Free Printable MATH pt1

1.

Given a function z = $5x^3y^2 - x^4y^8$ . Find the partial derivative of the function with respect to $z_x$ :

a)

$15x^2y^2 - 4x^3y^8$

b)

$15x^2y + 3x^3y^8$

c)

$-15x^2y - 3x^2y^8$

d)

$15x^2 - 3x^2$

e)

$15x^2 + 3x^2$

2.

Given a function z = $5x^3y^2 - x^4y^8$ . Find the partial derivative of the function with respect to $z_{xy}$ :

a)

$10x^3 - 56x^4y^6$

b)

$10x^3y + 8x^4y^7$

c)

$30x^2y - 12x^3y^8$

d)

$30x^2y - 32x^3y^6$

e)

$10x^3 + 56x^4y^6$

3.

A first order ordinary differential equation has the form

a)

F(x, y, y') = 0

b)

F(x, y) = 0

c)

F(x, y, y', y'') = 0

d)

F(y, y', y'') = 0

e)

F(x', y', y') = 0

4.

A second order differential equation has the form

a)

F(y'', y', y, x) = 0

b)

F(x, y') = 0

c)

F(x, y, y', y'') = 0

d)

F(y, y', y'') = 0

e)

F(x', y', y') = 0

5.

A first order linear equation (n = 1) looks like

a)

y' + P(x)y = Q(x)

b)

y' + P(x) = Q(x)

c)

y' + P(x)y'' = Q(x)

d)

y' + P(x)y = 0

e)

y' + y = Q(x)

6.

The Bernoulli equation is given by

a)

$dy/dx + P(x)y = Q(x)y^n$

b)

$dy/dx + y = Q(x)y^n$

c)

$\frac{dy}{dx} = Q(x)y^n$

d)

$dy/dx + P(x)y = y^n$

7.

Properties of double integrals, monotonicity: Which of the following statements is true?

a)

∫∫_K f(x, y)dA ≥ ∫∫_K g(x, y)dA if f(x, y) ≥ g(x, y) on D

b)

∫∫_K f(x, y)dA = ∫∫_K g(x, y)dA if f(x, y) = g(x, y) on D

c)

∫∫_K f(x, y)dA = ∫∫_K f(x, y)dA ± ∫∫_K g(x, y)dA

d)

∫∫_K f(x, y)dA ≥ ∫∫_K g(x, y)dA if f(x, y) = g(x, y) on K

e)

∫∫_K f(y', x, y)dA ≥ ∫∫_K g(x, y)dA if f(y') ≥ g(x, y) on D

8.

Properties of double integrals, constant multiple rule: Which of the following statements is true?

a)

∫∫_K λf(x, y)dA = λ∫∫_K f(x, y)dA for any const λ

b)

∫∫_K λf(x, y)dA = C∫∫_K f(x, y)dA for any const λ

c)

∫∫_K f(x, y)dA = ∫∫_K f(x, y)dA ± ∫∫_K g(x, y)dA

d)

∫∫_K λf(x, y)dA > λ∫∫_K f(x, y)dA

e)

∫∫_K λf(x, y)dA < λ∫∫_K f(x, y)dA for any const λ

9.

A function F(x, y) is called homogeneous of degree n if

a)

$F(\lambda x, \lambda y) = \lambda^n F(x, y)$

b)

$F(x, y) = \lambda^n F(x, y)$

c)

$F(\lambda y) = \lambda^n F(x, y)$

d)

F(λx, λy) = F(y)

e)

F( $\lambda x$ ) = $\lambda^n F(y)$

10.

What is the total differential of the function z = f(x, y)?

a)

dz = (∂z/∂x) dx + (∂z/∂y) dy

b)

z' = lim_{y→0} (f(x, y + Δy) - f(x, y))/Δy

c)

dz = lim_{Δy→0} (f(x, y + Δy) - f(x, y))/Δy

d)

dz = lim_{Δx→0} (f(x + Δx, y) - f(x, y))/Δx

e)

z' = lim_{Δx→0} (f(x + Δx, y) - f(x, y))/Δx

11.

Evaluate the iterated integrals $\int_{2}^{3} \int_{2}^{4} (40 - 2xy) dy dx$

a)

112

b)

10

c)

19

d)

113

e)

114

12.

Evaluate the iterated integrals ∫ from y=1 to y=3 ∫ from x=2 to x=4 (40 - 2xy) dxdy

a)

112

b)

10

c)

19

d)

113

e)

114

13.

Evaluate the iterated integrals ∫ from y=0 to y=1 ∫ from x=0 to x=2 (4 - x - y) dxdy

a)

5

b)

10

c)

19

d)

125

e)

25

14.

Evaluate the iterated integrals ∫ from x=0 to x=2 ∫ from y=0 to y=1 (4 - x - y) dydx

a)

A) 5

b)

B) 10

c)

C) 19

d)

D) 125

e)

E) 25

15.

Evaluate the iterated integrals ∫ from x=0 to x=2 ∫ from y=0 to y=1 (10 - 8x² - 2y²) dydx

a)

23/3

b)

10

c)

19

d)

16/3

e)

25

16.

Evaluate the triple integrals ∫ from x=0 to x=2 ∫ from y=0 to y=3 ∫ from z=-1 to z=2 (2xy²z³) dzdydx

a)

648

b)

500

c)

19

d)

16/3

e)

25

17.

Given a function z = $5x^3y^2 - x^4y^8$ . Find the partial derivative of the function with respect to $z_{xx}$ .

a)

$30x^2y^2 - 12x^2y^8$

b)

$30x^2y^2 + 12x^2y^8$

c)

$10x^3 - 56x^4y^7$

d)

$30x^2y - 32x^3y^7$

e)

$10x^3y - 8x^4y^7$

18.

Properties of double integrals, sum and difference rule:

a)

∬_R (f(x, y) ± g(x, y))dA = ∬_R f(x, y)dA ± ∬_R g(x, y)dA

b)

∬_R f(x, y)dA = ∬_R f(x, y)dA ± ∬_R g(x, y)dA

c)

∬_R k f(x, y)dA = ∬_R f(x, y)dA ± ∬_R g(x, y)dA

d)

∬_R g(x, y)dA = ∬_R f(x, y)dA ± ∬_R g(x, y)dA

e)

∬_R f(x, y)dA = ∬_R f(x, y)dA - ∬_R g(x, y)dA

19.

Given a function z = $5x^3y^2 - x^4y^8$ . Find the partial derivative of the function with respect to $z_y$ .

a)

$10x^3y - 8x^4y^7$

b)

$10x^3y + 8x^4y^7$

c)

$30x^3y - 24x^3y^7$

d)

$15x^2y - 3x^2y^8$

e)

$-10x^3y - 8x^3y^7$

20.

Given a function z = $3x^2y^4 + e^{2y}$ . Find the partial derivative of the function with respect to $z_y$ .

a)

$12x^2y^3 + 2e^{2y}$

b)

$xy^4 + e^{2y}$

c)

$6xy^4 + e^{2y}$

d)

$6xy^4 - e^{2y}$

e)

$24xy^4 + 2e^{2y}$

21.

Given a function z = $5x^3y^2 - x^4y^8$ . Find the partial derivative of the function with respect to $z_x$ .

a)

$15x^2y^2 - 4x^3y^8$

b)

$15x^2y + 3x^3y^8$

c)

$-15x^2y - 3x^2y^8$

d)

$15x^2 - 3x^2$

e)

$15x^2 + 3x^2$

22.

Given a function z = $5x^3y^2 - x^4y^8$ . Find the partial derivative of the function with respect to $z_{xy}$ .

a)

30x²y-32x³y⁷

b)

10x³-56x⁴y⁷

c)

30xy²-12x²y⁸

d)

30x²y+32x³y⁷

e)

10x³+56x⁴y⁷

23.

A first order ordinary differential equation has the form:

a)

F(x, y, y') = 0

b)

F(x, y') = 0

c)

F(x, y, y', y'') = 0

d)

F(y, y', y'') = 0

e)

F(x', y, y') = 0

24.

A second order differential equation has the form:

a)

F(y'', y', y, x) = 0

b)

F(x, y') = 0

c)

F(x, y, y', y'') = 0

d)

F(y, y', y'') = 0

e)

F(x', y, y') = 0

25.

A first order linear equation (n = 1) looks like:

a)

y' + P(x)y = Q(x)

b)

y' + P(x) = Q(x)

c)

y' + P(x)y'' = Q(x)

d)

y' + P(x)y = 0

e)

y' + y = Q(x)

26.

The Bernoulli equation is given by:

a)

$dy/dx + P(x)y = Q(x)y^n$

b)

$\frac{dy}{dx} + y = Q(x)y^n$

c)

$\frac{dy}{dx} = Q(x)y^n$

d)

$dy/dx + P(x)y = y^n$

e)

$dy/dx + P(x) = Q(x)y^n$

27.

Properties of double integrals, monotonicity:

a)

∫∫_K f(x, y)dA ≥ ∫∫_K g(x, y)dA if f(x, y) ≥ g(x, y) on D

b)

∫∫_K f(x, y)dA = ∫∫_K g(x, y)dA if f(x, y) ≥ g(x, y) on D

c)

∫∫_K f(x, y)dA = ∫∫_K f(x, y)dA ± ∫∫_K g(x, y)dA

d)

∫∫_K f(x, y)dA ≥ ∫∫_K g(x, y)dA if f(x, y) = g(x, y) on K

e)

∫∫_K f(y', x, y)dA ≥ ∫∫_K g(x, y)dA if f(y) ≥ g(x, y) on D

28.

Properties of double integrals, constant multiple rule: Which of the following is correct?

a)

∫_A λf(x, y)dA = λ ∫_A f(x, y)dA for any const λ

b)

∫_A λf(x, y)dA = C ∫_A f(x, y)dA for any const λ

c)

∫_A kf(x, y)dA = ∫_A f(x, y)dA ± ∫_A g(x, y)dA

d)

∫_A λf(x, y)dA > λ ∫_A f(x, y)dA

e)

∫_A λf(x, y)dA < λ ∫_A f(x, y)dA for any const λ

29.

Given a function z = 5x³y² - x⁴y⁸. Find the partial derivative of the function with respect to z_y.

a)

10x³y - 8x⁴y⁷

b)

10x³y + 8x³y⁷

c)

30x³y - 24x³y⁷

d)

15x²y - 3x²y⁸

e)

-10x³y - 8x³y⁷

30.

What is the total differential of the function z = f(x, y)?

a)

dz = (∂z/∂x)dx + (∂z/∂y)dy

b)

z'_y = lim_{y→0} [(f(x, y + Δy) - f(x, y))/Δy]

c)

dz_y = lim_{y→0} [(f(x, y + Δy) - f(x, y))/Δy]

d)

dz = lim_{x→0} [(f(x + Δx, y) - f(x, y))/Δx]

e)

z'_x = lim_{Δx→0} [(f(x + Δx, y) - f(x, y))/Δx]

31.

Evaluate the iterated integrals $\int_{i=2}^{4} \int_{j=3}^{4} (40 - 2xy) dy dx$

a)

112

b)

10

c)

19

d)

113

e)

114

32.

Evaluate the iterated integrals $\int_{x=1}^{3} \int_{y=2}^{4} (40 - 2xy) dx dy$

a)

112

b)

10

c)

19

d)

113

e)

114

33.

Evaluate the iterated integrals ∫₀²∫₀¹(4 - x - y)dxdy

a)

5

b)

10

c)

19

d)

125

e)

25

34.

Evaluate the iterated integrals ∫₀¹∫₀²(4 - x - y)dydx

a)

5

b)

10

c)

19

d)

125

e)

25

35.

Evaluate the iterated integrals ∫₀¹∫₀²(10 - 8x² - 2y²)dydx

a)

23/3

b)

10

c)

19

d)

16/3

e)

25

36.

Evaluate the triple integrals ∫₀²∫₀³∫₋₁²2xy²z²dzdydx

a)

648

b)

500

c)

19

d)

16/3

e)

25

37.

A necessary criterion for the convergence of a series

a)

limₙ→∞ aₙ = 0

b)

limₙ→∞ (uₙ₊₁/uₙ) = l

c)

limₙ→∞ (√uₙ) = l

d)

limₙ→∞ Sₙ = S

e)

limₙ→∞ aₙ = 1

38.

The sum of a series $\sum_{n=1}^\infty a_n$ .

a)

S = a_1 + a_2 + ... + a_n+...;

b)

S = a_1 - a_2 - ... - a_n;

c)

S_n = a_1 + a_3;

d)

S = a_1 + a_2 + ... + a_n;

e)

S_n = a_1;

39.

Properties of double integrals, sum and difference rule:

a)

∬_A [f(x, y) ± g(x, y)] dA = ∬_A f(x, y) dA ± ∬_A g(x, y) dA

b)

∬_A [f(x, y) - g(x, y)] dA = ∬_A f(x, y) dA ± ∬_A g(x, y) dA

c)

∬_A f(x, y) dA = ∬_A f(x, y) dA ± ∬_A g(x, y) dA

d)

∬_A [g(x, y)] dA = ∬_A f(x, y) dA ± ∬_A g(x, y) dA

e)

∬_A f(x, y) dA = ∬_A f(x, y) dA - ∬_A g(x, y) dA

40.

Given a function z = $5x^3 y^2 - x^4 y^8$ . Find the partial derivative of the function with respect to $z'_y$ .

a)

$10x^3 y - 8x^4 y^7$

b)

$10x^3 y + 8x^4 y^7$

c)

$30x^3 y - 24x^3 y^7$

d)

$15x^2 y - 3x^7 y^8$

e)

$-10x^3 y - 8x^4 y^7$

41.

Given a function z = $3x^2 y^4 + e^{2y}$ . Find the partial derivative of the function with respect to $z'_y$ .

a)

$12x^2y^3+2e^{2y}$

b)

$x y^4 + e^{2y}$

c)

$6x y^4 + e^{2y}$

d)

$6x y^4 - e^{2y}$

e)

$24x y^4 + 2e^{2y}$

42.

Given a function z = $5x^3 y^2 - x^4 y^8$ . Find the partial derivative of the function with respect to $z''_{xx}$ .

a)

$30x y^2 - 12x^2 y^8$

b)

$30x y^2 + 12x^2 y^8$

c)

$10x^3 - 56x^4 y^7$

d)

$30x^2 y - 32x^3 y^7$

e)

$10x^3 y - 8x^4 y^7$

43.

Given a function z = $5x^3 y^2 - x^4 y^8$ . Find the partial derivative of the function with respect to $z''_{yy}$ .

a)

$10x^3 - 56x^4 y^6$

b)

$10x^3 y + 8x^4 y^7$

c)

30xy²-12x²y⁸

d)

30x²y-32x³y⁷

e)

10x³+56x⁴y⁶

44.

Given a function z = $5x^3y^2 - x^4y^8$ . Find the partial derivative of the function with respect to $z_{xy}$ .

a)

$30x^2y−32x^3y^7$

b)

30xy²-12x²y⁸

c)

$10x^3+56x^4y^7$

d)

30x²y+32x³y⁷

e)

10x³-56x⁴y⁷

45.

Given a function z = $5x^3y^2 - x^4y^8$ . Find the partial derivative of the function with respect to $z_y$ .

a)

$10x^3y - 8x^4y^7$

b)

$10x^3y + 8x^3y^7$

c)

$30x^3y - 24x^3y^7$

d)

$15x^2y - 3x^2y^8$

e)

$-10x^3y - 8x^3y^7$

46.

Given a function z = $3x^2y^4 + e^{2y}$ . Find the partial derivative of the function with respect to $z_x$ .

a)

$6xy^4$

b)

$2axy + e^{2y}$

c)

$2e^{2y}$

d)

$4xy^4 + e^{2y}$

e)

$x^4y + 2e^{2y}$

47.

Given a function z = $3x^2y^4 + e^{2y}$ . Find the partial derivative of the function with respect to $z_y$ .

a)

$12x^2y^3 + 2e^{2y}$

b)

$xy^4 + e^{2y}$

c)

$6xy^4 + e^{2y}$

d)

$6xy^4 - e^{2y}$

e)

$24xy^4 + 2e^{2y}$

48.

Given a function z = $3x^2y^4 + e^{2y}$ . Find the partial derivative of the function with respect to $z_{xx}$ .

a)

$6y^4$

b)

$6xy + e^{2y}$

c)

$2e^{2y}$

d)

$xy^4 + e^{2y}$

e)

$y^4 + 2e^{2y}$

49.

Given a function z = $sin(x^2 + y^2)$ . Find the partial derivative of the function with respect to $z''_{xx}$ .

a)

A) $2cos(x^2 + y^2) · 4x^2sin(x^2 + y^2)$

b)

B) $2cos(x^2 + y^2) + 4x^2sin(x^2 + y^2)$

c)

C) -4x^2 $sin(x^2 + y^2)$

d)

D) $2cos(x^2 + y^2)$

e)

E) $2sin(x^2 + y^2)$

50.

Given a function z = $3x^2y^4 + e^{2y}$ . Find the partial derivative of the function with respect to $z''_{yy}$ .

a)

$36x^2y^2 + 4e^{2y}$

b)

$y^4 + e^{2y}$

c)

C) $36x^2y^2 + e^{2y}$

d)

D) $36x^2y^2 - 4e^{2y}$

e)

E) $24xy^4 + 2e^{2y}$

51.

Given a function z = $3x^2y^4 + e^{2y}$ . Find the partial derivative of the function with respect to $z'_y$ .

a)

$24xy^3$

b)

B) $24xy + e^{2y}$

c)

C) $24xy^3 + e^{2y}$

d)

D) $4xy^4 + e^{2y}$

e)

E) $x^2y^4 + 2e^{2y}$

52.

Given a function z = $sin(x^2 + y^2)$ . Find the partial derivative of the function with respect to $z'_x$ .

a)

$2x\cos(x^2+y^2)$

b)

B) $2x\sin(x^2+y^2)$

c)

C) $2ycos(x^2 + y^2)$

d)

D) $2xsinx^2$

e)

E) $2y\sin(x^2+y^2)$

53.

Given a function z = $sin(x^2 + y^2)$ . Find the partial derivative of the function with respect to $z'_y$ .

a)

A) $2ycos(x^2 + y^2)$

b)

B) $2xcos(x^2 + y^2)$

c)

C) $2xsin(x^2 + y^2)$

d)

D) $2ysin(x^2 + y^2)$

e)

E) $2ysin y^2$

54.

Given a function z = $sin(x^2 + y^2)$ . Find the partial derivative of the function with respect to $z''_{xx}$ .

a)

$2\cos(x^2+y^2)-4x^2\sin(x^2+y^2)$

b)

B $2cos(x^2 + y^2) + 4x^2sin(x^2 + y^2)$

c)

-4y²sin(x²+y²)

d)

2cos(x²+y²)

e)

2sin(x²+y2)

55.

Given a function z = $sin(x^2 + y^2)$ . Find the partial derivative of the function with respect to z'_yy.

a)

$-4x^2sin(x^2 + y^2)$

b)

$2cos(x^2 + y^2)$

c)

$2sin(x^2 + y^2)$

d)

2cos(x²+y²)-4y²sin(x²+y²)

e)

2cos(x²+y²)+4y²sin(x²+y²)

56.

Given a function z = $5x^3y^2 - x^4y^8$ . Find the partial derivative of the function with respect to $z''_{yy}$ .

a)

$10x^3 - 56x^4y^6$

b)

$10x^3y + 8x^3y$

c)

$30x^2 - 12x^3y^8$

d)

$30x^2y - 32x^3y^6$

e)

$10x^3 + 56x^4y^6$

57.

Given a function z = $sin(x^2 + y^2)$ . Find the partial derivative of the function with respect to $z''_{xy}$ .

a)

$-4xy \sin(x^2 + y^2)$

b)

$-4y^2 sin(x^2 + y^2)$

c)

$-4x^2 \sin(x^2 + y^2)$

d)

$2cos(x^2 + y^2)$

e)

$2sin(x^2 + y^2)$

58.

Given a function z = $cos(x^2 + y^2)$ . Find the partial derivative of the function with respect to $z'_x$ .

a)

$-2sin(x^2 + y^2)$

b)

$2sin(x^2 + y^2)$

c)

$-4x \sin(x^2 + y^2)$

d)

$2cos(x^2 + y^2)$

e)

$2sin(x^2 + y^2)$

59.

Given a function z = $cos(x^2 + y^2)$ . Find the partial derivative of the function with respect to $z'_y$ .

a)

$-2ysin(x^2 + y^2)$

b)

$-2xsin(x^2 + y^2)$

c)

$-4y \sin(x^2 + y^2)$

d)

$2cos(x^2 + y^2)$

e)

$2sin(x^2 + y^2)$

60.

Given a function z = $cos(x^2 + y^2)$ . Find the partial derivative of the function with respect to $z''_{xx}$ .

a)

A) $-4x^2sin(x^2 + y^2)$

b)

B) $2cos(x^2 + y^2)$

c)

C) $2sin(x^2 + y^2)$

d)

-2sin(x²+y²)-4y²cos(x²+y²)

e)

-2sin(x²+y²)-4x²cos(x²+y²)

61.

Given a function z = $cos(x^2 + y^2)$ . Find the partial derivative of the function with respect to $z_{yy}$ .

a)

-2 $sin(x^2 + y^2)$ - 4 $y^2\cos(x^2+y^2)$

b)

-2 $sin(x^2 + y^2)$ - 4 $x^2 cos(x^2 + y^2)$

c)

$2cos(x^2 + y^2)$

d)

$2sin(x^2 + y^2)$

e)

4xy cos(x^2 + y^2)

62.

Given a function z = $cos(x^2 + y^2)$ . Find the partial derivative of the function with respect to $z_{yy}$ .

a)

A) $-2sin(x^2 + y^2) - 4y^2 cos(x^2 + y^2)$

b)

B) -2 $sin(x^2 + y^2)$ - 4x^2 $cos(x^2 + y^2)$

c)

C) $2cos(x^2 + y^2)$

d)

D) $2sin(x^2 + y^2)$

e)

E) $4xy cos(x^2 + y^2)$

63.

Given a function z = $x^5 y^2 + x^2 ln y$ . Find the partial derivative of the function with respect to $z_x$ .

a)

$5x^4 y^2 + 2x \ln y$

b)

B) $20x^3 y^2 + 2 \ln y$

c)

C) $2x y + x^2 / y$

d)

D) $2x^5 - x^2 / y^2$

e)

E) $10x^4 y^2 + 2x / y$

64.

Given a function z = $x^5 y^2 + x^2 ln y$ . Find the partial derivative of the function with respect to $z_y$ .

a)

$2x^5 y + x^2 / y$

b)

B) $2x^5 - x^2 / y^2$

c)

C) $5x^4 y^2 + 2x \ln y$

d)

D) $20x^3 y^2 + 2 ln y$

e)

E) $10x^4 y^2 + 2x / y$

65.

Given a function z = $x^5 y^2 + x^2 ln y$ . Find the partial derivative of the function with respect to $z_{xx}$ .

a)

A) $20x^3 y^2 + 2 ln y$

b)

B) $5x^4 y^2 + 2x \ln y$

c)

C) $2x^5 y + x^2 / y$

d)

D) $2x^5 - x^2 / y^2$

e)

E) $10x^{4^{ }}$

66.

Given a function z = $x^5y^2+x^2\ln y$ . Find the partial derivative of the function with respect to $z'_{yy}$ .

a)

$2x^5 - x^2 / y^2$

b)

$2x^5 y + x^2 / y$

c)

$5x^4 y^2 + 2x \ln y$

d)

20x^3 y^2 + 21 ln y

e)

$10x^4 y + 2x / y$

67.

Given a function z = $x^2 / y + x y^7$ . Find the partial derivative of the function with respect to $z'_x$ .

a)

$2x / y + y^7$

b)

$-x^2 / y^2 + 7x y^6$

c)

C) 2 / y

d)

D) $2x^2 / y^3 + 42x y^5$

e)

E) $-2x / y^2 + 7y^6$

68.

Given a function z = $x^2 / y + x y^7$ . Find the partial derivative of the function with respect to $z'_y$ .

a)

$-x^2 / y^2 + 7x y^6$

b)

B) $2x / y + y^7$

c)

C) 2 / y

d)

D) $2x^2 / y^3 + 42x y^5$

69.

Given a function z = $x^2/y + xy^7$ . Find the partial derivative of the function with respect to $z'_{xx}$ .

a)

A) 2/y

b)

$x^2/y^2 + 7xy^6$

c)

C) $2x/y + y^7$

d)

D) $2x^2/y^3 + 42xy^5$

e)

E) $2x/y^2 + 7y^6$

70.

Given a function z = $x^5y^2 + x^2 \ln y$ . Find the partial derivative of the function with respect to $z'_y$ .

a)

$2x^5y+\frac{x^2}{y}$

b)

B) $2x^5 - \frac{x^2}{y^2}$

c)

C) $5x^5y + 2x \ln y$

d)

D) $20x^5y + 2ln y$

e)

E) $10x^5y + 2x/y$

71.

Given a function z = $x^2/y + xy^7$ . Find the partial derivative of the function with respect to $z'_{yy}$ .

a)

$2x^2/y^3 + 42xy^5$

b)

B) $2x/y + y^7$

c)

C) 2/y

d)

D) $x^2/y^2 + 7xy^6$

e)

E) $2x/y^2 + 7y^6$

72.

Given a function u(M) = $e^{x^2 + y^2 + z^2}$ . Find the partial derivative of the function with respect to u'_x.

a)

A) $2xe^{x^2 + y^2 + z^2}$

b)

$e^{x^2}$

c)

C) $2xe^{x^2}$

d)

D) $e^{x^2 + y^2 + z^2}$

e)

E) $2xyze^{x^2 + y^2 + z^2}$

73.

u(M) = $e^{x^2 + y^2 + z^2}$ find the partial derivative of the function with respect to u'_y.

a)

A) $2ye^{x^2 + y^2 + z^2}$

b)

B) $2xyze^{x^2 + y^2 + z^2}$

c)

C) $2ye^{y^2}$

d)

D) $e^{x^2 + y^2 + z^2}$

e)

$e^{y^2}$

74.

Given a function u(M) = $\sqrt{x^2 + y^2 + z^2}$ . Find the partial derivative of the function with respect to $u'_z$ .

a)

$\frac{z}{\sqrt{x^2 + y^2 + z^2}}$

b)

$\frac{y}{\sqrt{x^2 + y^2 + z^2}}$

c)

$\frac{x}{\sqrt{x^2 + y^2 + z^2}}$

d)

$\frac{2}{\sqrt{x^2 + y^2 + z^2}}$

e)

$\frac{1}{2\sqrt{x^2 + y^2 + z^2}}$

75.

u(M) = $e^{x^2 + y^2 + z^2}$ find the partial derivative of the function with respect to u'_z.

a)

A) $2ze^{x^2 + y^2 + z^2}$

b)

B) $2ze^{z^2}$

c)

C) $e^{x^2 + y^2 + z^2}$

d)

$e^{z^2}$

e)

E) $2xyze^{x^2 + y^2 + z^2}$

76.

Given a function u(M) = $sqrt{x^2 + y^2 + z^2}$ . Find the partial derivative of the function with respect to $u'_x$ . sqrt это корень

a)

A) $x / \sqrt{x^2 + y^2 + z^2}$

b)

B) $y / \sqrt{x^2 + y^2 + z^2}$

c)

C) $z / \sqrt{x^2 + y^2 + z^2}$

d)

D) $2 / \sqrt{x^2 + y^2 + z^2}$

e)

E) $\frac{1}{2} \sqrt{x^2 + y^2 + z^2}$

77.

Given a function u(M) = $sqrt{x^2 + y^2 + z^2}$ . Find the partial derivative of the function with respect to $u'_y$ .

a)

A) $y / \sqrt{x^2 + y^2 + z^2}$

b)

B) $x / \sqrt{x^2 + y^2 + z^2}$

c)

C) $z / \sqrt{x^2 + y^2 + z^2}$

d)

D) $2 / \sqrt{x^2 + y^2 + z^2}$

e)

E) $\frac{1}{2} \sqrt{x^2 + y^2 + z^2}$

78.

Given a function u(M) = $\sqrt{x^2 + y^2 + z^2}$ . Find the partial derivative of the function with respect to $u'_z$ .

a)

A) $z / \sqrt{x^2 + y^2 + z^2}$

b)

B) $y / \sqrt{x^2 + y^2 + z^2}$

c)

C) $x / \sqrt{x^2 + y^2 + z^2}$

d)

D) $2 / \sqrt{x^2 + y^2 + z^2}$

e)

E) $\frac{1}{2} \sqrt{x^2 + y^2 + z^2}$

79.

Given a function u(M) = $e^{x^2 + y^2 + z^2}$ . Find the partial derivative of the function with respect to u'_y.

a)

A) $2x e^{x^2 + y^2 + z^2}$

b)

B) $2x y e^{x^2 + y^2 + z^2}$

c)

C) $2y e^{x^2 + y^2 + z^2}$

d)

D) $e^{x^2 + y^2 + z^2}$

80.

Find the general solution of equation $y' = \frac{1}{\cos^2 3x}$

a)

A) y = (1/3) tg 3x + C

b)

B) y = tg 3x + C

c)

C) y = (1/3) ctg 3x + C

d)

D) y = - tg 3x + C

e)

E) y = - ctg 3x + C

81.

Find the general solution of equation $y' = \frac{1}{\sqrt{9 - x^2}}$

a)

A) y = arcsin(x/3) + C

b)

B) y = arctg(x/3) + C

c)

C) y = $ln(x + \sqrt{x^2 - 9})$ + C

d)

D) y = (1/3) arcsin(x/3) + C

e)

E) y = 3 arcsin(x/3) + C

82.

Find the general solution of equation y' = $\frac{1}{\sqrt{x^2 + 7}}$

a)

y = $ln(x + \sqrt{x^2 + 7}$ + C

b)

y = (1/7) arctg(x/7) + C

c)

y = - $ln(x + \sqrt{x^2 + 7}$ + C

d)

y = arctg(x/7) + C

e)

y = 7 arctg(x/7) + C

83.

Find the derivative of the function y = $(x^2 + 5)e^{3x}$

a)

y' = $(3x^2 + 2x + 15)e^{3x}$

b)

y' = (2x + 15) $e^{3x}$

c)

y' = $(3x^2 + 15)e^{3x}$

d)

y' = $(3x^2 - 2x + 15)e^{3x}$

e)

y' = $(3x^2 + 2x - 15)e^{3x}$

84.

Find the derivative of the function y = $(\frac{2}{5})x^2\sqrt{x}$

a)

y' = x√x

b)

y' = 2x√x

c)

$y' = x^2\sqrt{x}$

d)

y' = 5x√x

85.

Find the second derivative of the function y = $(\frac{2}{5})x^2\sqrt{x}$ .

a)

y'' = (3/2)√x

b)

y'' = √x

c)

y'' = 2√x

d)

y'' = 3√x

e)

y'' = -√x

86.

Find the derivative of the function y = $2^{5x+1}$ .

a)

$y' = 5·2^{5x+1}·ln 2$

b)

y' = $2^{5x+1}·ln 2$

c)

$y' = 5·2^{5x+1}$

d)

$y' = 2^{5x+1}$

e)

y' = $10^{5x+1}·ln 2$

87.

Find the derivative of the function y = $(x + 6)e^{3x}$ .

a)

$y' = (3x + 19)e^{3x}$

b)

$y' = (2x + 15)e^{3x}$

c)

y' = $(3x^2 + 15)e^{3x}$

d)

y' = $(3x + 7)e^{3x}$

e)

$y' = (3x + 18)e^{3x}$

88.

Find the derivative of the function y = x·tg4x.

a)

y' = $tg4x + \frac{4x}{\cos^2 4x}$

b)

$y' = \frac{4x}{\cos^2 4x}$

c)

$y' = \frac{1}{\cos^2 4x}$

d)

y' = tg4x

e)

y' = $tg4x - \frac{4x}{cos^2 4x}$

89.

Given a function u(M) = $e^{2x'+y'+z'}$ . Find the partial derivative of the function with respect to u'z

a)

A) $2ze^{2x'+y'+z'}$

b)

B) $2ze^{z'}$

c)

C) $e^{x'+y'+z'}$

d)

$e^{x'}$

e)

E) $2xyze^{x'+y'+z'}$

90.

Find the general solution of equation $\frac{1}{x^2 + 16}$

a)

y = (1/4) arctg(x/4) + C

b)

y = $ln|x + \sqrt{x^2 + 16}| + C$

c)

y = arctg(x/4) + C

d)

y = (1/16) arctg(x/16) + C

e)

y = arcsin(x/4) + C

91.

Find the general solution of equation y' = 1/(3x + 7)

a)

y = (1/3) ln|3x + 7| + C

b)

y = ln|3x + 7| + C

c)

y = (1/3) ln|x| + C

d)

y = ln|x| + C

e)

y = -(1/3) ln|3x + 7| + C

MATH pt1

Total questions: 91

Worksheet time: 46mins

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