MATH 273 Lecture Notes - Lecture 3: Contraposition

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scarletserval756

21 Nov 2016

School

University of Calgary

Department

Mathematics

Course

MATH 273

Professor

Karen Seyffarth

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Document Summary

Mathematical and logical statements must have a determinable truth value. The statement 1^2 + 2^2 = 3^2 is false. X + y = 3: not a statement, since truth value cannot be determined due to variables. Negation of a universal quantifier is an existential quantifier and vice versa. The negation of all birds can fly is there exists a bird that cannot fly . A common mistake is to say the negation is all birds cannot fly . Let p (a hypothesis) and q (a conclusion) denote mathematical statements. The negation of p is true if p is false. Conditional: if p, then q (p implies q) There is generally no relationship between a statement and its converse. If p then q becomes if not q, then not p . The truth value of a statement is the same as the truth value of its contrapositive. If x and y are real numbers and x>y, then x^2 > y^2.

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[1] 1. (a) Solve 2 – 31 < 7. Solution: We have – 7<-3 < 7 which gives -4 0 on (-0, 1] U [2,). Next, we need to see where Vr2 – 3r +2 -1 > 0. If r2 - 3.c + 2 – 1 > 0, then V.22 – 3.0 +2 > . Observe that this is definitely true if r < 0. If r > 0, then squar- ing both sides we get 22 - 3.+2 > 1 2 > 3. V A VICO Thus, the domain of f is (-00, ). [2] (c) Find the exact value of tan(sec-14). Solution: Let y = sec-14. Then, sec y = 4. From the diagram we get that tan(sec-? 4) = tan y = 15 [2] (d) Find all r such that sin(2.x) = COS I. Solution: We have 2 sin r cos r = COS I, SO 0 = 2 sin Cos I -Cos I = cos .r (2 sin c - 1). Hence, sin(2x) = cost when cos x = 0, or when sin r = Thus, r= +ik, kez, I= +2nk, ke Z, or r= +2nk, kez [2] (e) State the precise mathematical definition of lim f(x) = 0. Solution: For every e > 0 there exists a 8 >0 such that if 0 < x-al <, then f(c) >

Q1. Determine, without graphing, whether the quadratic function f(x) = x2 + 2x - 6 has a maximum value or a minimum value and then find that value. a. minimum; -7 b. minimum; -1 c. maximum; -1 d. maximum; -7 Q2. Find (f - g)(4) when f(x) = 5x2 + 6 and g(x) = x + 2. a. 80 b. 88 c. 84 d. -90 Q3. The graph of a function is given. Decide whether it is even, odd, or neither. a. even b. odd c. neither Q4. Find the average rate of change for the function f(x) = -3x2 - x from the values 5 to 6. a. 1/2 b. -2 c. -34 d. -1/6 Q5. Find the vertex and axis of symmetry of the graph of the function f(x) = 3x2 + 36x. a. (-6, -108); x = -6 b. (-6, 0); x = -6 c. (6, -108); x =6 d. (6, 0); x = 6 Q6. The graph of a function is given. Decide whether it is even, odd, or neither. a. even b. odd c. neither Q7. Use the graph to find the intervals on which it is increasing, decreasing, or constant. a. Increasing on (-3, -2) and (2, 4); decreasing on (-1, 1); constant on (-2, -1) and (1, 2) b. Decreasing on (-3, -2) and (2, 4); increasing on (-1, 1); constant on (-2, -1) and (1, 2) c. Decreasing on (-3, -2) and (2, 4); increasing on (-1, 1) d. Decreasing on (-3, -1) and (1, 4); increasing on (-2, 1) Q8. Graph the function F(x) = -5. State whether it is increasing, decreasing, or constant. a. b. c. d. Q9. Determine algebraically whether f(x) = -2x3 is even, odd, or neither. a. even b. odd c. neither Q10. Find the average rate of change for the function f(x) = 3/(x - 2) from the values 4 to 7. a. 7 b. 1/3 c. -3/10 d. 2 Q11. For the graph of the function y = f(x), find the absolute maximum and the absolute minimum, if it exists. a. Absolute maximum: f(4) = 7; Absolute minimum: f(1) = 2 b. Absolute maximum: f(4) = 7; Absolute minimum: none c. Absolute maximum: none; Absolute minimum: none d. Absolute maximum: none; Absolute minimum: f(1) = 2 Q12. The graph of a function f is given. Find the numbers, if any, at which f has a local minimum. What are the local minima? a. f has a local minimum at x = -2; the local minimum is 0 b. f has a local minimum at x = 0; the local minimum is 3 c. f has a local minimum at x = -2 and 2; the local minimum is 0 d. f has no local minimum Q13. Graph the function h(x) = -2x + 3. State whether it is increasing, decreasing, or constant. a. b. c. d. Q14. Find -f(x) when f(x) = -2x2 + 5x + 2. a. 2x2 - 5x + 2 b. -2x2 - 5x - 2 c. -2x2 - 5x + 2 d. 2x2 - 5x - 2 Q15. Determine algebraically whether f(x) = 1/x2 is even, odd, or neither. a. even b. odd c. neither Q16. Match the graph to one of the listed functions. a. f(x) = x2 - 8 b. f(x) = -x2 - 8x c. f(x) = x2 - 8x d. f(x) = -x2 - 8 Q17. In a certain city, the cost of a taxi ride is computed as follows: There is a fixed charge of $2.90 as soon as you get in the taxi, to which a charge of $1.70 per mile is added. Find an equation that can be used to determine the cost, C(x), of an x-mile taxi ride. a. C(x) = 1.70 + 2.90x b. C(x) = 3.10x c. C(x) = 4.60x d. C(x) = 2.90 + 1.70x Q18. Graph the function f(x) = -x2 + 3 by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). a. b. c. d. Q19. Graph the function f(x) = x2 + 8x + 7 using its vertex, axis of symmetry, and intercepts. a. b. c. d. Q20. Determine where the function f(x) = -x2 + 8x - 7 is increasing and where it is decreasing. a. increasing on (4, â) and decreasing on (-â, 4) b. increasing on (-â, 4) and decreasing on (4, â) c. increasing on (9, â) and decreasing on (-â, 9) d. increasing on (-â, 9) and decreasing on (9, â) Q21. Find the domain of the rational function f(x) = (x + 9)/(x2 - 4x). a. {x|x â -2, x â 2} b. {x|x â -2, x â 2, x â -9} c. all real numbers d. {x|x â 0, x â 4} Q22. For the polynomial f(x) = 4(x - 5)(x - 6)3, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x-intercept. a. 5, multiplicity 1, touches x-axis; 6, multiplicity 3 b. -5, multiplicity 1, touches x-axis; -6, multiplicity 3 c. 5, multiplicity 1, crosses x-axis; 6, multiplicity 3, crosses x-axis d. -5, multiplicity 1, crosses x-axis; -6, multiplicity 3, crosses x-axis Q23. Find all zeros of the function f(x) = 2x4 + 3x3 + 16x2 + 27x - 18 and write the polynomial as a product of linear factors. a. f(x) = (2x - 1)(x + 2)(x + 3)(x - 3) b. f(x) = (2x + 1)(x - 2)(x + 3)(x - 3) c. f(x) = (2x - 1)(x + 2)(x + 3i)(x - 3i) d. f(x) = (2x + 1)(x - 2)(x + 3i)(x - 3i) Q24. For the polynomial f(x) = (1/5)x(x2 - 5), list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x-intercept. a. 0, multiplicity 1, touches x-axis; â5, multiplicity 1, touches x-axis; -â5, multiplicity 1, touches x-axis b. 0, multiplicity 1, crosses x-axis; â5, multiplicity 1, crosses x-axis; -â5, multiplicity 1, crosses x-axis c. â5, multiplicity 1, touches x-axis; -â5, multiplicity 1, touches x-axis d. 0, multiplicity 1 Q25. List the potential rational zeros of the polynomial function f(x) = x5 - 6x2 + 5x + 15. Do not find the zeros. a. Â± 1, Â± 1/5, Â± 1/3 Â± 1/15 b. Â± 1, Â± 5, Â± 3, Â± 15 c. Â± 1, Â± 5, Â± 3 d. Â± 1, Â± 1/5, Â± 1/3, Â± 1/15, Â± 5, Â± 3,Â± 15 Q26. The function f(x) = x4 - 5x2 - 36 has the zero -2i. Find the remaining zeros of the function. a. 2i, 6, -6 b. 2i, 3i, -3i c. 2i, 3, -3 d. 2i, 6i, -6i Q27. Use transformations of the graph of y = x4 to graph the function f(x) = 5x4. a. b. c. d. Q28. State whether the function f(x) = x(x - 9) is a polynomial function or not. If it is, give its degree. If it is not, tell why not. a. Yes; degree 2 b. No; it is a product c. Yes; degree 0 d. Yes; degree 1 Q29. Solve the inequality algebraically. Express the solution in interval notation. (x - 2)2(x + 9) < 0 a. (-â, -9) or (9, â) b. (-â, -9] c. (-â, -9) d. (-9, â) Q30. Use the Intermediate Value Theorem to determine whether the polynomial function f(x) = -4x4 - 9x2 + 4; has a zero in the interval [-1, 0]. a. f(-1) = -9 and f(0) = -4; no b. f(-1) = 9 and f(0) = 5; no c. f(-1) = 9 and f(0) = -4; yes d. f(-1) = -9 and f(0) = 4; yes Q31. Use the Factor Theorem to determine whether x + 5 is a factor of f(x) = 3x3 + 13x2 - 9x + 5. a. Yes b. No Q32. Find a bound on the real zeros of the polynomial function f(x) = x4 - 8x2 - 9. a. -17 and 17 b. -9 and 9 c. -18 and 18 d. -10 and 10 Q33. Find the x-intercepts of the graph of the function f(x) = (x - 3)(2x + 7)/(x2 + 9x - 8). a. (3, 0), (-7, 0) b. (-3, 0), (7/2, 0) c. (3, 0), (-7/2, 0) d. none Q34. Solve the inequality algebraically. Express the solution in interval notation. (9x - 5)/(x + 2) â¤ 8 a. (-2, 13] b. (-2, 21) c. (-2, 21] d. (-2, 13) Q35. A polynomial f(x) of degree 3 whose coefficients are real numbers has the zeros -4 and 4 - 5i. Find the remaining zeros of f. a. 4, -4 + 5i b. 4, 4 + 5i c. 4 + 5i d. -4 + 5i Q36. Find the intercepts of the function f(x) = x3 + 3x2 - 4x - 12. a. x-intercepts: -2, 2, 3; y-intercept: -12 b. x-intercepts: -3, -2, 2; y-intercept: -12 c. x-intercept: -3; y-intercept: -12 d. x-intercept: -2; y-intercept: -12 Q37. Find the real solutions of the equation 3x3 - x2 + 3x - 1 = 0. a. {-3, 1/3, -1} b. {1/3} c. {1/3, -1} d. {-3, -1/3, -1} Q38. Solve the inequality algebraically. Express the solution in interval notation. x3 â¥ 27 a. (-â, 3] b. (-â, -3] âª [3, â) c. [-3, 3] d. [3, â) Q39. Find the intercepts of the function f(x) = x2(x - 1)(x - 6). a. x-intercepts: 0, -1, -6; y-intercept: 0 b. x-intercepts: 0, 1, 6; y-intercept: 6 c. x-intercepts: 0, 1, 6; y-intercept: 0 d. x-intercepts: 0, -1, -6; y-intercept: 6 Q40. Find the domain of the rational function R(x) = (-3x2)/(x2 + 2x - 15). a. {x|x â 5, 3} b. {x|x â 5, -3} c. {x|x â - 15, 1} d. {x|x â -5, 3}

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MATH 273 Lecture Notes - Lecture 3: Contraposition

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