61. What is curl of the vector field 2x2yi + 5z2j - 4uyzk?
62. A rail engine accelerates from its stationary position for 8 seconds and travels a distance of 280 m. According to the Mean Value Theorem, the speedometer at a certain time during acceleration must read exactly
63. Let f(x) = x e-x. The maximum value of the function in the interval (0, \[\infty \]) is
64. The contour on the x - y plane, where the partial derivative of x2 + y2 with respect to y is equal to the partial derivative of 6y + 4x with respect
to x, is
65. Consider the following equations
\[\begin{gathered}
\frac{{\partial {\text{V}}\left( {{\text{x, y}}} \right)}}{{\partial {\text{x}}}} = {\text{p}}{{\text{x}}^2} + {{\text{y}}^2} + 2{\text{xy}} \hfill \\
\frac{{\partial {\text{V}}\left( {{\text{x, y}}} \right)}}{{\partial {\text{y}}}} = {{\text{x}}^2} + {\text{q}}{{\text{y}}^2} + 2{\text{xy}} \hfill \\
\end{gathered} \]
where p and q are constants. V(x, y) that satisfies the above equations is
\[\begin{gathered} \frac{{\partial {\text{V}}\left( {{\text{x, y}}} \right)}}{{\partial {\text{x}}}} = {\text{p}}{{\text{x}}^2} + {{\text{y}}^2} + 2{\text{xy}} \hfill \\ \frac{{\partial {\text{V}}\left( {{\text{x, y}}} \right)}}{{\partial {\text{y}}}} = {{\text{x}}^2} + {\text{q}}{{\text{y}}^2} + 2{\text{xy}} \hfill \\ \end{gathered} \]
where p and q are constants. V(x, y) that satisfies the above equations is
66. $$\mathop {\lim }\limits_{{\text{x}} \to 0} \frac{{{{\text{e}}^{\text{x}}} - \left( {1 + {\text{x}} + \frac{{{{\text{x}}^2}}}{2}} \right)}}{{{{\text{x}}^3}}} = ?$$
67. The quadratic approximation of f(x) = x3 - 3x2 - 5 a the point x = 0 is
68. The following inequality is true for all x close to
0.
\[2 - \frac{{{{\text{x}}^2}}}{3} < \frac{{{\text{x}}\sin {\text{x}}}}{{1 - \cos {\text{x}}}} < 2\]
What is the value of \[\mathop {\lim }\limits_{{\text{x}} \to 0} \frac{{{\text{x}}\sin {\text{x}}}}{{1 - \cos {\text{x}}}}?\]
\[2 - \frac{{{{\text{x}}^2}}}{3} < \frac{{{\text{x}}\sin {\text{x}}}}{{1 - \cos {\text{x}}}} < 2\]
What is the value of \[\mathop {\lim }\limits_{{\text{x}} \to 0} \frac{{{\text{x}}\sin {\text{x}}}}{{1 - \cos {\text{x}}}}?\]
69. The series \[\sum\limits_{{\text{n}} = 0}^\infty {\frac{1}{{{\text{n}}!}}} \] converges to
70. The quadratic approximation of f(x) = x3 - 3x2 - 5 a the point x = 0 is
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