Practice Problems for Test II

Transcription

1 Math 117 Practice Problems for Test II 1. Let f() = 1/( + 1) 2, and let g() = (a) Calculate (b) Calculate f ( h) f ( ) h g ( z k) g( z) k. Simplify your answer as much as possible. Simplify your answer as much as possible. 2. Find an equation of the inverse of the function: 7 y For each function below, determine if the function is even, odd, or neither. Eplain! (a) F() = (1 + 2 ) (b) G( ) 2 4 (c) H() = ( ) 5

2 4 4. Suppose that h ( ) (a) If f ( ) find a function g such that h f g (b) If f ( ) 13 find a function g such that h f g 1 (c) If g( ) 2 find a function f such that h f g Suppose that y F( ). Find a formula for F Find the range of the function y = 31 ( 2015) Eplain! 7. Let G() = (a) Eplain why G() is one-to-one. (b) What are the domain and range of G? (c) Compute the value of 5G -1 (4) + (G -1 (4)) (G -1 (4)) Find the equation of the circle that has center A = (-5, 3) and radius Find the radius and center of the circle ( + 11) 2 + (y 7) 2 = Consider the circle having equation y 2 + 5y = 3. Find the: (a) center (b) Radius

3 11. Find the verte of the parabola: y = 3( + 13) Using the method of completing the square, find the verte of the parabola: y = Find the maimum value of the following epression (using the method of completing the square) Find the verte of the given parabola: (a) y = 2 14 (b) y = (c) y = (d) y = ( 7) 2 + ( 4) 2 (e) y = 3( + 8) (f) y = ( 2) Find the maimum (or minimum) value of each of the following epressions by completing the square. Also, determine the -value at which the ma or min is achieved. (a) (b) 2 16( 1) + 5 (c) (d) (e)

4 16. Find the verte of each of the following parabolas. Also, sketch each curve and locate the y-intercept, any -intercepts, and label the verte. In addition, determine the range of the function. (a) y = (b) y = (c) y = Find the center and radius of each of the following circles. (a) y 2 = 6 (b) y 2 3y = 1 (c) 2 + y 2 4y = Solve each of the following equations by completing the square or eplain why no solution eists: (a) = 0 (b) = 0 (c) 5 = Eplain the significance of the discriminant of a quadratic epression A + B + C. Give eamples of each of the three types of discriminants and their relationship to the corresponding graph of the parabola. 20. Find two numbers, t, such that the points A = (3, 5), B = (t + 1, 2t), and C = (2, t) are collinear.

5 21. Find two points where the circle of radius 3 centered at the origin intersects the circle of radius 4 centered at P = (5, 0). 22. Find the equation of the circle in the y-plane centered at C = (3, -7) that has circumference Find the equation of the circle in the y-plane centered at (-1, -2) that has area Find the point on the straight line y = 4 1 that is closest to the point P = (1, -1). 25. Find a constant c such that the graph of y = c has its verte on the -ais. 26. Find a point of the form (t, 4t) that is closest to the point (1, 4). 27. Find two numbers whose sum equals 10 and whose product equals How many real roots does each of the following equations possess? (Hint: These questions require very little calculation.) (a) ( 1) ( + 5) ( ) 4 = 0 (b) = 0 (c) = 0 (d) = 0 (e) ( 4 + 2)( + 1)( 2 9) = 0

6 29. Find the points of intersection (if any) of the circle ( 1) 2 + (y 2) 2 = 41 and the line 4y 5 = Show that there do not eist two real numbers whose sum is 7 and whose product is Find a constant c such that the graph of the parabola y = c has its verte on the line y =. 32. Find the intersection of the line containing the points A = (3, 4) and B = (1, 8) and the circle with radius 3 centered at C = (2, 9). 33. If an object is thrown straight up into the air from height h0 feet at time t= 0 with initial velocity v0 ft/sec, then at time t seconds the height of the object from the ground is given by y(t) = -16.1t 2 + v0 t + h0 feet. Suppose that a ball is tossed straight up into the air from a height 8 feet with initial velocity 30 ft/sec. (a) How long will it take until the ball reaches the ground? (b) How long will it take for the ball to reach its maimum height? (c) Determine the maimum height reached by the ball? 34. Suppose that a ball is tossed straight up into the air from height 4 feet. What should the initial velocity be for the ball to reach its maimum height after 2 seconds?

7 35. Calculate the perimeter of a triangle with vertices A = (1, 0), B = (2, -5) and C = (7, 11). 36. Given A = (1, 1), B = (5, 13), C = (7, 23), D = (-8, -26), find three of the four points that are collinear. 37. Find the length of the graph given by ( 2) 2 + (y + 5) 2 = Suppose that a rope is just long enough to cover the equator of the Earth. About how much longer would the rope need to be so that it could be suspended 19 meters above the entire equator? 39. Find the slope of each of the following lines, or eplain why the slope does not eist. (a) 3 4y = 12 (b) 3y + 4 = 7 3 (c) 2( 1) + 5(y 2) = 13 (d) a + b(y 1) = c (e) = 3 (f) y = Given that 0 degrees Celsius corresponds to 32 degrees Fahrenheit and that 100 degrees Celsius corresponds to 212 degrees Fahrenheit, find a linear relation between C (degrees Celsius) and F (degrees Fahrenheit).

8 41. Find the equation of a straight line that passes through the points P = (7, 18) and Q = (-4, -3). 42. Find the equation of a straight line that has slope 5 and -intercept Find the equation of a straight line that is parallel to the line 5y = 131 and has y- intercept equal to Find the midpoint of the line segment joining P = (0, 4) and Q = (8, 19). 45. Find the equation of a line that is perpendicular to the line 8y = 123 and passes through the point R = (14, 11). 46. Find the equation of a straight line whose slope is 3 and whose y-intercept is the same as that of the line whose equation is y = Find the point of intersection of the lines y = 3 2 and y = Find numbers and y such that (-2, 8) is the midpoint of the line segment connecting (5, 3) and (, y). 49. Verify that the triangle with vertices A = (1, 1), B = (-1, 4), and C = (5, 8) is a right triangle. 50. Find functions f and g, each simpler than the given function h, such that h f g. ( a) h( ) ( b) h( ) ( 4) 13 ( c) ( d ) 5 h( ) 99 4 h( ) 3

9 51. Consider the circle ( + 11) 2 + (y + 999) 2 = 4. If the radius is increased by 5 cm, how much does the area change? Give an eact answer, using appropriate units. 52. Find the circumference of the circle ( 88) 2 + (y ) 2 = 225. Assume that units are measured in meters. Give an eact answer. 53. Find the distance between the points A = (2, -7) and B = (-1, 2). Assume that units are measured in feet. Give an eact answer Let f ( ) 1 3. (a) Evaluate f -1 (2). (b) Evaluate (f(2)) -1. (c) Evaluate f(2-1 ). 55. Let G( ) 1. 7 Evaluate each of the following: (a) G -1 (8) (b) G(8-1 ) (c) (G(8)) If y = g() has domain [0, 1] and range of [2, 3], find the domain and range of y = 1 +3g(4 5)?

10 57. Let y = F() be a function with domain [-6, 2] and range [-4, 4]. The graph of F is displayed below. 58. Find the inverse of the function g( ) 5 1 3

11 59. Let g() = 3( 2 + 1)( 5). Compute (without simplifying) (a) g(0) (b) g(2) (c) g( + 4) (d) g( 1) (e) g(2/) 60. Let y = f() = 2. Eplain what happens to the graph of f if we perform the following transformations. Sketch the graphs of the transformed functions. (a) y = 3f() 1 (b) y = 8 2f() (c) y = f( 5) (d) y = 3f( 1) + 4 (e) y = -2f(5) (f) y = 4f(/2) 8 (g) y = f(2 4) Has the domain or range changed for any of the above? Eplain! 61. Let f. Compute and simplify. ( ) 1 f ( ) f ( h) h

12 62. Which of the following functions, if any, is one-to-one? (Identify any and all that are one-to-one.) Justify your answers. (a) y = ( 5) ( 3) (b) y = (c) y = ( + 1) 2 (d) y = 2( + 3) (e) y = /5 63. Let f ( ) 1. Find f -1 (3) Let y = f() = 2. Eplain what happens to the graph of f if we perform the following transformations. Sketch the graphs of the transformed functions. (a) y = 3f() = 1 (b) y = 8 2f() (c) y = f( 5) (d) y = 3f( 1) + 4 (e) y = -2f(5) (f) y = 4f(/2) 8 (g) y = f(2 4) 65. Simplify fully: 2 a a 2 a ( a) Suppose that the graph of f is a parabola with verte at (7, 11). Let g() = 1 3. Find the verte of the parabola f g.

13 67. Suppose that the graph of f is a parabola with verte at (1, 3). Let g() = Find the verte of the parabola y g f. 68. Find the domain of each of the following functions: 69. Find the range of each of the following functions: (a) y = (b) y = ( 2) (c) y = 9 ( 4) 2 (d) y = Which pizza is least epensive. Justify your answer! (a) radius of 8 inches, $ 5.99 (b) radius of 10 inches, $7.97 (c) diameter of 23 inches, $ Find the area of the trapezoid with coordinates A = (13, 11), B = (13, 19), C = (8, 4), D = (8, 35). Assume that units are measured in cm. Sketch! (Hint: split the trapezoid into 2 triangles.)

14 72. Find the area of the annulus bounded by the two curves below. Sketch. ( + 2) 2 + (y 99) 2 = 1 ( + 2) 2 + (y 99) 2 = Find the domain of the function 5 3 ( 99) g ( ) ( 1) ( 88) ( 1) 74. Find the domain and range of the function G ( ) If y = f() has domain [-1, 1] and range of [2, 3], find the domain and range of y = 1 +3 f(4 5). 76. State the quadratic formula. Using the quadr4atic formula solve for in each of the following: (a) = 0 (b) = 0 (c) = Simplify the following comple fraction: 4a 3b 3 8a b 2

15 78. Suppose that the radius of a circle is increased by 50%. By what percentage does (a) the area change? (b) the circumference change? The universe stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth. - Galileo Galilei