MTH 121 Fall 2007 Essex County College Division of Mathematics and Physics Worksheet #1 1

Transcription

1 MTH Fall 007 Essex County College Division of Mathematics and Physics Worksheet # Preamble It is extremely important that you complete the following two items as soon as possible. Please send an message to rbannon@mac.com and I will add your address to our class list the message subject should read MTH-, and the body should have your full name as it appears on your student records at ECC. You will receive a reply that will include a link to the course web page. The course web page will be used throughout the semester and will be updated weekly. If you re reading this there s a good chance that you ve completed this exercise successfully. Visit and read this week s blog entry, and leave a comment. This blog will be updated weekly and all course files will be posted on this blog. Again, if you re reading this there s a good chance that you ve completed this exercise successfully. Diagnostic Review Your eventual success in future mathematics courses, including this course, depends to a large extent on knowledge. The following example questions are intended to diagnose weaknesses that you might have in the following areas: algebra, analytic geometry, and functions. This is not an all inclusive diagnostic, but it should nonetheless give you an idea of what you should be able to do at this point in time. For each of the following questions you need to show all work clearly and in order, and then box your final answers. Justify your answers algebraically whenever possible.. Find the complement of π 6. π π 6 π This document was prepared by Ron Bannon (rbannon@mac.com) using L A TEX ε. The URL is:

2 . Find the first five terms of the recursively defined sequence. a 5, a k+ a k. a 5, a a 5 4, a a 4 7, a 4 a 7 79, a 5 a Using your calculator to make relevant graphs, determine which of the following is the equation of the line tangent to the curve y x sin (πx) when x /. y x or y x or y 0 or y or y x y x 4. Use the properties of inverse functions to find the exact value of ( arccos cos 7π ). ( arccos cos 7π ) arccos (0) π 5. Solve the triangle, given a.5 inches b 7.8 inches, and c 9.0 inches. You should, of course, draw an approximating triangle first. A 08.8, B.45 and C Evaluate the given infinite geometric sum. 0.9 n n Here s one possible answer. 0.9 n n

3 7. Let What are the domain and range of f? Domain: R; Range: (0, ]. f (x) x Which of the following functions are increasing for all x? Which are decreasing for all x? f (x) x or g (x) sin x + or h (x) ln x or i (x) e x f (x) is always increasing. h (x) is increasing on its domain. i (x) is always decreasing. 9. Let f (x) x and f (x + h) f (x) g (x). h What is g () when h 0.? What is g (x) when h 0.?.6; x + 0.x Find the domain of the function. h (x) 4 x + x The domain of 4 x is (, 4]; and the domain of x is (, ] [, ). The domain if h (x) is the intersection of these two domains. Hence: (, ] [, 4]. If f (x) x + x and g (x) x, find each of the following. (a) f g (f g) (x) f (g (x)) f (x ) (x ) + (x ) 4x 8x +

4 (b) g f (g f) (x) g (f (x)) g ( x + x ) ( x + x ) x + 4x 5 (c) g g g (g g g) (x) g (g (g (x))) g (g (x )) g ( (x ) ) g (4x 9) (4x 9) 8x. Solve each inequality, and use interval notation to express the solution. (a) x x x x x x 0 x x 0 (x ) (x + ) x 0 Using a number line, you ll get [, 0) [, ). (b) ( x) 0 x ( x) 0 x x + x 0 x x 9 0 (x ) (x + ) 0 Using a number line, you ll get [, ]. 4

5 . Solve for x. (a) log 5 x log 5 x x 5 x 5 (b) log x 64 log x 64 x 64 x 4 (c) log 9 x log 9 x x 9 x 4. Graph h (x) 4 x + x You may have a tough time with this one, and an electronic aid may in fact be necessary to graph many functions Figure : Partial graph of h (x). 5. Find the equation of the line that passes through the point (, 5) and is perpendicular to the line x 4y 5. y + 5 (x ) 5

6 6. Find the equation of the circle that has center (, 4) and passes through the point (, ). Using (x h) + (y k) r to find the radius, where (h, k) is the center, r is the radius, and (x, y) is any point on the circle. ( + ) + ( 4) r r 5 r (x + ) + (y 4) 5 x + y + x 8y Find the radius and center of the circle with equation x + y 6x + 0y x + y 6x + 0y x 6x + y + 0y 9 x 6x y + 0y (x ) + (y + 5) 5 So the center is (, 5) and the radius is Verify the identity. sin θ sec θ sec θ Select the right side. sec θ sec θ sec θ sec θ sec θ cos θ sin θ Q.E.D. 9. Solve for x. x x + x x x x x + x x x (x + ) (x ) x x x + x 0 x x 6

7 So, it appears that x is a solution. Checking in the original equation, we have, which is clearly true., 0. Solve for x. x 4 x 4 x 0 x 4 x 4 x 0 x 4 x 4 x The solution (please check) is x 5. x (4 x) x x 5x x 5. If f (x) x, evaluate the difference quotient f ( + h) f (), h 0 h Do I need to say it? Yes, the following is true if and only if h 0. f ( + h) f () h ( + h) () h 8 + h + 6h + h 8 h + 6h + h. Find the domain of f (x) x 49 x x + 9 is always positive so we don t have to worry about the square root. However, to find the domain we need to solve x x x x x 6 x ±4 7

8 So, the domain is R, x ±4.. Solve the inequality. 4 x < 7 4 x < 7 4 x < 0 We know that for a > 0, that x < a is equivalent to a < x < a, so 4 x < 0 0 < 4 x < 0 4 < x < 6 4 > x > 6 6 < x < 4 Here the proper interval is: ( 6, 4). 4. Solve by using an augmented matrix and elementary row operations. x + y z 7 x y + z x + 4y + z Augmented matrix form of the system: 7 4 Elementary row operations, in order given: R + R R R + R R Produces: The second row gives x ; using x in row three, gives y ; finally, using x and y in row one, gives z. 8

9 5. Use long division to find the quotient and remainder when x 4 4x + x + 5 is divided by x. Doing the division gives a remainder of 9 and a quotient of x + x Is x a root of the polynomial function f (x) x 5x 4x +? Yes, because f ( ) Factor f (x) x 5x 4x +. Using the fact that x is a root, we can divide f (x) by x +, getting x 5x 4x + x + so the complete factorization of f (x) is x 7x + (x ) (x ), (x + ) (x ) (x ). 8. Solve the inequality. x x + 0 x x + 0 x + (x ) x 0 x + x 0 x x 0 Using simple sign analysis, the proper interval is: (, ). 9. Solve for x. log (x + ) + log (x ) log (x + ) + log (x ) log (x + ) (x ) ( log 4x ) 4x 4x 4 0 x 0 x ± Previous problem may be helpful. 9

10 The solution 4 is x and you should check this. 0. Given find and simplify the difference quotient f (x) x x, f (x + h) f (x), h 0. h Do I need to say it? Yes, the following is true if and only if h 0. f (x + h) f (x) h x + h x h + x x h (x + h ) ( x) (x ) ( x h) h ( x h) ( x) 4x + 4h 6x 6xh + x 4x + + 6x x + 6xh h h ( x h) ( x) h h ( x h) ( x) ( x h) ( x). Find the domain of f (x) x + x 8. Finding the domain requires solving x 8 > 0 for x. x 8 > 0 x 9 > 0 (x ) (x + ) > 0 Simple sign analysis gives (, ) (, ).. Given f (x) (x + ), x, find f (x). Graphing may be helpful, but is not required. 4 If you re limited to using real numbers only, whereas complex numbers can look mighty strange, such as the famous or infamous identity e iπ

11 The domain of f (x) is [, ) and the range is [0, ), so the domain of f (x) is [0, ) and the range is [, ). However, since y we have f (x) (x + ) y (x + ) x (y + ) ± x y + ± x y f (x) x, x 0.. Given f (x) x + 6 and g (x) x 5, answer each of the following questions. (a) Find (f g) (x) (f g) (x) f (g (x)) f ( x 5 ) x x + (b) The domain of (f g) (x) R. (c) Find (g f) (x) (g f) (x) g (f (x)) g ( x + 6 ) ( x + 6 ) 5 x x + (d) The domain of (g f) (x) x Given f (x) x 4 + 7x 4x 7x 8, and f () f ( ) 0 answer the following questions.

12 (a) Use the given roots, and long division, to completely factor f (x). Since we are given two roots, we know two factors of f (x) are (x ) and (x + ). Dividing f (x) by the product of these two factors gives So the complete factorization of f (x) is x + 5x + (x + ) (x + ). (x + ) (x + ) (x ) (x + ) (b) Graph f (x) using the roots, y-intercept and sign-analyses. Your graph does not need to have such precise detail as mine, but it should still reflect the key pre-calculus analysis Figure : Graph of f (x). 5. Given answer the following questions. f (x) x + x x +, (a) x-intercept(s) in point form. Set f (x) 0 and solve for x. Clearly, (, 0) and (0, 0). 0 x + x x + 0 x (x + ) x +

13 (b) y-intercept in point form. Set x 0 and evaluate. Clearly, (0, 0). (c) All linear asymptotes in equation form. Since the degree of the numerator is one more than the denominator, we have the possibility of getting a slant asymptote. The long division gives so the slant asymptote is y x +. f (x) x + x x + x + x + x +, (d) Graph f (x) using the information above and sign-analyses. Your graph does not need to have such precise detail as mine, but it should still reflect the key pre-calculus analysis Figure : Graph of f (x). 6. Given find and simplify the difference quotient f (x) x x +, f (x + h) f (x), h 0. h Do I need to say it? Yes, the following is true if and only if h 0. f (x + h) f (x) (x + h) (x + h) + ( x x + ) h h x + xh + h x h + x + x h xh + h h h x + h

14 7. Solve the inequality. 4 x 5 > 4 x 5 > 4 x > We know that x > a is equivalent to x > a or x < a, so 4 x > 4 x > or 4 x < x < / or 7/ < x Here the proper intervals are: (, /) (7/, ). 8. Write the system of linear equations represented by the augmented matrix. Then use back-substitution to find the solution. (Use the variables x, y, and z.) x y + z 4 y z z The last line gives z, then using the value for z in line two, we get y 0, finally using these two values in line one we get x Condense the expression to the logarithm of a single quantity. [ ( ln (x + ) + ln x ln x )] [ ( ln (x + ) + ln x ln x )] [ln (x + ) + ln x ln ( x )] [ ln ] x (x + ) x ln x (x + ) x 4

15 40. Solve for x. ( ) x 8 6 ( ) x 8 ( ) x ( ) x 6 ( ( ) 4 ) 4 So, x Solve the inequality. x + 6 x + < 0 x + 6 x + < 0 x + 6 (x + ) x + x + 4 x x + < 0 < 0 Using simple sign analysis, the proper intervals are: (, ) (4, ). 4. Solve algebraically (exact answer) and then approximate to three decimal places. + ln x 7 + ln x 7 ln x 9 ln x 9 x e 9/ x e9/ 4,

16 4. Given f (x) x x + (x ) (x ) x + 5x + (x + ) (x + ), answer the following questions: (a) x-intercepts in point form. (, 0) ; (, 0) (b) y-intercept in point form. (0, /) (c) Equation of the horizontal asymptote. y / (d) Equation of the vertical asymptotes. x ; x / (e) Graph f (x) using the above information and sign analysis Figure 4: Graph of f (x). 44. Find the rational number representation of the given repeating decimal..87 Let x.87. 0x x x 0x x x

17 45. Use summation notation to write the given sum Here s one possible answer n n 46. Use your calculator to approximate each of the following to three decimal places. (a) arcsin ( 0.987).409 or (b) arctan ( ).560 or (c) arccos ( 0.009).580 or Find the exact value of the following. ( ) (a) arccos π 4 or 45 [ ( )] ( ) 5π (b) arccos sin arccos 5π 6 or Simplify the factorial expression. (n + )! (n)! (n + )! (n)! (n + ) (n + ) (n)! (n)! (n + ) (n + ) 4n + 6n A ramp feet in length rises to a loading platform that is 5 feet off the ground. Assuming that the ground is level, what is the angle (to the nearest whole degree) between the ramp and the ground? You should draw a diagram first. sin θ 5 ( ) 5, arcsin 4 7

18 50. Write an algebraic expression that is equivalent to sec [arcsin (x )]. You should draw a triangle first and use the Pythagorean Theorem to determine the missing side. sec [arcsin (x )] x x 5. Find the numerical coefficient of the term whose variable part is x 6 y in the expansion of (x y) 9. ( 9 6 So, the numerical coefficient is 67. ) (x) 6 ( y) 84 ( 8) x 6 y 67x 6 y 5. Write an expression for the n th term., 5, 5 6, 5 4, 65 0,... a n 5n n! 5. Convert each of the following angle measures to radian measure. (a) 60 ( π ) π (b) 90 ( π ) π (c) 50 ( π ) π 8 8

19 54. Solve for x. (a) sin x + sin x in the interval [0, π). sin x + sin x sin x sin x. The reference is 45 and the solutions occur in the third and fourth quadrant, so x π 4 and x π 4 (b) sin x sin x 0 in the interval [0, π). sin x sin x 0 ( sin x + ) (sin x ) 0. Which gives two equations to solve. sin x and sin x. The reference for the first equation is 0 and the solutions occur in the third and fourth quadrant, so x π 6 and x π 6, and the solution to the second equation is x π 55. Use the Binomial Theorem to expand and simplify ( x y). First the expansion ( ) ( x ) + ( ) ( ) x ( ( y) + ) ( ) ( x ( y) + 0 ) ( y), then the simplification 7x 54 x 4 y + 6 x y 8y. 9

20 56. Evaluate (exact values) all six trigonometric functions for x 0. (a) sin x sin x (b) cos x cos x (c) tan x tan x (d) cot x cot x (e) sec x sec x (f) csc x csc x 57. Find the sum. 70 n48 (n ) Expand first. You should also notice that there are terms. 70 n48 (n ) ( 48 ) + ( 49 ) + ( 50 ) + + ( 70 ) ( ) ( ) ( ) Use mathematical induction to prove the formula for every positive integer n. ( n ) n Verify for n. P : () 0

21 Assume P k and show that P k P k+. ( P k : k ) k Add k to both sides. P k : ( k ) k ( k ) + k k + k ( k + k) k ( k + k) k+ This last line is exactly what we wanted, P k+. Q.E.D. 59. When an airplane leaves the runway, its angle of climb is 9 and its speed is 00 feet per second. Find the plane s altitude after 0 seconds. The plane will travel , 000 feet (this is the hypotenuse of a right triangle), so the plane s altitude (this is the side opposite the 9 angle) is 9000 sin 9 90 feet. 60. Use your calculator to evaluate the trigonometric function. Round your answers to five decimal places. (a) sin.67 sin (b) cos 0.45 cos ( (c) tan 8π ) 9 ( tan 8π ) (d) cot.79 cot.79 tan (e) csc (.689) csc (.689) sin(.689).8677 (f) sec 45 sec 45 cos

22 (g) arcsin arcsin or arcsin (h) arccos ( 0.67) arccos ( 0.67) or arccos ( 0.67).507 (i) arctan. arctan or arctan Find the infinite geometric sum a 8 and r S Let a n n n (a) Write the first five terms of the sequence, starting with a. a, a 8, a 4, a , a (b) Is this sequence arithmetic, geometric, or neither. Neither. 6. Find θ in degrees (0 < θ < 90 ) and radians (0 < θ < π ), if cot θ. Draw a triangle first, clearly θ 60 π. 64. Find the sum. 00 n5 6n

23 Expand first. 00 n5 6n ( ) 6 50 (00 + 5) Sketch one period of the graph of the function. f (x) sin (πx + π) + I m mainly looking for five points. (, ), (, ), (, ), (, 0 ), (0, ). They should be plotted and then connected using a sine wave Figure 5: Graph of f (x) sin (πx + π) Use csc θ and sec θ 4 (a) The quadrant that θ is in. First (I) quadrant. (b) sin θ to find the exact value of each of the following. sin θ

24 (c) tan θ tan θ 4 (d) cos θ cos θ 4 (e) sec (90 θ) sec (90 θ) 67. Perform the indicated addition and use the fundamental identities to simplify. sin x + + sin x sin x + + sin x sin x + sin x + sin x + + sin x sin x sin x + sin x sin x + sin x sin x + sin x + sin x sin x sin x cos x sec x 68. Write an expression for the n th term.,, 8 9,, 5,... By inspection. a n n n 69. If sin α 5 and cos β 5, with both α and β are in the fourth quadrant. Find the exact values of the following. 4

25 Preamble: Since α is in the fourth quadrant α will be in the second quadrant, which will determine the signs of the half-angle formulas. sin α 5 sin β 4 5 and cos α and cos β 5 (a) sin (α β) sin (α β) sin α cos β sin β cos α ( 5 ) ( ) ( 4 ) ( ) (b) cos (α β) cos (α β) cos α cos β + sin α sin β ( ) ( ) ( + 5 ) ( 4 ) (c) tan (α β) tan (α β) sin (α β) cos (α β) ( α ) (d) sin 5

26 α is in the second quadrant, so the sine will be positive. ( α ) sin cos α / 6 ( α ) (e) cos α is in the second quadrant, so the cosine will be negative. ( α ) cos + cos α + / 5 6 ( α ) (f) tan Just take the ratio of sine over cosine. ( α ) ( α ) sin tan ( α ) cos Find all solutions. sin x + cos x 0 You ll need to write it in terms of cosines first. sin x + cos x 0 ( cos x ) + cos x 0 cos x + cos x 0 Now multiply both sides of this equation by and factor. cos x cos x + 0 ( cos x ) (cos x ) 0 6

27 Set each factor equal to zero and solve. cos x x π + πk, k Z 5π + πk, k Z cos x x πk, k Z. 7. If z i, find the trigonometric form of z and z 9, also find z 9 in standard form. Here θ 5 or θ 5π 4, and r. z cos 5 + ( sin 5 ) i z 9 6 ( cos sin 05 ) i z 9 6 6i 7. Verify the identity. + sin x sin x + sin x cos x Select the left side. + sin x sin x + sin x sin x + sin x + sin x ( + sin x) sin x ( + sin x) cos x + sin x cos x Q.E.D Rewrite the expression so that it is not in fractional form tan x csc x +. q 5 Here it might be nice to mention that +sin x 0 so ( + sin x) +sin x. However, since cos x, the cos x cos x. 7

28 tan x csc x + tan x sin x sin x + sin x tan x sin x + sin x sin x sin x tan x sin x ( sin x) sin x tan x sin x ( sin x) cos x tan x sin x ( sin x) sec x sin x ( sin x) sec 4 x 74. Find the sum. π + π 4 + π π + π 4 + π 8π + π + π π 75. Find all solutions. cos x + sin x 0 You ll need to write it in terms of cosines first. cos x + sin x 0 ( sin x ) + sin x 0 sin x + sin x 0 Now multiply both sides of this equation by and factor. sin x sin x + 0 ( sin x ) (sin x ) 0 Set each factor equal to zero and solve. sin x x π 6 + πk, k Z 5π 6 + πk, k Z sin x x π + πk, k Z. 8

29 76. Solve the triangle, given a. inches b 8.4 inches, and B You should, of course, draw a triangle first. This one actually gives two distinct answers. I ll give you credit for either one. A 4.70, C and c 5.80 inches. or A 7.0, C.86 and c.69 inches. 77. Determine two coterminal angles in radian measure (one positive, one negative) for θ π. π + π 7π and π π 5π 78. Find the supplement of Find the length of arc on a circle of radius 4 inches and a central angle of 60. So, the length is 4π inches. S 4 π 80. If the sec θ 7 and 70 < θ < 60, find the following. From the information given we can conclude that x, r 7, and that y < 0. To find the value of y, solve So, y y y ± 48 ±4. (a) sin θ 4 7 (b) cos θ 7 (c) tan θ 4 (d) cot θ 4 9

30 (e) csc θ Given that P k k [5k ], find P k+. Just replace k with (k + ). P k+ (k + ) [5 (k + ) ]. 0