1. When differentiating the implicit function \(x^2 y^2 - \cos^2 y = \sin y\) with respect to \(x\), which of the following terms appears?

A. \(2x y^2\) B. \(-2x y^2\) C. \(2x^2 y\) D. \(-2x^2 y\)

2. If a function f(x, y) is homogeneous of degree n, what does Euler’s theorem state?

A. f(tx, ty) = tf(x, y) B. x fx + y fy = nf C. f(tx, ty) = t + f(x, y) D. x fx + y fy = f / n

3. What is the key step in transforming the Laplace equation ∂²f/∂x² + ∂²f/∂y² = 0 using new variables u = eˣ cos y and v = eˣ sin y?

A. Rewrite as f_uu + f_vv = 0 using factor (u² + v²) B. Replace variables x, y by u and v directly C. Ignore mixed partial derivatives D. Use new variables but keep original Laplace form

4. Which of the following correctly describes the relationship between partial derivatives in the u-v coordinate system when x and y are expressed as x = (1/2) ln(u² − v²), y = (1/2) ln((u+v)/(u−v))?

A. ∂f/∂u = fx (u / (u² − v²)) + fy (−v / (u² − v²)) B. ∂f/∂u = fx (−v / (u² − v²)) + fy (u / (u² − v²)) C. ∂f/∂v = fx (u / (u² − v²)) + fy (v / (u² − v²)) D. ∂f/∂v = fx (v / (u² − v²)) − fy (u / (u² − v²))

5. In verifying Rolle’s theorem for \(f(x) = (x - a)^m (x - b)^n\) on \([a,b]\), where is the critical point \(c\) where \(f'(c) = 0\) located?

A. \(c = \frac{m a + n b}{m + n}\) B. \(c = \frac{m b + n a}{m + n}\) C. \(c = \frac{a + b}{2}\) D. \(c = \frac{a b}{m + n}\)

6. Which of the following is the correct recursive formula for the integral \(I_n = \int x^n e^x dx\) for \(n > 0\)?

A. \(I_n = x^n e^x + n I_{n-1}\) B. \(I_n = x^n e^x - n I_{n-1}\) C. \(I_n = x^n e^x - I_{n-1}/n\) D. \(I_n = n x^{n-1} e^x - I_{n-1}\)

7. In Euler’s theorem applied to \(f(x,y) = x^3 + 4xy^2 - 3y^3\), what is the degree of homogeneity?

A. 1 B. 2 C. 3 D. 4

8. Applying the Mean Value Theorem to a function \(f\) with \(f(3) = 2\) and \(3 \leq f'(x) \leq 4\), what can be said about \(f(5)\)?

A. \(f(5) = 8\) B. \(8 \leq f(5) \leq 10\) C. \(f(5) \geq 10\) D. \(f(5) \leq 8\)

9. Given \(u = f(x,y)\) with \(x = s^2 - t^2\) and \(y = 2 s t\), what is the expression for \(s\frac{\partial u}{\partial s} - t\frac{\partial u}{\partial t}\)?

A. \(2(s^2 - t^2) \frac{\partial f}{\partial y}\) B. \(2(s^2 + t^2) \frac{\partial f}{\partial x}\) C. \((s - t) \frac{\partial f}{\partial x}\) D. \((s^2 + t^2) \frac{\partial f}{\partial y}\)

10. What is the value of \(I_2 = \int x^2 e^x dx\) based on the reduction formula?

A. \((x^2 - 2x + 2) e^x\) B. \((x^2 + 2x + 2) e^x\) C. \((x^2 - x + 1) e^x\) D. \((x^2 - 4x + 6) e^x\)

11. When parametrizing the semicircle \(x^2 + y^2 = 1\) for \(x \geq 0\) from \(A(0,1)\) to \(B(0,-1)\) with \(x = \cos \theta, y = \sin \theta\), what are the limits for \(\theta\)?

A. \(0\) to \(\pi\) B. \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) C. \(\frac{\pi}{2}\) to \(-\frac{\pi}{2}\) D. \(\pi\) to \(2\pi\)

12. Given the line integral \( I = \int_C (x^2 dx - 2xy dy) \) over a triangle with vertices \((0,0),(1,0),(0,1)\), which method is suitable for evaluating it?

A. Direct evaluation using parametric equations for a circle B. Splitting the integral into line integrals over each segment and calculating separately C. Using Green’s theorem directly D. Applying the Divergence theorem

13. To maximize the function f(x, y) = x² y² subject to x + y = 8 using Lagrange multipliers, what is the point where maximum occurs?

A. (8, 0) or (0, 8) B. (4, 4) C. (0, 0) D. (2, 6)

14. In the given line integral ∫C (x² dx − 2xy dy) over the triangle with vertices O(0,0), A(1,0), and B(0,1), the integral over segment AB can be written as integration over which variable to simplify?

A. y B. x C. z D. t

15. What is the relationship between \(\frac{\partial w}{\partial r}\) and the Cartesian partials \(f_x, f_y\) for \(w = f(x,y)\), with \(x = r \cos \theta\), \(y = r \sin \theta\)?

A. \(\frac{\partial w}{\partial r} = f_x \sin \theta + f_y \cos \theta\) B. \(\frac{\partial w}{\partial r} = f_x \cos \theta + f_y \sin \theta\) C. \(\frac{\partial w}{\partial r} = -f_x \sin \theta + f_y \cos \theta\) D. \(\frac{\partial w}{\partial r} = -f_x \cos \theta - f_y \sin \theta\)

16. For the function \( F(x, y) = \sqrt{x^2 + 2y^{0.7}} \), what is the approximate value of \( F(3.02, 2.1) \) using the given partial derivatives at (3, 2)?

A. 3.4994 B. 3.5328 C. 3.264 D. 3.60

17. When transforming variables with \(x = \ln\sqrt{u^2 - v^2}\) and \(y = \frac{1}{2} \ln\frac{u+v}{u-v}\), which kind of transform is this?

A. Linear B. Nonlinear logarithmic change of variables C. Polar coordinate transform D. Inverse function transform

18. When applying the Mean Value Theorem, given f'(x) is between 3 and 4, and f(3) = 2, what are the bounds for f(5)?

A. 8 ≤ f(5) ≤ 10 B. 5 ≤ f(5) ≤ 8 C. 10 ≤ f(5) ≤ 12 D. 3 ≤ f(5) ≤ 5

19. For the parametric substitution \( x = s^2 - t^2 \), \( y = 2st \), and a function \( u = f(x,y) \), what does the expression \( s \frac{\partial u}{\partial s} - t \frac{\partial u}{\partial t} \) simplify to?

A. \( 2 (s^2 - t^2) \frac{\partial f}{\partial y} \) B. \( 2 (s^2 + t^2) \frac{\partial f}{\partial x} \) C. \( (s^2 + t^2) \frac{\partial f}{\partial y} \) D. Zero

20. How is the area between the curves \( y_1 = (x-1)^2 \) and \( y_2 = 4 - (x-3)^2 \) between their intersection points computed?

A. By integrating \( y_1 - y_2 \) from 1 to 3 B. By integrating \( y_2 - y_1 \) from 1 to 3 C. By summing the integrals of \( y_1 \) and \( y_2 \) separately D. By calculating the difference of definite integrals of \( y_1 \) only

21. What is the formula for the volume of the solid bounded by the quarter circle \(x^2 + y^2 \leq 4\), \(x,y \geq 0\), and \(z = 6 - x y\), \(z \geq 0\)?

A. \(6 \pi\) B. \(6 \pi - 2\) C. \(4 \pi\) D. \(8 \pi - 4\)

22. The integral ∫ (x² + 2y) dx + xy dy over a path where y = 4x² can be simplified by substituting dy as what expression?

A. 8x dx B. 2x dx C. 4x dx D. 16x dx

23. For the volume V of the region bounded by the quarter circle x² + y² ≤ 4 with z = 6 − xy, how is the volume expression given as an iterated integral?

A. V = ∫₀² ∫₀^{√(4−x²)} (6 − xy) dx dy B. V = ∫₀² ∫₀^{√(4−x²)} (6 − xy) dy dx C. V = ∫₀^{√(4−x²)} ∫₀² (6 − xy) dy dx D. V = ∫₀^{√(4−x²)} ∫₀² (6 − xy) dx dy

24. Which of the following represents Euler’s theorem for a function \( f(x, y) \) homogeneous of degree \( n \)?

A. \( \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} = n f \) B. \( x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = f \) C. \( x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f \) D. \( f(tx, ty) = t f(x, y) \)

25. What is the value of the integral ∫₀² √(4 − x²) dx representing a quarter circle's area of radius 2?

A. π/2 B. π C. 2π D. 4π

26. Which of the following statements about the function f(x) = √x is correct regarding its Maclaurin expansion?

A. It can be expanded as a Maclaurin series around x=0 because derivative exists there B. It cannot be expanded as a Maclaurin series because derivative at x=0 is undefined C. It converges everywhere D. It has an infinite radius of convergence

27. When minimizing \( f = x^2 + y^2 + z^2 \) subject to \( ax + by + cz = p \) using Lagrange multipliers, what is the formula for the minimizing values of \( x, y, z \)?

A. \( x = \frac{a}{p}, y = \frac{b}{p}, z = \frac{c}{p} \) B. \( x = \frac{p a}{a^2 + b^2 + c^2}, y = \frac{p b}{a^2 + b^2 + c^2}, z = \frac{p c}{a^2 + b^2 + c^2} \) C. \( x = \lambda a, y = \lambda b, z = \lambda c \) D. \( x = \frac{p}{a^2 + b^2 + c^2}, y = \frac{p}{a^2 + b^2 + c^2}, z = \frac{p}{a^2 + b^2 + c^2} \)

28. In the problem involving partial derivatives and the transformation \( x + y = \ln(u+v) \), \( x - y = \ln(u-v) \), what forms do \( x \) and \( y \) take in terms of \( u \) and \( v \)?

A. \( x = \ln(u), y = \ln(v) \) B. \( x = \frac{1}{2} \ln(u^2 - v^2), y = \frac{1}{2} \ln\left(\frac{u+v}{u-v}\right) \) C. \( x = u+v, y = u-v \) D. \( x = \ln(u+v) - \ln(u - v), y = \ln(u + v) + \ln(u - v) \)

29. What is the radius of convergence of the Maclaurin series expansion of the function \( f(x) = \frac{1}{\sqrt{4 - x}} \)?

A. 1 B. 2 C. 4 D. Infinity

30. What is the condition on the derivative \( f'(x) \) for the Mean Value Theorem application given \( f(3) = 2 \), that allows to estimate the range of \( f(5) \)?

A. \( f'(x) \) is constant between 3 and 5 B. \( 3 \leq f'(x) \leq 4 \) for \( x \in [3,5] \) C. \( f'(x) \) is zero in the interval D. No condition is needed on \( f'(x) \) for MVT

31. For a homogeneous function \(f(x,y)\) of degree \(n\), which identity holds true?

A. \(x \frac{\partial f}{\partial x} - y \frac{\partial f}{\partial y} = n f\) B. \(\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} = n f\) C. \(x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f\) D. \(x \frac{\partial f}{\partial y} + y \frac{\partial f}{\partial x} = n f\)

32. Which of the following expressions represents the second derivative of function f in new variables u and v indicating harmonic functions?

A. (u² − v²)(f_uu − f_vv) B. (u² + v²)(f_uu + f_vv) C. (u + v)(f_uu + f_vv) D. (u − v)(f_u − f_v)

33. Given the function \(c(p) = p^{0.5p^3 + 17}\) and \(p(t) = 3.1 + 0.1 t^2\), what rule is primarily used to find \(\frac{dc}{dt}\)?

A. Product rule B. Chain rule C. Quotient rule D. Integration by parts

34. For the integral \(\int (x^2 + 2y) dx + xy dy\) over the curve \(y = 4x^2\) from \((0,0)\) to \((1,4)\), which of the following expresses the correct substitution for \(dy\)?

A. \(dy = 8x dx\) B. \(dy = 8x^2 dx\) C. \(dy = 2x dx\) D. \(dy = 4x dx\)

35. Given the change of variables x + y = ln(u + v) and x − y = ln(u − v), what is the expression of y in terms of u and v?

A. y = (1/2) ln(u² − v²) B. y = (1/2) ln((u + v)/(u − v)) C. y = ln(u + v) − ln(u − v) D. y = ln(u − v)/2

36. Why can't the function \(f(x) = \sqrt{x}\) be expanded by Maclaurin’s theorem?

A. It is not continuous at zero. B. It is not differentiable at zero. C. It has an essential singularity at zero. D. The function is not defined on negative x.

37. When using integration by parts on \( I_n = \int x^n e^x dx \), what is the reduction formula relating \( I_n \) to \( I_{n-1} \)?

A. \( I_n = x^n e^x + n I_{n-1} \) B. \( I_n = x^n e^x - n I_{n-1} \) C. \( I_n = e^x - n I_{n-1} \) D. \( I_n = n x^{n-1} e^x - I_{n-1} \)

38. What is the final simplified result of the volume V = ∫∫R (6 − xy) dA over the quarter circle in the example?

A. 6π + 2 B. 6π − 2 C. 2π − 6 D. 2π + 6

39. What is the general form of the integral I₄ for the integral In = ∫ xⁿ eˣ dx using integration by parts?

A. (x⁴ − 4x³ + 12x² + 24x − 24)eˣ B. (x⁴ − 4x³ + 12x² − 24x + 24)eˣ C. (x⁴ + 4x³ − 12x² − 24x + 24)eˣ D. (x⁴ − 4x³ − 12x² + 24x + 24)eˣ

40. The area bounded by the curves \(y_1 = (x - 1)^2\) and \(y_2 = 4 - (x - 3)^2\) between their intersection points is:

A. \(\frac{8}{3}\) B. \(2\pi\) C. \(4\) D. \(6\)

41. What is the radius of convergence for the Maclaurin series of f(x) = 1/√(4 − x)?

A. 2 B. 4 C. 1 D. 8

42. In the application of Rolle’s theorem to f(x) = (x − a)^m (x − b)^n on [a, b], how is the critical point c where f'(c) = 0 expressed?

A. c = (m + n)(a + b) B. c = (mb + na) / (m + n) C. c = (m a + n b) / (m − n) D. c = (m b − n a) / (m + n)

43. Which statement correctly describes the relationship between the second derivatives \(\partial_x^2 f - \partial_y^2 f\) and transformations in variables \(u,v\) where \(x = \frac{u}{u^2 - v^2}\) and \(y = \frac{-v}{u^2 - v^2}\)?

A. The relationship is independent of \(u,v\) B. \( \partial_x^2 f - \partial_y^2 f = (u^2 - v^2)( \partial_u^2 f - \partial_v^2 f) \) C. \( \partial_x^2 f - \partial_y^2 f = \partial_u^2 f + \partial_v^2 f \) D. The relation does not involve \(u\) and \(v\)

44. For the function \(T(x) = 100 - 30x + 3x^2\), what is the value of \(x\) minimizing \(T\)?

A. \(x = 3\) B. \(x = 5\) C. \(x = 10\) D. \(x = 15\)

45. How is the volume of the solid bounded by the quarter circle \( x^2 + y^2 \leq 4 \), \( x \geq 0 \), \( y \geq 0 \), and \( z = 6 - xy \), with \( z \geq 0 \), expressed?

A. \( V = 6\pi + 2 \) B. \( V = 6\pi - 2 \) C. \( V = 2 \pi \) D. \( V = \pi \)

46. What is the correct form for the derivative of FA(x) = A eˣ cos x + B eˣ sin x?

A. F′A(x) = eˣ[(A + B) cos x + (B − A) sin x] B. F′A(x) = eˣ[(A − B) cos x + (B + A) sin x] C. F′A(x) = eˣ[(A − B) sin x + (A + B) cos x] D. F′A(x) = eˣ[(A + B) sin x + (B − A) cos x]

47. Given the function \( f(x, y) \) is homogeneous of degree \( n \), Euler’s theorem states that:

A. \( \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} = n f \) B. \( x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = f \) C. \( x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f \) D. \( x \frac{\partial f}{\partial y} + y \frac{\partial f}{\partial x} = n f \)

48. In the integral \( \int_0^1 (x^2 + 2 y) dx + \int_0^1 xy dy \) with \( y = 4x^2 \), which substitution simplifies evaluation?

A. Substitute \( dy = 8x dx \) and integrate with respect to \( x \) B. Substitute \( dx = 2y dy \) and integrate with respect to \( y \) C. Use Green's theorem D. Use polar coordinates

49. When minimizing f = x² + y² + z² subject to ax + by + cz = p, what is the expression for the minimum point x?

A. p a / (a² + b² + c²) B. p / a C. a / p D. p (a² + b² + c²)

50. For the function u = x³ + y³ with parametric form x = a cos t, y = b sin t, how do you express du/dt?

A. 3a³ cos² t sin t + 3b³ sin² t cos t B. 3a³ sin² t cos t − 3b³ cos² t sin t C. −3a³ cos² t sin t + 3b³ sin² t cos t D. −3a³ sin² t cos t − 3b³ cos² t sin t