MATH 1301, Practice problems

Transcription

1 MATH 1301, Practice problems 1. For each of the following, denote Ann s age by x, choose the equation(s) (may be more than one) that describes the statement, and find x. (a) In three years, Ann will be twice as old as she was two years ago. (A) x 3 = 2(x + 2) (B) 2(x + 3) = x 2 (C) (x + 3)/2 = x 2 (D) x + 3 = 2(x 2) (b) Five years ago, Ann was ten years younger than she will be when she is twice as old as she was a year ago. (A) x 5 = 2(x 1) 10 (B) x 5 = 2x + 10 (C) x 5 = 2(x 1) + 10 (D) x + 5 = 2(x + 1) (c) Eleven years from now, Ann will be twice as old as she was three years ago. (A) x 11 = 2(x 3) (B) x + 11 = 2(x 3) (C) x = 2(x 3) + 11 (D) (x + 11)/2 = x + 3 (E) x + 11 = 2x 3 2. A square with side x is 3 area units larger than a rectangle with sides x + 1 and 3. Which of the following equations is correct? (a) (x 3) 2 = 3(x + 1) (b) x 2 3 = 3(x + 1) (c) x 2 = 3(x + 1) 3 (d) x 2 32 = 3(x + 1) (e) x = 3(x + 1)

2 3. For each of the following, state the quadratic equation and find x. (a) A square with side x has an area that is 3 units smaller than the area of a rectangle with sides x and 4. (b) A square with side x has an area that is 3 units larger than the area of a rectangle with sides x and 2. (c) A square with side 2x has the same area as a square with side x For which values of a does the equation x 2 + 2x + a = 0 have (a) 2 solutions, (b) 1 solution, (c) no solutions? 5. Ann and Bob use two different business strategies. With Ann s strategy, the expected fortune grows as a linear function of time and with Bob s strategy, the expected fortune grows as a quadratic function of time. Ann starts with a fortune of 5 (million dollars) and Bob starts with a fortune of 2. Both Ann and Bob double their fortunes after one year, and Bob s fortune reaches 10 after 2 years. After how many years are their fortunes equal? 6. If the temperature is said to be in the twenties in degrees Celsius C, what range does this correspond to in degrees Fahrenheit, F? State the inequalities for F and solve. 7. Solve the following inequalities. (a) x 2 2x + 5 3x 1. (b) 2x 2 4x + 2 > 0 (c) 3x 2 + 5x + 10 < 0 8. A square with side x has an area that is at least 6 units larger than the area of a rectangle with sides x and 5. What are the possible values of x? 9. Solve the inequalities x (a) 1 + x x x (b) 1 + x < x 10. Let P 1 and P 2 be two points where P 1 is located at ( 8, 5) and the

3 midpoint M is located at ( 3, 3). Find the taxi distance between P 1 and P 2. 11(a) Let P 1 : (x 1, y 1 ) and P 2 : (x 2, y 2 ) be two points and let Q be the point that is located 1/4 of the way from P 1 to P 2. Argue by repeated use of the midpoint formula that the coordinates of Q are ( 3x1 + x 2 4, 3y ) 1 + y 2 4 (b) If P 1 is located at (0, 5) and Q is at (1, 1), where is P 2? 12. Which of the following lines is perpendicular to the line y = 2x + 2? (a) y = 0.5x + 2 (b) y = 2x 2 (c) y = 0.5x (d) y = 2x + 4 (e) y = x Which of the following quadratic functions has a graph that has its vertex at the point (0, 3)? (a) f(x) = 3x 2 (b) f(x) = (x 3) (c) f(x) = x 2 3 (d) f(x) = (x 3) 2 (e) f(x) = x What is the slope m of the line that goes through the points ( 1, 3) and ( 2, 2)? 15. Which of the following are possible graphs of polynomials? Of those that are polynomials, determine whether the degree is odd or even (assuming that there are no critical points other than those shown).

4 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) 16. Find the degree, the leading term, and the leading coefficient of the following polynomials. (a) f(x) = x 3 + x + 2 (b) f(x) = 1 + x 3x 2 + 5x 5 2x For each of the following polynomials, determine the degree and leading coefficient and use these to decide what happens to f(x) as x and as x.

5 (a) f(x) = x 2 + x (b) f(x) = 4x + 1 (c) f(x) = 2x 8 + 3x 7 (d) f(x) = 1 x 5 + x Which of the following scenarios are possible for a cubic polynomial? (a) No critical points (b) No zeros (c) One zero and one critical point (d) One zero and one turning point (e) One zero and two critical points (f) One zero and one inflexion point (g) Two zeros and no turning points (h) Three zeros and two critical points (i) Three critical points 19. Which of the following graphs can not be of a cubic function? (a) (b) (c) (d) 20. For the cubic polynomials below, show that the given z is a zero and use it to write f(x) on the form f(x) = a(x z)q(x). (a) f(x) = x 3 6x x 6, z = 1 (b) f(x) = x 3 + x + 2, z = 1 (c) f(x) = x 3 + x 2 x 1, z = For the cubic polynomials below, find the number and types of critical points. (a) f(x) = 2x 3 + 3x 2 + 2x 1

6 (b) f(x) = x 3 1 (c) f(x) = x 3 + x For the following cubic polynomials, find all zeros, the number and types of critical points, and sketch their graphs. (a) f(x) = x 3 4x 2 + 6x 3 (b) f(x) = x 3 + 3x 2 4 (c) f(x) = x 3 x 2 10x Consider the function f(x) = 2x 1, x R. Find the inverse function 5 f 1 (x). 24. Which of the following are graphs of one-to-one functions? Dashed lines denote asymptotes. (a) (b) (c) (d)

7 (e) (f) 25. Consider the function f(x) = x 2 1. Suggest a domain that makes f one-to-one (and try to choose the domain as large as possible). 26. The function f in the figure below has domain R and horizontal asymptotes at y = 0 and y = 2. (a) Is it possible for f to be a rational function of the type f(x) = g(x) h(x)? If so, what can you say about the degrees of g(x) and h(x)? (b) Find the domain and range of the inverse function f 1. (c) Sketch the graph of f

8 27. Let f(x) = x 1 x + 1, x 0. Find the inverse function f 1 (x). 28. If a rational function f(x) = g(x) has no vertical asymptotes, what can h(x) you conclude about the degree of h(x)? 29. Consider the rational function f(x) = x2 4x + 3. Do a sign diagram x 2 2x to decide what happens to f(x) as x 2 and as x For each of the following rational functions, decide if there is a horizontal asymptote and if so, what it is. (a) f(x) = 1 + x + x x x4 (b) f(x) = 1 x 5 6x x x x (c) f(x) = 2x x x x 5 + x 4 + x Suggest a rational function f(x) = g(x) that satisfies all of the following h(x) criteria: the degree of h(x) is at least 1 f(x) crosses the x-axis exactly once f(x) has no asymptotes. 32. Consider the function f(x) = 3x 1, x R. Find the inverse function 2 f 1 (x) (including its domain). 33. Let f(x) = 2x 1 x + 1, x 0. Find the inverse function f 1 (x) (including its domain). 34. This plot is the graph of a one-to-one function f:

9 Which of the plots in (a)-(f) is the graph of the inverse function f 1? (a) (b) (c) (d) (e) (f) 35. A population grows according to the exponential function q(t) = 1000e 0.1t, t 0 where q(t) is the size after t years.

10 (a) What is the initial population size? (b) What is the size after 2 years? (c) After how many years has the population increased by 50%? ( ) Write 2 ln x 3 ln 2 ln y as one logarithm. y ( x 2 ) y 37. Express ln in terms of ln x, ln y and ln z. z Compute log 3 (3 log 3 9 ). 39. Convert to radians: (a) 180 (b) 90 (c) 720 (d) Let θ be an angle such that sin θ = 0.6. What is cosθ? 41. Let θ be an angle such that cosθ = 0.6. What is tanθ? 42. Consider the angle counter-clockwise from the positive x-axis to the line that goes through the origin and the point ( 3, 4). Find sin θ, cosθ and tanθ. 43. Which of the following is the graph of the function f(x) = cosx? (a) (b)

11 (c) (d) 44. You roll a die twice. (a) What is the probability that you get at least one 4? (b) What is the probability that the sum equals 8? (c) What is the probability that the sum is at least 8? (d) What is the probability that the difference between the largest and smallest number equals 1? 45. If you toss 20 coins, you have about a one-in-a-million chance that they all land heads. If one million people toss 20 coins each, what is the probability that somebody manages to get 20 heads? 46. European roulette tables have the numbers 0-37 but no 00. The payout if you win a straight bet is 35 dollars, like in American roulette. If you wager one dollar on a straight bet in European roulette, what is the probability that you win and what is your expected gain? 47. Back to American roulette. In a five number bet you win if any of the numbers 00, 0, 1, 2, or 3 come up. (a) In order to get the usual expected gain of 2/38, what should the payout be? (b) The actual payout in a casino on this bet is 6:1. What is your expected gain?