ENM Review 503 Midterm

Running the Bases 1. Find a 3-digit number in base 7 that is equivalent to 147 in base 10 representation. Q R 7 | 147 21 0 21 3 0 3 0 3 3007 2. ACDC16 = ________10 44252 3. Write the first 10 primes in base 7.

Functions 4. For a and b > 0, prove that geometric mean , < arithmetic mean (a+b)/2. Assume ab < [(a + b)/2]2 Then 4ab < a2 + 2ab + b2 => a2 -2ab + b2 = (a-b)2 > 0. 5. Determine conditions on a, b, and c so that the function f(x) = ax2 + bx + c satisfies f(x) = x2f(1/x) for all x  0. ax2+ bx + c = x2(a/x2 + b/x + c) = a + bx + cx2 => a = c; b arbitrary 6. Suppose a function f satisfies f(1) = 1 and f(x) = f(x - 1) + 2 for all x. Find f(3) and f(4). f(2) = f(1) + 2 = 3; f(3) = f(2) + 2 = 5; f(4) = f(3) + 2 = 7. 7. If f(x) = x2 and x = g(t) = t + 2, find f[g(3)]. f(g(t) = (t + 2)2 => f(g(3) = 25 8/30/2019 rd

Graphs 8. Sketch Domain, range, symmetry, intercepts, and asymptotes. 9. Sketch |x| + |y| = 1. 10. Describe location in xy-plane of points P(x, y) for which a. x > 0 b. x > 0 and y < 0 c. x = 0 d. y = 0 e. x and y have same sign. f. y = -|x| g. xy = 0 a. rt hand plane b. R-Bot plane c. y-axis d. x-axis e. quads 1 & 3 f. Inverted v g. xy-axes 8/30/2019 rd

Lines 11. Find a) inclination and slope of the line y = x; b) equation of line through (-2, 0) with inclination 45°. Inclination is 45° and slope is tan (45°) = 1; y = x + 2 12. Points (4, -2) and (8, 6) are endpoints of a diameter. Write the equation of the circle. d = => d2 = 80 => 4r2 = 80 => r2 = 20. midpoint = (6, 2) => circle equation is: (x - 6)2 + (y - 2)2 = 20. 13. Show that the points (-3, 2), (4, 1), and (5, 8) are vertices of a right triangle. Find the center and radius of the circumscribed circle. D2: (-3, 2) & (4, 1) = 50; D2: (-3, 2) & (5, 8) = 100; D2: (4, 1) & (5, 8) = 50 => isosceles right triangle (x - 1)2 + (y - 5)2 = 50 with center at (1, 5) with radius = 50½ 8/30/2019 rd

Lines 14. Show that the line through points (4, -1) and (5, 1) is perpendicular to the line through points (11, 1) and (9, 2). slope m1 = 2 and slope m2 = -1/2 and product = -1. 15. Find the slopes of the lines y = ax + b and ax + by + c = 0. a and -a/b 16. a. Show that the line 6x - y - 9 = 0 is tangent to y = x2. b. The slope of the curve y = |x| at x = p is 1. 6x - 9 = x2 => (x - 3)2 = 0 => line intersects parabola at x = 3 and y = 9 => the slope is y' = 2x = 2(3) = 6. 8/30/2019 rd

17. Find a real root solution to the equation e5x + e4x - 6e3x = 0. e3x(e2x + ex - 6) = 0 Let y = ex; Then y5 + y4 - 6y3 = 0 y3(y2 + y – 6) = 0 => (y – 2)(y + 3) = 0 y = 2 = ex => x = ln 2; ex > 0

Solving logarithmic Equations 18. log 3x + log 3 = 2 => log 9x = 2 => 9x = 102 => x = 100/9 19. e3x = 14 => 3x = Ln 14 = 2.639057 => x = 0.8796857 20. Ln (logx3) = 2 => logx3 = e2 => x7.389 = 3 => 7.389 Ln x = Ln 3 => x = 1.160 8/30/2019 rd

Solving logarithmic Equations 21. 63-4x = 15 => (3 – 4x) Ln 6 = Ln 15 => 3 – 4x = 1.51139 => x = 0.37215210 22. Ln (x + 2) = x2 – 7 x + 2 = e(x^2 - 7) x = 2.932 8/30/2019 rd

Inverses 23. Find the inverse of y = x2 – 16. x =  (y – 16)½ and the function y is not 1-1 and the inverse does not exist. 24. Find and plot the inverse of y = 3x + 6. 25. Use truth tables to prove De-Morgan's Law. Include a Venn diagram to clarify. of ~A~B = ~(A + B)

26. Solve for x. Log5 1027 = logx 4532 27. A 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 D 1 1 0 0 1 1 0 0 D = abc + abC + Abc + AbC = ab + Ab = b

Tableaus Max 3x1 + 4x2 s.t. 2x1 + 3x2 <= 12 AX = B 5x1 + 3x2 <= 15 Initial Tableau at corner point (0, 0) -3 -4 0 0 0 2 3 1 0 12 5 3 0 1 15 Tableau at corner point (0, 4) -1/3 0 4/3 0 16 2/3 1 1/3 0 4 3 0 -1 1 3 Final Tableau Tableau at corner point (3, 0) 0 0 11/9 1/9 49/3 0 -7/5 0 ¾ 9 0 1 5/9 -2/9 10/3 0 7/5 1 -2/5 6 1 0 -1/3 1/3 1 1 3/5 0 1/5 3 X2 5 4 X1 0 3 6 8/30/2019 rd

Dual Solutions Max Z = 10X1 + 24X2 Min W = 120Y1 + 180Y2 s.t. 1X1 + 2X2 <= 120 s.t 1Y1 + 1Y2 >= 10 1X1 + 4X2 <= 180 2Y1 + 4Y2 >= 24 (LP '((-10 -24 0)(1 2 120)(1 4 180)))  0 0 8 2 1320 1 0 2 -1 60 0 1 -1/2 1/2 30 W = 120(8) + 180(2) = 1320 8/30/2019 rd

Converting (<= >=) Constraints Min W = 4X1 + 3X2 s.t. 3X1 - 1X2 >= 2  3X1 - 1X2 >= 2 1X1 + 1X2 <= 1  -1X1 - 1X2 >= -1 -4X1 + 1X2 <= 3  4X1 - 1X2 >= -3 LP '((-2 1 3 0)(3 -1 4 4)(-1 -1 -1 3)))  0 1/3 17/3 2/3 0 8/3 1 -1/3 4/3 1/3 0 4/3 0 -4/3 1/3 1/3 1 13/3 W = 4(2/3) + 3(0) = 8/3 8/30/2019 rd

Converting Equality Constraints 2X1 + 3X2 = 6 becomes 2X1 + 3X2 <= 6 2X1 + 3X2 >= 6 8/30/2019 rd

Inverse Matrix 28. Find the inverse and determinant of matrix … and write its characteristic equation. Show that the matrix satisfies its characteristic equation and again find its inverse. 3 4 -1 7 The characteristic equation is 1t2 -10t + 25 = 0 7/25 -4/25 1/25 3/25 A2 – 10A + 25I = 0 5 40 - 30 40 + 25 0 = 0 0 -10 45 -10 70 0 25 0 0 -