q is a rational zero of 1 ( ) n n f x a x a x a x n... a x a x a, n n1 n 1 0 where a0, a1..., an are integers and an 0. Then p divides a0 and q divides a n. The Rational Zero Test provides a list of possible rational zeros. Examples: Find all the rational zeros of f( x) then factor the polynomial completely. a) f x x x x x 4 3 ( ) 3 10 3 8. b) f x x x x x 4 3 ( ) 3 3
AMHS Precalculus - Unit 4 61 Complex Roots of Polynomials Consider factoring the function: f x 3 ( ) x 1 The Square Root of -1 We define i 1 so that i 1. Complex Numbers A complex number is a number of the form a bi where a and b are real numbers. The number a is called the real part and the number b is called the imaginary part. Complex Arithmetic Ex. 1: a) ( 3 i) (6 i) b) ( 3 i)(4 i) c) (36 i)(3 6 i) d) (4 5 i)(4 5 i) Complex Conjugates The complex conjugate for a complex number z a bi is z a bi. In general, ( a bi)( a bi)
AMHS Precalculus - Unit 4 6 Ex. : Simplify. a) ( 3 i) (1 6 i) ( i) (1 7 i) Ex. 3: Simplify. a) 4 b) 8 Ex. 4: Determine all solutions to the equation x 4x13 0 Ex. 5: Completely factor f x 3 ( ) x 1.
AMHS Precalculus - Unit 4 Ex. 6: Find the complete factorization of multiplicity two. 63 4 3 f ( x) x 1x 47x 6x 6 given that 1 is a zero of Conjugate Pairs of Zeros of Real Polynomials If the complex number z a bi is a zero of some polynomial px ( ) with real coefficients, then its conjugate z a bi is also a zero of px ( ). Ex. 7: Find a 3 rd degree polynomial gx ( ) with real coefficients and a leading coefficient of 1 with zeros 1 and 1 i. 4 3 Ex. 8: 1 i is a zero of f ( x) x x 4x 18x 45. Find all other zeros and then give the complete factorization of f( x ).
AMHS Precalculus - Unit 4 64 Rational Functions A rational function y f ( x) is a function of the form functions. px ( ) f( x), where p and q are polynomial qx ( ) Ex. 1: Recall the parent function f( x) 1. Use transformations to sketch x gx ( ) x 1 Asymptotes of Rational Functions The line x ais a vertical asymptote of the graph of f( x) if f( x) or f( x) as x a (from the right) or x a (from the left). Vertical Asymptotes px ( ) The graph of f( x) has vertical asymptotes at the zeros of qx ( ) after all of the common factors qx ( ) of px ( ) and qx ( ) have been canceled out; the values of x where qx ( ) 0 and px ( ) 0. Holes The graph of px ( ) f( x) has a hole at the values of x where qx ( ) 0 and px ( ) 0. qx ( )
AMHS Precalculus - Unit 4 65 Horizontal Asymptotes The line y bis a horizontal asymptote of the graph of f( x) if f ( x) b when x or x. In particular, with a rational function There are three cases: f( x) n px ( ) anx a x... a x a m q( x) b x b x... b x b m n1 n1 1 0 m1 m1 1 0 1. If n m, then y 0is the horizontal asymptote. 3x Ex: f( x) 3 13x 7x. If n m, then a y b n m is the horizontal asymptote. Ex: f( x) 3 3x 6x 3 4x x 3 3. If n m, then there is no horizontal asymptote. Ex: f( x) 4 3 3x x 5x 1 3x 4x Slant Asymptote If the degree of numerator is exactly one more than the degree of the denominator, the graph of f( x) has a slant asymptote of the form y mx b. The slant asymptote is the linear quotient found by dividing px ( ) by qx ( ) and essentially disregarding the remainder. Example: f( x) x 3 8x1 x 1
AMHS Precalculus - Unit 4 66 Ex. : Find all asymptotes and intercepts and sketch the graphs of the given rational functions: a) f( x) x 1 Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x - intercepts: y - intercept: b) 3x f( x) x 4 Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x - intercepts: y - intercept:
AMHS Precalculus - Unit 4 67 x 3 c) f( x) ( x)( x5) Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x - intercepts: y - intercept: d) f( x) x x x 1 Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x - intercepts: y - intercept: x 1 e) f( x) x x
AMHS Precalculus - Unit 4 68 Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x - intercepts: y - intercept: f) (3x1)( x) f( x) ( x)( x1) Domain: Range: Equation(s) of vertical asymptotes: Equation(s) of horizontal asymptotes: Equation of slant asymptote: x - intercepts: y - intercept: Ex. 3: Sketch the graph of a rational function that satisfies all of the following conditions: f( x) as x 1 and f( x) as x 1 f( x) as x and f( x) as x f( x) has a horizontal asymptote y 0 f( x) has no x -intercepts Has a local maximum at ( 1, ) Ex. 4: The product of two non-negative numbers is 60. What is the minimum sum of the two numbers?
AMHS Precalculus - Unit 4 69
AMHS Precalculus - Unit 4 Honors Precalculus Academic Magnet High School 70 Name Mandelbrot Set Activity using Fractint fractal generator STEP 1 - CREATE, SAVE, and PRINT an inspirational, visually pleasing area of the Mandelbrot set. Important Menu Items: VIEW- Image Settings, Zoom In/ Out box, Coordinate Box FRACTALS- Fractal Formula, Basic Options, Fractal Parameters COLORS- Load Color- Map FILE- Save As 1) Start Fractint by clicking on the desktop icon. Fractint always starts with the Mandelbrot set, but in case things get weird, ALWAYS make sure mandel is selected in the Fractal-Fractal Formula menu item. Use the Image Settings box to set the size of the picture (800 x 600 should work fine). ) Use the Zoom In/Out feature along with the Colors-Load Color Map to create a variation of the Mandelbrot set. If the color palettes do not load, double click on the box that is labeled Pallette Files (*.Map) If you zoom in a few times you lose detail, you can increase the iterations in the Fractals-Basic Options Box- Remember that the more iterations the computer has to perform, the longer it will take 3) Use the Fractals-Fractal Params window to record the x and y mins and maxs of the viewing rectangle on the imaginary plane. 4) Using the Coordinates box, point your arrow to a point you think is in the Mandelbrot set and record the x and y values. 5) Repeat #4 for a point you think is NOT in the set. 6) SAVE the fractal. Write down the coordinates (x and y mins and maxs) and number of iterations of your current position in the Mandelbrot set. 7) Print your fractal. STEP - Create a typed text document (1 page or so) including, but not limited to: The NAME of your group s fractal and the name of everyone in your group A short story about your creation (what it makes you think of, color, choice, etc.)
AMHS Precalculus - Unit 4 71 STEP 3 Typed: 1) List the x and y mins and maxs for your viewing rectangle from Step 1 ) Recall the coordinates of the point you thought was in the Mandelbrot set from Step 1. Let x = a and y = b for the complex number a + bi Let this number a + bi = c iterate this value 100 or more times using the Mandelbrot sequence: x 0 = c x 1 = x 0 + c x = x 1 + c Etc You will be using decimals and your calculator. Unlike the fractals, these calculations will not be pretty. Let your TI-84 do the work for you (i is above the decimal point). 3) Record the last 0 iterations for analysis. Remember that you may need to scroll the TI- 84 to the right to get the entire number 4) Were your predictions right about this point? Do you need more information to determine if it is in the set? 5) Repeat for the point you thought was not in the set. 6) Summarize your findings. TURN IN ALL 3 STEPS PAPER-CLIPPED together in order. Extra Credit: Create your own color map. http://www.nahee.com/spanky/www/fractint/fractint.html - for info on Fractint
AMHS Precalculus - Unit 5 7 Unit 5: Graphs of Functions Revisited Solving Equations Graphically The Intersection Method To solve an equation of the form f ( x) g( x) : 1. Graph y1 f ( x) and y g( x) on the same screen.. Find the x - coordinate of each point of intersection. Ex. 1: Solve. a) x 1 x 3 4 b) x 3x 4 c) 3 x x x x 4 3 6 The x - intercept Method To solve an equation of the form f ( x) g( x) : Ex.: Solve. 1. Write the equation in the equivalent form f( x) 0.. Graph y f ( x). 3. The x - intercepts of the graph are the real solutions to the equation. a) x 1 x 3 4 b) x 3x 4 c) 3 x x x x 4 3 6 d) 5 3 x x x 5
AMHS Precalculus - Unit 5 73 Technological Quirks 1. Solve f( x) 0 by solving f( x) 0.. Solve Ex. 3: Solve. f( x) 0 by solving f( x) 0 (eliminate any values that also make gx ( ) 0 ). gx ( ) a) 4 x x x 1 0 b) x x 1 9x 9x 0 Applications Ex. 1: According to data from the U.S. Bureau of the Census, the approximate population y (in millions) of Chicago and Los Angeles between 1950 and 000 are given by: Chicago: Los Angeles: y x x x 3.0000304.003.004 3.6 y x x x 3.0000113.00099.0538 1.97 where x 0 corresponds to 1950. In what year did the two cities have the same population? Ex. : The average of two real numbers is 41.15, and their product is 1683. Find the two numbers. Ex. 3: A rectangle is twice as wide as it is high. If it has an area of 4.5 square inches, what are the dimensions of the rectangle?
AMHS Precalculus - Unit 5 74 3 Ex. 4: A rectangular box with a square base and no top is to have a volume of 30,000 cm. If the surface area of the box is 6000 cm, what are the dimensions of the box? Ex. 5: A box with no top that has a volume of 1000 cubic inches is to be constructed from a x 30-inch sheet of cardboard by cutting squares of equal size from each corner and folding up the sides. What size square should be cut from each corner? Ex. 6: A pilot wants to make 840-mile trip from Cleveland to Peoria and back in 5 hours flying time. There will be a headwind of 30 mph going to Peoria, and it is estimated that there will be a 40-tail wind on the return trip. At what constant engine speed should the plane be flown?
AMHS Precalculus - Unit 5 75 Solving Inequalities Graphically 1. Rewrite the inequality in the form f( x) 0or f( x) 0.. Determine the zeros of f. 3. Determine the interval(s) where the graph is above ( f( x) 0 ) or below ( f( x) 0 ) the x -axis. Ex. 1: Solve each inequality graphically. Express your answer in interval notation. a) x( x 4)( x 3) 0 b) x 3x 4 x 3 c) 0 x 4 d) 3 x4 x1 e) x + 3x 4 < 6 f) 4 3 x x x x 6 5 Ex. : A company store has determined the cost of ordering and storing x laser printers is: 300, 000 cx x If the delivery truck can bring at most 450 printers per order, how many printers should be ordered at a time to keep the cost below $1600.00?
AMHS Precalculus - Unit 5 76 Increasing, Decreasing and Constant Functions A function f is increasing on an interval when, for any x1 and x in the interval, x 1 < x implies f ( x ) f ( x ). 1 A function f is decreasing on an interval when, for any x1 and x in the interval, x 1 < x implies f ( x ) f ( x ). 1 A function f is constant on an interval when, for any x1 and x in the interval, f ( x1) f ( x). Ex.1: Determine the open intervals on which each function is increasing, decreasing or constant. a) f ( x) x 1 x 3 b) 3 f ( x) x 3x c) f ( x) x 3
AMHS Precalculus - Unit 5 77 Relative Minimum and Maximum Values (Relative Extrema) A function value f( a) is called relative minimum of f when there exists an interval ( x1, x) that contains a such that x1 x ximplies f ( a) f ( x). A function value f( a) is called relative maximum of f when there exists an interval ( x1, x) that contains a such that x1 x ximplies f ( a) f ( x). Ex. : Determine the relative minimum and x -intercepts of f x x x ( ) 3 4 Ex. 3: Use a graphing utility to determine the relative minimum and x -intercepts of f x x x ( ) 3 4 Ex. 4: Use a graphing utility to determine any relative minima or maxima for 3 f ( x) x x
AMHS Precalculus - Unit 5 78 Ex. 5: During a 4-hour period, the temperature tx ( )(in degrees Fahrenheit) of a certain city can be 3 approximated by the model t( x).06x 1.03x 10.x 34, 0 x 4 where x represents the time of day, with x 0 corresponding to 6 A.M. Approximate the maximum and minimum temperatures during this 4-hour period. Optimization: Translating Words into Functions revisited Ex.1: The sum of two nonnegative numbers is 15. Express the product of one and the square of the other as a function of one of the numbers. Use a graphing utility to find the maximum product. Ex.: A rectangle has an area of 400 in. Express the perimeter of the rectangle as a function of the length of one of its sides. Use a graphing utility to find the minimum perimeter. Ex. 3: An open box is made from a rectangular piece of cardboard that measures 30cm by 40cm by cutting a square of length x from each corner and bending up the sides. Express the volume of the box as a function of x. Use a graphing utility to find the dimensions of the box with the maximum volume.
AMHS Precalculus - Unit 5 79 Ex. 4: Express the area of the rectangle as a function of x. The equation of the line is xy 4.The lower left-hand corner is on the origin and upper right-hand corner of the rectangle with coordinate ( xy, ) is on the line. Use a graphing utility to find the rectangle with the maximum area. Concavity and Inflection Points Concavity is used to describe the way a curve bends. For any two points in a given interval that lie on a curve, if the line segment that connects them is above the curve, then the curve is said to be concave up over the given interval. If the segment is below the curve, then the curve is said to be concave down over the interval. A point where the curve changes concavity is called an inflection point.
AMHS Precalculus - Unit 5 80 Ex. 1 For the following functions, estimate the following: 1. All local maxima and minima (relative extrema) of the function. Intervals where the function is increasing and/or decreasing 3. All inflection points of the function 4. Intervals where the function is concave up and when it is concave down a) 3 f ( x) x 6x x 3 b) g x x x 3 ( ) 4 3 c) f( x) ( x ) d) f( x) x x x 1
AMHS Precalculus - Unit 5 81 Rational exponents Unit 6: Exponential and Logarithmic Functions If b is a real number and n and m are positive and have no common factors, then n m m b = b ( b) m n n Laws of exponents a) b) c) d) e) f) g) Ex. 1: Simplify a) 8 ( ) 7 1 3 b) 5 9 Exponential Function If b 0and b 1, then an exponential function y f ( x) is a function of the form f ( x) The number b is called the base and x is called the exponent. x b.
AMHS Precalculus - Unit 5 8 Ex. : Graph each of the given functions a) f( x) x and f( x) 4 x b) x 1 x f( x) ( ) and x 1 f( x) 4 ( ) 4 x
AMHS Precalculus - Unit 5 83 In general f ( x) b x x 1 f ( x) b ( ) b x Domain: Range: Intercept: Horizontal Asymptote: Domain: Range: Intercept: Horizontal Asymptote: Ex. 3: Sketch each of the given functions a) hx ( ) 3 x 1 x b) hx ( ) 5
AMHS Precalculus - Unit 5 The Natural Base e 84 Use you calculator to explore 1 lim(1 ) n. n n Conclusion: Ex. 4: Sketch each of the given functions x a) f ( x) e b) h( x) e x c) f ( x) e x For c) State the Domain and Range
AMHS Precalculus - Unit 5 85 Ex. 5: Solve a) x3 x1 8 b) ( x1) 7 343 c) 3x x x d) 64 10(8 ) 16 0 x 4 Compound Interest The amount of money At () at some time t (in years) in an investment with an initial value, or principle of P with an annual interest rate of r (APR given as a decimal), compounded n times a year is: A( t) P1 r n nt Ex. 6: Determine the value of a CD in the amount of $1000.00 that matures in 6 years and pays 5% per year compounded a) Annually b) Monthly c) Daily
AMHS Precalculus - Unit 5 86 Continuously Compounded Interest. If the interest is compounded continuously ( n ), then the amount of money after t years is: A() t Pe rt Ex. 7: Determine the amount in the CD from example 6 if the interest is compounded continuously. Ex. 8: Which interest rate and compounding period gives the best return? a) 8% compounded annually b) 7.5% compounded semiannually c) 7% compounded continuously Ex. 9: What initial investment at 8.5 % compounded continuously for 7 years will accumulate to $50,000?
AMHS Precalculus - Unit 5 87 Logarithmic Functions Set up Sketch f( x) x. Give the domain and range. Then find f 1 ( x). f 1 ( x) = Domain: Range: Intercept: V.A.: Definition y For each positive number a 0 and each x in (0, ), y log a x if and only if x a. y x a is the corresponding exponential form of the given logarithmic form y log a x. Ex. 1: Evaluate each expression. a) log101000 b) log10 0.1 c) log 3 d) log 4 e) log88 f) log3 1 g) log3 8 log5 5 h) 3
AMHS Precalculus - Unit 5 88 Properties of the Logarithm function with base a. a) log a1 0 b) log a a 1 c) log a x a x d) log a x a x Ex. : On the same coordinate plane, sketch the following functions. f( x) 3 x and g( x) log3 x 1 x f( x) ( ) and g( x) log1/ x In general. g( x) log a x, a 1 g( x) log a x, 0a 1 Domain: Range: Intercept: V.A.: Domain: Range: Intercept: V.A.:
AMHS Precalculus - Unit 5 89 Ex. 3: Sketch the following functions. g( x) log ( ) 3 x g x 1/ ( ) log x 1 The Natural Logarithm Function The function defined by f ( x) log x ln x and y ln x iff e x y e. Ex. 4: On the same coordinate plane, sketch the following functions. x f ( x) e and g( x) ln x
AMHS Precalculus - Unit 5 90 Properties of the Logarithm function with base e. a) b) c) d) Arithmetic Properties of Logarithms For each positive number a 1, each pair of positive real numbers U and V, and each real number n we have: Base a Logarithm Natural Logarithm a) a) b) b) c) c) Ex. 5: Evaluate each expression a) c) 4 ln e b) 1 ln e d) ln 45 e e (1/)ln16 e) log 6 log 15 log 0 3ln8 e f) Change-of-Base Formula For a 0, a 0, x 0... log a log x ln x x log a ln a Ex. 6: Use your calculator to evaluate log6 13.
AMHS Precalculus - Unit 5 91 Ex.7: Use the properties of logarithms to simplify each expression so that the ln y does not contain products, quotients or powers. a) y (x1)(3 x) 4x 3 b) y x x x 6 3 64 1 Solving Exponential and Logarithmic Equations Ex. 8: Solve each of the given equations x a) e 83 b) x 4e 7 c) x 13 d) ln(3 x) 6 e) ln( x1) ln( x3) 1 f) ln( x ) ln(x 3) ln x
AMHS Precalculus - Unit 5 9 g) log ( x 3) 4 h) 1 5 3 x e x i) x x x e x e x 6e j) xln x x 0 Ex. 9: Given the function function. f x 3 1 ( ) e x 5, Find f 1 ( x) and state the domain and range of the inverse
AMHS Precalculus - Unit 5 93 Exponential Growth and Decay In one model of a growing (or decaying) population, it is assumed that the rate of growth (or decay) of the population is proportional to the number present at time t (rate of growth = kp() t ). Using calculus, it can be shown that this assumption gives rise to: kt P() t P e where k is the rate of growth ( k 0 ) or decay ( k 0 ). 0 Ex.1: The number of a certain species of fish is given by nt () is measured in millions. n( t) 1e 0.01t where t is measured in years and a) What is the relative growth rate of the population? b) What will the fish population be after 15 years? Ex.: A bacteria culture starts with 500 bacteria and 5 hours later has 4000 bacteria. The population grows exponentially. a) Find a function for the number of bacteria after t hours. b) Find the number of bacteria that will be present after 6 hours. c) When will the population reach 15000?
AMHS Precalculus - Unit 5 94 Ex. 3: A culture of cells is observed to triple in size in days. How large will the culture be in 5 days if the population grows exponentially? Ex. 4: Carbon-14, one of the three isotopes of carbon, has a half-life of 5730 years. If 10 grams were present originally, how much will be left after 000 years? When will there be grams left? Ex. 5: On September 19 th, 1991, the remains of a prehistoric man were found encased in ice near the border of Italy and Switzerland. 5.4% of the original carbon 14 remained at the time of the discovery. Estimate the age of the Ice Man. Ex. 6: The radioactive isotope strontium 90 has a half-life of 9.1 years. a) How much strontium 90 will remain after 0 years from an initial amount of 300 kilograms? b) How long will it take for 80% of the original amount to decay?
AMHS Precalculus - Unit 7 95 Unit 7: Trigonometry Part 1 Right Triangle Trigonometry Hypotenuse a) Sine sin( ) d) Cosecant csc( ) Opposite Adjacent b) Cosine cos( ) c) Tangent tan( ) e) Secant sec( ) f) Cotangent cot( ) Ex. 1: Find the values of the six trigonometric functions of the angle. 3 Ex.: Find the exact values of the sin,cos, and tan of 45 45
AMHS Precalculus - Unit 7 96 Ex. 3: Find the exact values of the sin,cos, and tan of 60 and 30 30 Ex. 4: Find the exact value of x (without a calculator). 5 30 x Ex. 5: Find all missing sides and angles (with a calculator). 33 1
AMHS Precalculus - Unit 7 97 Applications An angle of elevation and an angle of depression can be measured from a point of reference and a horizontal line. Draw two figures to illustrate. Ex. 6: (use a calculator) A surveyor is standing 50 feet from the base of a large building. The surveyor measures the angle of elevation to the top of the building to be 71.5. How tall is the building? Draw a picture. Ex. 7: A ladder leaning against a house forms a 67 angle with the ground and needs to reach a window 17 feet above the ground. How long must the ladder be?