Homework Answers
Let f(x) = x^(1/3) with domain (0,infinity). Prove, by epsilon-delta language, that f is continuous at...
Let f(x) = x^(1/3) with domain (0,infinity). Prove, by
epsilon-delta language, that f is continuous at c in an element of
(0, infinity).
2. Let f(0) = 25 with domain (0,00). Prove, by the e-8 language, that f is continuous at CE (0,0)
Let f (x) = x2 – 6 and g(x) = V6 – 2. Determine the domain...
Let f (x) = x2 – 6 and g(x) = V6 – 2. Determine the domain of f (g(x)). O [-6,6] o [6,00) 0 (-00,-6] U [6,00) o(-00,00) 01-00,6] Determine the domain restriction, if one is necessary, so that f (x) = 212 + 1 is one-to-one. O [3,00) o(-0,3) O No restriction is necessary. O (1,00) O (3,00) Determine the vertex of the parabola described by f (x) = 3 – 12x + 2x2. O (2, -13) O (3,...
15. Question Details Let f(x) = x2-4 and g(x) = In x. The domain of f(g(x))...
15. Question Details Let f(x) = x2-4 and g(x) = In x. The domain of f(g(x)) is (0,e-20 [e?,..) (-2,-2) (e2,00) [0,e2] [e 2e2] [-2,2] 0 O Type here to search
Q1. Let z = f(x,y) -√4x² – 2y² Find (i). domain of f(x,y) (ii). range of...
Q1. Let z = f(x,y) -√4x² – 2y² Find (i). domain of f(x,y) (ii). range of f(x,y) (iii). f(1,1) (iv). The level curves of f(x,y) for k = 0,1,2 4x2y Q3. Let f(x,y) = x2+y2 if (x,y) = (0,0) 1 if (x,y) = (0,0) Find (i) lim limf(x,y) (x,y)-(0,0) (ii). Is f(x, y) continuous at (0,0)? (iii). Find the largest set S on which f(x,y) is continuous.
3. (25 points) Let f(x) = 2/2_8 (a) Find the domain of f. (b) Find the...
3. (25 points) Let f(x) = 2/2_8 (a) Find the domain of f. (b) Find the equation of all vertical asymptotes or explain why none exist. For each vertical asymptote = a, calculate both the one-sided limits limo+a+ f(x) and limo-ha-f(T). (c) Find the equation of all horizontal asymptotes or explain why none exist. C Bollett. ollett (d) Let g(x) = f(x) if x 70 For what value of b would g(x) be continuous at I=0? (or if no 91...
Exercice 2 (5pts) Let f given by f(x, y) Isinyif (x, y) (0,0) and f(0,0) 0...
Exercice 2 (5pts) Let f given by f(x, y) Isinyif (x, y) (0,0) and f(0,0) 0 1V224 1. Is f continuous at (0,0). 2. Compute the partial derivatives of f at any (x, y) E R2. Are the partial derivatives continuous (0,0). at (0,0) (0,0) and 3. Compute the second derivatives 4. Compute the linear approzimant of f at (0,0).
Exercice 2 (5pts) Let f given by f(x, y) Isinyif (x, y) (0,0) and f(0,0) 0 1V224 1. Is f...
(0, 1) given by f (x) - sin (). Is f Let f b e the...
(0, 1) given by f (x) - sin (). Is f Let f b e the function t on the domain uniformly continuous? Explain. (You may take it as given that sin is a continuous function) Suppose that f [0, oo) -R is a continuous function, and suppose also that lim, ->oo f (x)- 0. Prove that f is uniformly continuous Just to be clear: to say that lim,->o f (x) - 0 means that
1 2. Let: f(x):= (x - 2)2 – 31 (a) Discuss domain of definition and image...
1 2. Let: f(x):= (x - 2)2 – 31 (a) Discuss domain of definition and image of this function. (b) Find the values of a E R such that: f(x)dx is convergent. (c) Compute the value of the integral above for a = 6.
1. Let X be an RV with density f(x) = ¼arosinx + c, x E [-1,11 (f(x) = 0 elsewhere). (a) Compute ...
1. Let X be an RV with density f(x) = ¼arosinx + c, x E [-1,11 (f(x) = 0 elsewhere). (a) Compute the constant c. (b) Compute the DF of X. (c) Compute the DF of the RV Y d) Compute P( <0.5) X2.
1. Let X be an RV with density f(x) = ¼arosinx + c, x E [-1,11 (f(x) = 0 elsewhere). (a) Compute the constant c. (b) Compute the DF of X. (c) Compute the DF of...
Let X and Y be joint continuous random variables with joint density function f(x, y) =...
Let X and Y be joint continuous random variables with
joint density function
f(x, y) = (e−y
y
0 < x < y, 0 < y, ∞
0 otherwise
Compute E[X2
| Y = y].
5. Let X and Y be joint continuous random variables with joint density function e, y 0 otwise Compute E(X2 | Y = y]
Let f(x) = x^(1/3) with domain (0,infinity). Prove, by
epsilon-delta language, that f is continuous at c in an element of
(0, infinity).
2. Let f(0) = 25 with domain (0,00). Prove, by the e-8 language, that f is continuous at CE (0,0)
Let f (x) = x2 – 6 and g(x) = V6 – 2. Determine the domain of f (g(x)). O [-6,6] o [6,00) 0 (-00,-6] U [6,00) o(-00,00) 01-00,6] Determine the domain restriction, if one is necessary, so that f (x) = 212 + 1 is one-to-one. O [3,00) o(-0,3) O No restriction is necessary. O (1,00) O (3,00) Determine the vertex of the parabola described by f (x) = 3 – 12x + 2x2. O (2, -13) O (3,...
15. Question Details Let f(x) = x2-4 and g(x) = In x. The domain of f(g(x)) is (0,e-20 [e?,..) (-2,-2) (e2,00) [0,e2] [e 2e2] [-2,2] 0 O Type here to search
Q1. Let z = f(x,y) -√4x² – 2y² Find (i). domain of f(x,y) (ii). range of f(x,y) (iii). f(1,1) (iv). The level curves of f(x,y) for k = 0,1,2 4x2y Q3. Let f(x,y) = x2+y2 if (x,y) = (0,0) 1 if (x,y) = (0,0) Find (i) lim limf(x,y) (x,y)-(0,0) (ii). Is f(x, y) continuous at (0,0)? (iii). Find the largest set S on which f(x,y) is continuous.
3. (25 points) Let f(x) = 2/2_8 (a) Find the domain of f. (b) Find the equation of all vertical asymptotes or explain why none exist. For each vertical asymptote = a, calculate both the one-sided limits limo+a+ f(x) and limo-ha-f(T). (c) Find the equation of all horizontal asymptotes or explain why none exist. C Bollett. ollett (d) Let g(x) = f(x) if x 70 For what value of b would g(x) be continuous at I=0? (or if no 91...
Exercice 2 (5pts) Let f given by f(x, y) Isinyif (x, y) (0,0) and f(0,0) 0 1V224 1. Is f continuous at (0,0). 2. Compute the partial derivatives of f at any (x, y) E R2. Are the partial derivatives continuous (0,0). at (0,0) (0,0) and 3. Compute the second derivatives 4. Compute the linear approzimant of f at (0,0).
Exercice 2 (5pts) Let f given by f(x, y) Isinyif (x, y) (0,0) and f(0,0) 0 1V224 1. Is f...
(0, 1) given by f (x) - sin (). Is f Let f b e the function t on the domain uniformly continuous? Explain. (You may take it as given that sin is a continuous function) Suppose that f [0, oo) -R is a continuous function, and suppose also that lim, ->oo f (x)- 0. Prove that f is uniformly continuous Just to be clear: to say that lim,->o f (x) - 0 means that
1 2. Let: f(x):= (x - 2)2 – 31 (a) Discuss domain of definition and image of this function. (b) Find the values of a E R such that: f(x)dx is convergent. (c) Compute the value of the integral above for a = 6.
1. Let X be an RV with density f(x) = ¼arosinx + c, x E [-1,11 (f(x) = 0 elsewhere). (a) Compute the constant c. (b) Compute the DF of X. (c) Compute the DF of the RV Y d) Compute P( <0.5) X2.
1. Let X be an RV with density f(x) = ¼arosinx + c, x E [-1,11 (f(x) = 0 elsewhere). (a) Compute the constant c. (b) Compute the DF of X. (c) Compute the DF of...
Let X and Y be joint continuous random variables with
joint density function
f(x, y) = (e−y
y
0 < x < y, 0 < y, ∞
0 otherwise
Compute E[X2
| Y = y].
5. Let X and Y be joint continuous random variables with joint density function e, y 0 otwise Compute E(X2 | Y = y]
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