Study Guide for Benchmark #1 Window of Opportunity: March 4-11

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1 Study Guide for Benchmark #1 Window of Opportunity: March -11 Benchmark testing is the department s way of assuring that students have achieved minimum levels of computational skill. While partial credit is given on the other tests in this course, on the benchmark tests you are expected to demonstrate that you can do basic computations carefully and accurately. This benchmark covers skills that are considered prerequisite for this course, and computations that have been covered in the first two chapters of this course. To pass a benchmark test, you must get 17 (or more) of 0 points. If you pass on the first attempt, your score will be recorded as 100%. (So strive to pass on the first attempt...) If you do not pass the benchmark test on your first attempt, your score will initially be recorded as 0%. You can replace this 0% with a better grade by retaking the benchmark once or twice within the windowof-opportunity. You are encouraged to meet with either me or Ed to go over problems that are giving you difficulty. If you pass the Benchmark on a re-take, the score of 0% that was originally recorded will be changed to the average of your scores on your two or three attempts. If you re-take a benchmark but do not pass it within three attempts or by the due date, your score will be changed to the average of your scores together with a 0%. Each Benchmark grade will contribute 5% to your overall grade for the course. This test will have ten basic computational problems. You may not u use a calculator while you are working on the benchmark. If you pass the benchmark on your first attempt, I will record your grade as 100%. If you do not pass on your first attempt, you may meet with me or Ed to go over your mistakes and demonstrate that you have done some additional study. If you pass on a re-take, your grade will be recorded as the average of your scores on each attempt. If you do not pass the benchmark in three attempts or by March 11, your score will be recorded as the average of your scores together with 0. So the idea is to do as well as you can on your first attempt, and to make sure that you do pass by March 11. The first attempt will take place in class on March th. It will be the last 30 minutes of class. You will receive your exams back the following Tuesday. If you do not pass the first time, it will be on you to discuss with me when you will re-take the exam outside of class. If you do not then you will receive a score of 0% for the assessment. if you have an unexcused absence on the day of the in-class attempt on a benchmark assessment, you will not be allowed any additional attempts. This benchmark will have two problems from basic algebra, two from coordinate geometry, two from trigonometry, and four problems from symbolic computation of the derivative.

2 Basic Algebra ( problems) You should be able to demonstrate the following skills: Simplify arithmetic or algebraic expressions using correct order of operations and rules for working with exponents. Evaluate a given function, f(x), at particular values, such as x = 5, x = t + h, or x = g(t). Use rules of exponents to evaluate or simplify an algebraic equation. Remember: A negative exponent denotes a reciprocal, and a fractional exponent denotes a root. So what is 5 3? Calculate the sum or product of polynomials. Solve a linear equation for a specified value of x or y. Given an equation of the form y = f(x) = Ax n or y = f(x) = Ae bx, you should be able to take logs of both sides of the equation, and simplify the resulting expression. Coordinate Geometry ( problems) You should be able to demonstrate the following skills: Plot points in the coordinate plane using rectangular coordinates. Find the slope of a straight line given its graph, its algebraic expression, or a set of point which lie on the line. Use the slopes to determine whether two lines are parallel, perpendicular, or neither. Remember: Parallel lines have the same slope. If two lines are perpendicular, the product of their slopes will be 1. Find the equation of a line given two points, or a point and the slope of the line. Find the length and the midpoint of a line segment. Sketch the graph of a line given its equation, its slope and one point, or two points. You should know and be able to use formulas for area of a square, rectangle, triangle, and circle; volume of a rectangular box or right circular cylinder, perimeter of a polygon, and circumference of a circle. Trigonometry ( problems) You should be able to demonstrate the following skills: Use the Pythagorean Theorem to find the length of the third side of a right triangle, given the lengths of any two sides. Use the two standard triangles ( and ) to find the values of the six trigonometric relationships for the angles in the two standard triangles.

3 You should know the six trigonometric relationships: sine, cosine, tangent, cotangent, secant, and cosecant. Given a right triangle with lengths of any two of the sides indicated, you should be able to find the sine, cosine, or tangent of each of the angles. Given the sine, cosine, or tangent of an angle, you should be able to sketch a triangle and calculate the other trigonometric values of that angle. Convert between degrees and radians. In particular, you should know the radian measures for angles of 0, 30, 5, 60, 90, 180, 70, and 360. Use the unit circle to find the sine, cosine, or tangent of angles in the second, third, and fourth quadrants that are related to the angles in the two standard triangles. In particular, you should be able to find the sine, cosine or tangent of 0, 30, 5, 60, 90, 10, 135, 150, 180, 70, and 360. Symbolic computation of the Derivative ( problems - see Sections.3 and. of our text) You should be able to take the derivative of the following kinds of functions, where a, b, and c denote constants: Constant functions:f(x) = π, f(x) = 5,(or f(x) = any other constant) Linear functions: f(x) = ax + b (any linear function) Polynomial functions: f(x) = ax + bx + c (or any polynomial) Power functions: f(x) = ax n (any power function) The power could be negative, such as f(x) = a x = ax 1 Constant multiple rule: f(x) = c g(x) For example, f(x) = c (ax + bx + π) f(x) = b x (an exponential function) f(x) = e x (a special exponential function) Sum rule: functions formed by taking the sum of simpler functions For example: f(x) = x + x + ln()

4 Sample Questions for this Benchmark Be sure to review the sample questions from the Study Guide for the Quiz you took during the first week of class. In addition to those questions, the following questions are questions similar in nature to what you can expect. 1. Write the expression x 3 in the form x n.. Write an expression for the cube root of x 5 in the form x n. 3. Write the expression 1 x 3 in the form x n.. Write the expression x x 7 in the form x n. What do fractional exponents mean? What do negative exponents mean? 5. If g(x) = 3x, find an expression for g( 7). 6. If g(x) = 3x, find an expression for g(x + 5). 7. If g(x) = 3x, find an expression for g(π). 8. If g(x) = 3x, find an expression for g (cos (5)). 9. If g(x) = 3x, find an expression for g ( cos ( π )). Although you are not expected to evaluate cos(5) by hand, you should be able to evaluate the cosine or sine of 0, π 6, π, π 3, π, π, π without using a calculator. 10. Find the slope of the line which passes through the point (3, 5) and goes through the origin. 11. Find the equation of the line which passes through the point (3, 5) and goes through the origin. 1. Find the length of the line segment from the point (3, 5) to the origin. 13. Find the midpoint of the line segment from the point (3, 5) to the origin. 1. Find the slope of the line which passes through the points (3, 5) and ( 3, 8). 15. Find the equation of the line which passes through the points (3, 5) and ( 3, 8). 16. Find the length of the line segment from (3, 5) to ( 3, 8). 17. Find the midpoint of the line segment from (3, 5) to ( 3, 8). 18. Find the slope of the line which passes through the points (3, 5) and (3, ). 19. Find the equation of the line which passes through the points (3, 5) and (3, ). 0. Find the length of the line segment from (3, 5) to (3, ). 1. Find the midpoint of the line segment from (3, 5) to (3, ).. Does the line from (, 5) to (, 8) pass through the origin? How do you know?

5 3. Does the line from (3, 5) to ( 6, 10) pass through the origin? How do you know?. Find the equation of a line which passes through the point (3, 5), and is perpendicular to the line y = x. 5. Is the line y = 5x + [parallel to, perpendicular to, neither parallel nor perpendicular to] the line x + 5y = 0? Explain. 6. Is the line y = 5x + [parallel to, perpendicular to, neither parallel nor perpendicular to] the line 5y 10x = 0? How do you know? 7. Find an angle whose tangent is If sin(t) = a, find an expression for cos(t) in terms of a. Hint: Sketch a triangle which has an angle whose sine is a. 9. If sin(t) = a, find an expression for sec(t). 30. Evaluate (find the exact value of ) sin ( π ). (Remember: While taking the benchmark, you may not use a calculator. You should be able to answer questions like this using the unit circle and the two standard triangles.) 31. Evaluate cos ( π 3 ). 3. Evaluate tan ( π ). 33. Evaluate cos(π). 3. If an equilateral triangle has sides of length 3 units, what are the lengths of its altitude and its base? 35. If an equilateral triangle has sides of length 3 units, what is its area? 36. How many feet of fencing are needed to enclose a field which is 76 yards wide and 50 yards long? 37. A rectangular box has a volume of 600 cubic inches, and is 8 inches high. If the length is three times the width, what are the dimensions [length, width, height] of the box? 38. A rectangular box has a volume of 600 cubic inches, and is 8 inches high. If the length is three times the width, what is the surface area of the box? 39. What is the volume of a cylinder if its diameter and its height are both 3 inches? 0. What is the total surface area of a cylinder (including its top, bottom, and curved side) if its diameter and its height are both 3 inches? 1. Take the natural log of both sides of the equation y = 5e 3x, and simplify the resulting expression.. Use logs to transform the equation y = f(x) = 5x 3 to a linear equation. What is the slope of the resulting equation?

6 3. Use logs to transform the equation y = f(x) = 7x n to a linear equation. Is the resulting equation a semi-log equation or a log-log equation?. Consider the equation y = f(x) = 3e x. Transform this equation by taking the natural log on both sides of the equation. What is the slope of the transformed equation? 5. Find the derivative of the function y = f(x) = 3x Find the derivative of the function y = f(x) = 5x x Find the derivative of the function y = f(x) = 3x n 8. Find the derivative of the function y = f(x) = x 9. Find the derivative of the function y = f(x) = Find the derivative of the function y = f(x) = π 51. Find the derivative of the function y = f(x) = x 5. Find the derivative of the function y = f(x) = e x 53. Find the derivative of the function y = f(x) = 3 x + 5x 5. Find the derivative of the function y = f(x) = (5x x + 7) 55. Find the derivative of the function y = f(x) = 3x n + x 1 x Find the derivative of the function y = f(x) = x Find the derivative of the function y = f(x) = ln(5) 58. Find the derivative of the function y = f(x) = π x 59. Find the derivative of the function y = f(x) = x 3 x + 5x 60. Find the derivative of the function y = f(x) = 15e x 61. Find the derivative of the function y = f(x) = 3x x 15 x 6. Find the derivative of the function y = f(x) = 5x x Find the derivative of the function y = f(x) = 3x Find the derivative of the function y = f(x) = 65. Find the derivative of the function y = f(x) = 15(π x ) 66. Find the derivative of the function y = f(x) = πx 3 + x 67. Find the derivative of the function y = f(x) = 3 x + x Find the derivative of the function y = f(x) = e x + 7x 1 x

7 Answer Key for Benchmark Study Questions 1. x 3. x x 3. x 3 5. g( 7) = g(x + 5) = 3(x + 5) = 3x + 30x g(π) = 3π 8. g(cos(5)) = 3cos (5) 9. g ( cos ( π )) = 3 0 = m = y = 5 3 x = ( 3, 5 ) 1. m = = y = 1 x (3 ( 3) + (5 8) = 5 = 3 5 ( ) ( 3), 5+8 = ( 0, 13 ) 18. The slope is undefined because the line is vertical. 19. x = 3 0. (3 3) + (5 ( )) = 9 = 7 ( ) , 5+( ) = ( 3, 3 ). It does not because the slope of the line containing these two points is 3, so the function is decreasing and passes through points in the first quadrant. In order to pass through the origin, it would have to have a positive slope in order to pass through points in the first quadrant. 3. The slope of the line containing these points is 5 3 and the slope of the line from the origin to the point (3, 5) is also 5 3 (see question 10). So, this line does indeed pass through the origin.

8 . The line perpendicular to the line y = x has a slope of 1. So, we have y = x + b, and using the point (3, 5), we see that the equation we seek is y = x The equation x + 5y = 0 can be put into slope-intercept form to become y = 1 5x +. Since this line has a slope of 1 5 and the first line has a slope of 5, these lines are perpendicular. 6. The equation 5y 10x = 0 can be rewritten as y = x +. We have different slopes that are not negative reciprocals of each other, so the lines are neither parallel or perpendicular. 7. We know from our reference angles that tan ( π ) = 1. Since we want an angle whose tangent is 1, however, we need an angle in quadrant or. Since we are not asked for the arctangent, which would restrict the domain, both answers are necessary here and so the angles we want are π and 3π. 8. If we draw a right triangle, we see that sin(t) = a gives that the length of the opposite side from angle t is of length a as compared to the hypotenuse of length 1. If we apply the Pythagorean Theorem, we get that the length of the adjacent side is 1 a. So, cos(t) = 1 a. 9. Since sec(t) = 1 1 cos(t), we have that sec(t) =. 1 a 30. sin ( ) π = cos ( ) π 3 = 1 3. tan ( π ) = cos (π) = 1 3. The base is 3 units long. To find the length of the altitude, we need to set up a right triangle with the legs being half the base and the height being the altitude, and the hypotenuse as one of the sides of the equilateral triangle. We have to solve the Pythagorean Theorem to find the length of the other leg. 3 = a + 3 a = 9 9 a = 7 a = 3 3 ( 35. Using the base and altitude lengths from the past problem, we have that A = 1 ) (3) 3 3 = 9 3 square units. 36. The perimeter of the field is field is (76) + (50) = 5 yards. But, the problem asked for how many feet of fencing is required, so we need 3(5) = 756 feet of fencing.

9 37. The height is given the be 8 inches. The volume is 600 cubic inches, so the area of the base is = 75 square inches. Since length is three times the width, if the width is x inches, then the length is 3x inches and by solving A = 3x(x) = 75, we see that x = 5 and therefore the length is 15 inches and the width is 5 inches. 38. Using the dimensions we found in the last problem, the surface area is given by S = A B +P B h where A B is the area of the base and the perimeter of the base is given by P B. This gives S = 75 + (0)8 = 70 square inches 39. V = πr h We are given that the diameter is 3 inches, so the radius has a length of 3 inches. So, we have V = π ( ) 3 3 = 7 cubic inches. 0. S = πr + πrh. Using the dimensions from the last question, we have S = π ( 3 ) + π ( ) 3 (3) = 9π + 9π square inches. 1. We are working with y = f(x) = 5e 3x.. Here we are working with y = f(x) = 5x 3. y = 5e 3x ln(y) = ln(5e 3x ) ln(y) = ln(5) + ln(e 3x ) ln(y) = ln(5) + 3x y = 5x 3 ln(y) = ln(5x 3 ) ln(y) = ln(5) + ln(x 3 ) ln(y) = ln(5) + 3ln(x) The slope of this transformed linear equation is Here, we work on y = f(x) = 7x n. This is a log-log equation.. Here we are working on y = f(x) = 3e x. y = 7x n ln(y) = ln(7x n ) ln(y) = ln(7) + nln(x) y = 3e x ln(y) = ln(3e x ) ln(y) = ln(3) + ln(e x ) The slope of this transformed equation is. ln(y) = ln(3) + x

10 5. f (x) = 3 6. f (x) = 10x 7. f (x) = 3nx n 1 8. f (x) = x 9. f (x) = f (x) = f (x) = ln() x 5. f (x) = e x 53. f (x) = 3 + 0x 3 x 5. f (x) = 0x f (x) = 3nx n 1 + ln() x + 3 x 56. f (x) = 1 x 57. f (x) = f (x) = π x 59. f (x) = ln() x + 6 x 3 + 0x f (x) = 15e x 61. f (x) = x + 15 x 6. f (x) = 10x 63. f (x) = 5x 1 6. f (x) = f (x) = 15 ln(π)π x 66. f (x) = 3πx f (x) = ln(3) 3 x + 3x 68. f (x) = e x x

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