Question

1. Let f : ? ? ? be defined by f(x) = {x if x is rational, x" if x is irrational.} Using the ?-? definition of continuity, show that the function f(x) is continuous at x = 1. Write down the necessary details. [3 marks] 2. Suppose we want to construct a cylindrical canister of volume 30 cubic centimeters with a bottom but no top. Determine the dimensions of the canister that minimize the amount of the material needed to construct the canister. [4 marks] 3. Let f be a twice differentiable function on [0, 2]. Show that if f(0) = 0, f(1) = 2 and f(2) = 4, then there exists a point in (0, 2) at which the second derivative of f is zero. Justify your answer with details. [3 marks] 4. Does the series ?_{n=3}^? 1/(ln n)^{ln n} converge or diverge? Explain. [4 marks] 5. Determine the Taylor series for the function ?_0^x sin(t") dt around x = 0. The answer should contain the first two nonzero terms in the series explicitly. [5 marks] 6. Evaluate ?_{-1}^2 1/x? dx. [2 marks] 7. Let R be the region bounded by y = 2?(x - 1) and y = x - 1. Find the volume of the solid generated by revolving R about the line x = 7 using (a) the washer method and (b) the shell method. [2+2 marks] 8. Let f : ?" ? ? be a function defined by f(x, y) = {0 for xy ? 0, 1 + x for y = 0, 1 for x = 0,} and let C denote the set of points where f is continuous and D denote the set of points where f is discontinuous. (a) Find out the sets C and D. Give justification for your answer. [2 marks] (b) Is the set C an open subset of ?"? Justify your answer. [1 mark] (c) Is the set D a closed subset of ?"? Justify your answer. [1 mark] (d) Find out the partial derivatives f_x(0,0) and f_y(-1,0). [1 mark]

          1. Let f : ? ? ? be defined by
f(x) = {x if x is rational, x" if x is irrational.}
Using the ?-? definition of continuity, show that the function f(x) is continuous at x = 1. Write down the necessary details. [3 marks]
2. Suppose we want to construct a cylindrical canister of volume 30 cubic centimeters with a bottom but no top. Determine the dimensions of the canister that minimize the amount of the material needed to construct the canister. [4 marks]
3. Let f be a twice differentiable function on [0, 2]. Show that if f(0) = 0, f(1) = 2 and f(2) = 4, then there exists a point in (0, 2) at which the second derivative of f is zero. Justify your answer with details. [3 marks]
4. Does the series ?_{n=3}^? 1/(ln n)^{ln n} converge or diverge? Explain. [4 marks]
5. Determine the Taylor series for the function ?_0^x sin(t") dt around x = 0. The answer should contain the first two nonzero terms in the series explicitly. [5 marks]
6. Evaluate ?_{-1}^2 1/x? dx. [2 marks]
7. Let R be the region bounded by y = 2?(x - 1) and y = x - 1. Find the volume of the solid generated by revolving R about the line x = 7 using (a) the washer method and (b) the shell method. [2+2 marks]
8. Let f : ?" ? ? be a function defined by
f(x, y) = {0 for xy ? 0, 1 + x for y = 0, 1 for x = 0,}
and let C denote the set of points where f is continuous and D denote the set of points where f is discontinuous.
(a) Find out the sets C and D. Give justification for your answer. [2 marks]
(b) Is the set C an open subset of ?"? Justify your answer. [1 mark]
(c) Is the set D a closed subset of ?"? Justify your answer. [1 mark]
(d) Find out the partial derivatives f_x(0,0) and f_y(-1,0). [1 mark]

1. Let f : ? ? ? be defined by
f(x) = x if x is rational, x" if x is irrational.
Using the ?-? definition of continuity, show that the function f(x) is continuous at x = 1. Write down the necessary details. [3 marks]
2. Suppose we want to construct a cylindrical canister of volume 30 cubic centimeters with a bottom but no top. Determine the dimensions of the canister that minimize the amount of the material needed to construct the canister. [4 marks]
3. Let f be a twice differentiable function on [0, 2]. Show that if f(0) = 0, f(1) = 2 and f(2) = 4, then there exists a point in (0, 2) at which the second derivative of f is zero. Justify your answer with details. [3 marks]
4. Does the series ?n=3^? 1/(ln n)^ln n converge or diverge? Explain. [4 marks]
5. Determine the Taylor series for the function ?0^x sin(t") dt around x = 0. The answer should contain the first two nonzero terms in the series explicitly. [5 marks]
6. Evaluate ?-1^2 1/x? dx. [2 marks]
7. Let R be the region bounded by y = 2?(x - 1) and y = x - 1. Find the volume of the solid generated by revolving R about the line x = 7 using (a) the washer method and (b) the shell method. [2+2 marks]
8. Let f : ?" ? ? be a function defined by
f(x, y) = 0 for xy ? 0, 1 + x for y = 0, 1 for x = 0,
and let C denote the set of points where f is continuous and D denote the set of points where f is discontinuous.
(a) Find out the sets C and D. Give justification for your answer. [2 marks]
(b) Is the set C an open subset of ?"? Justify your answer. [1 mark]
(c) Is the set D a closed subset of ?"? Justify your answer. [1 mark]
(d) Find out the partial derivatives fx(0,0) and fy(-1,0). [1 mark]

Added by Trijal M.

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