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Math 2415 · Practice Final A PDF

Question 2. (30 pts) A rectangular box without a lid is made from $12 m^{2}$ of cardboard. Use Lagrange multipliers to find the maximum volume. $maximize V = x y z s.t. x y + 2 x z + 2 y z = 12$

Question 3. (30 pts) Evaluate $∭_{E} z d V$ by converting to spherical coordinates.

Question 4. (16 pts) Set up the triple integral for the volume bounded by $z = x^{2} + y^{2}$ and $z = 8 - x^{2} - y^{2}$ .

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Math 2415 · Practice Final B PDF

Question 1. (24 pts) Find the interval of convergence of the power series. $\sum_{n = 1}^{\infty} \frac{(x - 2)^{n}}{n \cdot 3^{n}}$

Question 4. (14 pts) Verify Green’s theorem both directly and via the double integral for $\oint_{C} y^{2} d x + x d y$ .

Question 5. (18 pts) Determine whether $\sum \frac{(- 1)^{n}}{\sqrt{n}}$ converges absolutely, conditionally, or diverges.

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Uploaded · practice-final-draft-3.pdf

Graded · 78/100

AI grading · instant feedback

Q1 ✓ Correct — full credit 10/10

Q2 ✕ Sign error in part (b) — the bound flips at x = 0 3/10

Q3 ◐ Partial credit — setup right, integration incomplete 6/10

Topic mastery updated

Lagrange multipliers5871 Convergence tests4439

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