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Advanced university · Mathematics

Advanced university Mathematics practice

15 questions on complex analysis, partial differential equations, real analysis, optimization. The answer key is a separate download.

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What is on the worksheet

  1. Find the residue at z = 8 of f(z) = (11z² + 3)/(z − (8)). Round the final result to 4 decimal places if needed.

  2. For u_t=0.7u_xx on 0<x<π with zero boundary values and u(x,0)=9sin(3x), find the amplitude of this mode at t=0.5. Round the final result to 4 decimal places if needed.

  3. For f_n(x) = 29x/n on [0,1], calculate sup over x of |f_n(x)| when n = 100. Round the final result to 4 decimal places if needed.

  4. Find the residue at z = 2 of f(z) = (5z³ + 5z)/(z − (2))². Round the final result to 4 decimal places if needed.

  5. For f(x,y)=10x² + (28)x + 12y², find the x-coordinate of the unique global minimizer. Round the final result to 4 decimal places if needed.

  6. For f(x,y)=3x² + (18)x + 6y², find the x-coordinate of the unique global minimizer. Round the final result to 4 decimal places if needed.

  7. Find the residue at z = -3 of f(z) = (12z² + 2)/(z − (-3)). Round the final result to 4 decimal places if needed.

  8. A fixed-end string of length 11 m has wave speed 23 m/s. Write the angular frequency of mode n=7 as aπ rad/s. Find a. Round the final result to 4 decimal places if needed.

  9. A fixed-end string of length 12 m has wave speed 24 m/s. Write the angular frequency of mode n=7 as aπ rad/s. Find a. Round the final result to 4 decimal places if needed.

  10. For f_n(x) = 20x/n on [0,1], calculate sup over x of |f_n(x)| when n = 83. Round the final result to 4 decimal places if needed.

  11. Find the residue at z = -9 of f(z) = (12z² + 1)/(z − (-9)). Round the final result to 4 decimal places if needed.

  12. Let V be a 29-dimensional vector space and W a 13-dimensional subspace. Find dim(V/W). Round the final result to 4 decimal places if needed.

  13. A fixed-end string of length 10 m has wave speed 22 m/s. Write the angular frequency of mode n=8 as aπ rad/s. Find a. Round the final result to 4 decimal places if needed.

  14. A fixed-end string of length 12 m has wave speed 26 m/s. Write the angular frequency of mode n=7 as aπ rad/s. Find a. Round the final result to 4 decimal places if needed.

  15. A fixed-end string of length 7 m has wave speed 36 m/s. Write the angular frequency of mode n=11 as aπ rad/s. Find a. Round the final result to 4 decimal places if needed.