Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Problem Set 2 ANSWERS Due 9:00AM Sep. 2 1. Graph the following functions Please come to my office hours if you have any questions on this. (a) f (x) = −1 if 0 ≤ x ≤ 1 2x if 1 < x < 3 (b) h(x) = x − 3 if 0 ≤ x < 2 2 − 3x if −3 < x ≤ −1 2. Compute the indicated values of the given function (a) f (x) = 3x2 + 5x − 2; f (1), f (0), f (−1) f (1) = 6, f (0) = −2, f (−1) = −4 (b) g(x) = x + x1 ; g(1), g(2) g(1) = 2, g(2) = 52 (c) h(x) = x − |x − 2|; h(1), h(2), h(3) h(1) = 0, h(2) = 2, h(3) = 2 (d) t(x) = x2 + 3x + 5; t(0), t(3), t(−3) t(0) = 5, t(3) = 23, t(−3) = 5 (e) f (x) = 3x2 + x1 + 5; f (x − 1) 1 f (x − 1) = 3(x − 1)2 + x−1 +5 3. We will practise some multiplication and division: (a) Multiply 5x2 y 4 by 4yx6 = 20x8 y 5 (b) Divide 6x2 y 3 by 2yx5 2 = 3y x3 (c) Divide = 6x y4 3x2 y (d) Simplify = by xy 3 2 . 12xy 3 . 2x2 y 2 6y x 1 4. Factorize the following, simplifying wherever possible: (a) 3x + 6xy = 3x(1 + 2y) (b) 2y 2 + 7y = 2y(2y + 7) (c) x(x2 + 8) + 2x2 (x − 5) − 8x = x2 (3x − 10) (d) 3x(x + x4 ) − 4(x2 + 3) + 2x = x(2 − x) 5. Factorize the following quadratics: (a) x2 + 5x + 6 = (x + 3)(x − 2) (b) y 2 − y − 12 = (y − 4)(y + 3) (c) 2x2 − 5x − 12 = (x − 4)(2x + 3) (d) x2 − 49 = (x − 7)(x + 7) 6. Fun with square roots: √ √ (a) Show that 2 × 18 = 6 = = √ √ 2 × 18 36 =6 (b) Show that √ √ 245 = 7 5 √ 49 × 5 √ =7 5 = (c) Show that 15 √ 3 √ =5 3 √ 15 3 = √3 =5 3 (d) Simplify = x2 √ 3 √2x . 8x 7. Laws of Indices and Logs 2 2 3 (a) Without using a calculator, simplify ( 27 8 ) 9 =4 (b) Without using a calculator, evaluate log2 64 =6 (c) Without using a calculator, evaluate log10 100 =3 (d) Simplify log10 a2 + 31 log10 b − 2 log10 ab = − 35 log10 b 8. Equations and Inequalities. Solve the following: (a) 5(3 − y) < 2y + 3 12 7 >y (b) |9 − 2x| = 11 x = −1, 10 (c) |x + a| < 2, where a is a parameter and 0 < a < 2 −2 − a < x < 2 − a (d) x2 − 8x + 12 < 0 2<x<6 9. Find the inverse function for the relationship between Celcius and Farenheight, where f (x) = 9 5 x + 32 f −1 (x) = 95 (x − 32). 10. S&B 2.8 (a) and (e) (p 21) (a) Use substitution to identify the y intercept: y = 2x + 3 Since the slope is 2, then we have m = 2 immediately. Since the y-intercept is (0, 3), we plug in this point to y = mx + b to obtain 3 = 2(0) + b, so b = 3. (e) Use y1 − y2 = m(x1 − x2 ) to identify the y intercept: y = x + 1 Plugging in points (2, 3) and (4, 5) to y = mx + b, we have 3 = 2m + b and 5 = 4m + b. Subtracting the first equation from the second, 5 − 3 = 4m + b − (2m − b) 2 = 2m m = 1. Using (2, 3) in y = 1x + b, we have 3 = 1(2) + b, which yields b = 1. 11. Find the value of r such that vA = vB 3 (1) (2) (3) zA = 1 − δvB zB = 1 − δvA 1 vA = pzA + prδvA + qδvA 2 vB = qzB + 2p(1 − r)δvB . Solving for vA and vB , we obtain p 1 + pδ − prδ − 2q δ q vB = . 1 + qδ + 2pδ(r − 1) vA = Setting these two values equal, we have vA = vB q p q = 1 + pδ − prδ − 2 δ 1 + qδ + 2pδ(r − 1) Now cross multiply so you get: q p[1 + qδ + 2pδ(r − 1)] = q[1 + pδ − prδ − δ] 2 And now solving for r is trivial q − p + 2p2 δ − r= 2p2 δ + pqδ 12. Solve the following equation (calculate a numeric answer): (ex )2 = 3e4 e2x = 3e4 e2x−4 = 3 2x − 4 = ln 3 ln 3 + 4 x = = 2.549 2 4 q2 δ 2 .