MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find y ‘.

1) y = (4x – 3)(3x + 1)

A) 24x – 5

B) 24x – 2.5

C) 24x – 13

D) 12x – 5

2) y = (4x – 5)(5x^{3} – x^{2} + 1)

A) 80x^{3} – 87x^{2} + 10x + 4

B) 60x^{3} + 87x^{2} – 29x + 4

C) 20x^{3} + 29x^{2} – 87x + 4

D) 80x^{3} – 29x^{2} + 87x + 4

3) y = (x^{2} – 2x + 2)(4x^{3} – x^{2} + 5)

A) 20x^{4} – 32x^{3} + 30x^{2} + 6x – 10

B) 4x^{4} – 32x^{3} + 30x^{2} + 6x – 10

C) 20x^{4} – 36x^{3} + 30x^{2} + 6x – 10

D) 4x^{4} – 36x^{3} + 30x^{2} + 6x – 10

4) y = (4x^{3} + 3)(4x^{7} – 4)

A) 160x^{9} + 84x^{6} – 48x^{2}

B) 160x^{9} + 84x^{6} – 48x

C) 16x^{9} + 84x^{6} – 48x

D) 16x^{9 }+ 84x^{6} – 48x^{2}

5) y = (x + (1/x)) (x – (1/x))

A) 2x + (1/x^{3})

B) 2x + (1/x^{2})

C) 2x – (1/x^{2})

D) 2x + (2/x^{3})

6) y = ((4/x) + x) ((4/x) – x)

A) (32/x^{3}) + 2x

B) – (16/x^{3}) – 2x

C) – (32/x) + 2x

D) – (32/x^{3}) – 2x

7) y = ((1/x^{2}) + 3) (x^{2} – (1/x^{2}) + 3)

A) (4/x^{3}) + 6x

B) – (1/x^{5}) + 6x

C) – (4/x^{5}) – 6x

D) (4/x^{5}) + 6x

8) y = ((1/x) + 1) (x – (1/x) + 1)

A) – (2/x^{3}) – 1

B) – (1/x^{3}) – 1

C) (1/x^{3}) + 1

D) (2/x^{3}) + 1

Find the derivative of the function.

9) y = (x^{2} – 3x + 2/x7 – 2)

A) y ‘ = (-5x^{8} + 18x^{7} – 13x^{6} – 4x + 6/(x^{7} – 2)^{2})

B) y ‘ = (-5x^{8} + 18x^{7} – 14x^{6} – 3x + 6/(x^{7} – 2)^{2})

C) y ‘ = (-5x^{8} + 19x^{7} – 14x^{6} – 4x + 6/(x^{7} – 2)^{2})

D) y ‘ = (-5x^{8} + 18x^{7} – 14x^{6} – 4x + 6/(x^{7} – 2)^{2})

10) y = (x^{3}/x – 1)

A) y ‘ = (-2x^{3} + 3x^{2}/(x – 1)^{2})

B) y ‘ = (2x^{3} – 3x^{2}/(x – 1)^{2})

C) y ‘ = (2x^{3} + 3x^{2}/(x – 1)^{2})

D) y ‘ = (-2x^{3} – 3x^{2}/(x – 1)^{2})

11) g(x) = (x^{2} + 5/x^{2} + 6x)

A) g ‘(x) = (2x^{3} – 5x^{2} – 30x/x^{2}(x + 6)^{2})

B) g ‘(x) = (4x^{3} + 18x^{2} + 10x + 30/x^{2}(x + 6)^{2})

C) g ‘(x) = (x^{4} + 6x^{3} + 5x^{2} + 30x/x^{2}(x + 6)^{2})

D) g ‘(x) = (6x^{2} – 10x – 30/x^{2}(x + 6)^{2})

12) y = (x^{2} + 8x + 3/√(x))

A) y ‘ = (3x^{2} + 8x – 3/x)

B) y ‘ = (2x + 8/x)

C) y ‘ = (3x^{2} + 8x – 3/^{2}x3/^{2})

D) y ‘ = (2x + 8/2x^{3}/^{2})

13) y = (x^{2} + 2x – 2/x^{2} – 2x + 2)

A) y ‘ = (4x^{2} + 8x/(x^{2} – 2x + 2)^{2})

B) y ‘ = (-4x^{2} + 8x/(x^{2} – 2x + 2)^{2})

C) y ‘ = (-4x^{2} – 8x/(x^{2} – 2x + 2)^{2})

D) y ‘ = (4x^{2} – 8x/(x^{2} – 2x + 2)^{2})

14) f(t) = (6 – t)(6 + t^{3})^{-1}

A) f ‘(t) = (2t^{3} – 18t^{2} – 6/6 + t^{3})

B) f ‘(t) = (2t^{3} – 18t^{2} – 6/(6 + t^{3})^{2})

C) f ‘(t) = (- 2t^{3} + 18t^{2} – 6/(6 + t^{3})^{2})

D) f ‘(t) = (- 4t^{3} + 18t^{2} – 6/(6 + t^{3})^{2})

15) r = (√(θ) – 5/√(θ) + 5)

A) r ‘ = (10/(θ + 5)√(θ^{2} – 25))

B) r ‘ = (5/√(θ)(θ + 5)^{2})

C) r ‘ = – (5/√(θ)(θ + 5)^{2})

D) r ‘ = (5/θ + 5)

16) y = ((x + 4)(x + 1)/(x – 4)(x – 1))

A) y ‘ = (10x^{2} – 40/(x – 4)^{2}(x – 1)^{2})

B) y ‘ = (10x – 40/(x – 4)^{2}(x – 1)^{2})

C) y ‘ = (-x^{2} + 8/(x – 4)^{2}(x – 1)^{2})

D) y ‘ = (-10x^{2} + 40/(x – 4)^{2}(x – 1)^{2})

17) y = (-9x^{3} + 12x + 2/x)

A) (dy/dx) = -18x + 12 – 2x^{-2}

B) (dy/dx) = -18 – 2x

C) (dy/dx) = -18x – 2x^{-2}

D) (dy/dx) = -9x + 12 + 2x^{-2}

18) f(x) = 5x^{2} + 4x – 1 – (3/x^{3})

A) f'(x) = 5x + 4 – (9/x^{4})

B) f'(x) = 10x + 4 + (9/x^{4})

C) f'(x) = 5x^{2} + 4 + (9/x^{4})

D) f'(x) = 10x^{2} + 4 – (3/x^{2})

Provide an appropriate response.

19) Find an equation for the tangent to the curve y = (10x/x^{2} + 1) at the point (1, 5).

A) y = 5x

B) y = 5

C) y = x + 5

D) y = 0

20) Find an equation for the tangent to the curve y = (27/x^{2} + 2) at the point (1, 9).

A) y = -3x + 12

B) y = -6

C) y = -6x + 15

D) y = 6x + 3

Solve the problem.

21) The area A = πr^{2} of a circular oil spill changes with the radius. At what rate does the area change with respect to the radius when r = 4 ft?

A) 8 ft^{2}/ft

B) 4π ft^{2}/ft

C) 8π ft^{2}/ft

D) 16π ft^{2}/ft

22) The number of gallons of water in a swimming pool t minutes after the pool has started to drain is Q(t) = 50(20 – x)^{2}. How fast is the water running out at the end of 13 minutes?

A) 350 gal/min

B) 700 gal/min

C) 1,225 gal/min

D) 2,450 gal/min

23) The size of a population of mice after t months is P = 100(1 + 0.2t + 0.02t^{2}). Find the growth rate at t = 21 months.

A) 204 mice/month

B) 104 mice/month

C) 208 mice/month

D) 52 mice/month

24) The size of a population of lions after t months is P = 100 (1 + 0.2t + 0.02t^{2}). Find the growth rate when P = 2500.

A) 160 lions/month

B) 180 lions/month

C) 10,020 lions/month

D) 140 lions/month

25) A charged particle of mass m and charge q moving in an electric field E has an acceleration a given by

a = (qE/m),

where q and E are constants. Find (da/dm).

A) (da/dm) = – (m/qE)

B) (da/dm) = (qE/m^{2})

C) (da/dm) = qEm

D) (da/dm) = – (qE/m^{2})

26) A charged particle of mass m and charge q moving in an electric field E has an acceleration a given by

a = (qE/m),

where q and E are constants. Find (d^{2}a/dm^{2}).

A) (d^{2}a/dm^{2}) = (qE/m^{3})

B) (d^{2}a/dm^{2}) = (2qE/m^{3})

C) (d^{2}a/dm^{2}) = (qE/2m)

D) (d^{2}a/dm^{2}) = – (qE/m^{2})

27) Under standard conditions, molecules of a gas collide billions of times per second. If each molecule has diameter t, the average distance between collisions is given by

L = (1/√(2)πt^{2}n),

where n, the volume density of the gas, is a constant. Find (dL/dt).

A) (dL/dt) = – (2/√(2)πt^{3}n)

B) (dL/dt) = – (1/√(2)πtn)

C) (dL/dt) = (1/2√(2)πt^{3}n)

D) (dL/dt) = (2/√(2)πt^{3}n)

28) Under standard conditions, molecules of a gas collide billions of times per second. If each molecule has diameter t, the average distance between collisions is given by

L = (1/√(2)πt^{2}n),

where n, the volume density of the gas, is a constant. Find (d^{2}L/dt^{2}).

A) (d^{2}L/dt^{2}) = – (2/√(2)πt^{2}n)

B) (d^{2}L/dt^{2}) = (6/√(2)πt^{4}n)

C) (d^{2}L/dt^{2}) = – (6/√(2)πt^{4}n)

D) (d^{2}L/dt^{2}) = – (2/√(2)πt^{3}n)

where is the heat absorbed in one cycle and , the heat released into a reservoir in one cycle, is a constant? Find .

A) = –

B) =

C) = –

D) =

30) A heat engine is a device that converts thermal energy into other forms. The thermal efficiency, e, of a heat engine is defined by

where is the heat absorbed in one cycle and , the heat released into a reservoir in one cycle, is a constant? Find .

A) =

B) =

C) =

D) =

30) A heat engine is a device that converts thermal energy into other forms. The thermal efficiency, e, of a heat engine is defined by

e = (Q_{h} – Q _{C}/Q_{h}),

where Q_{h} is the heat absorbed in one cycle and Q

_{C}, the heat released into a reservoir in one cycle, is a constant. Find (d^{2}e/dQ_{h}^{2}).

A) (d^{2}e/dQ_{h}^{2}) = (- Q _{C}/2Q_{h}^{2})

B) (d^{2}e/dQ_{h}^{2}) = (Q _{C}/Q_{h}^{2})

C) (d^{2}e/dQ_{h}^{2}) = (Q _{C}/Q_{h}^{3})

D) (d^{2}e/dQ_{h}^{2}) = (- 2Q _{C}/Q_{h}^{3})

Find the second derivative of the function.

31) y = (x^{4} + 7/x^{2})

A) (d^{2}y/dx^{2}) = 2 + (42/x^{4})

B) (d^{2}y/dx^{2}) = 2 – (42/x^{4})

C) (d^{2}y/dx^{2}) = 2x – (14/x^{3})

D) (d^{2}y/dx^{2}) = 1 + (42/x^{4})

32) s = (t^{7} + 3t + 5/t^{2})

A) (d^{2}s/dt^{2}) = 20t^{3} + (6/t^{3}) + (30/t^{4})

B) (d^{2}s/dt^{2}) = 5t^{3} – (3/t^{3}) – (10/t^{4})

C) (d^{2}s/dt^{2}) = 20t^{5} + (6/t) + (30/t^{2})

D) (d^{2}s/dt^{2}) = 5t^{4} – (3/t^{2}) – (10/t^{3})

33) y = ((x – 8)(x^{2} + 3x)/x^{3})

A) (d^{2}y/dx^{2}) = (5/x^{2}) + (48/x^{3})

B) (d^{2}y/dx^{2}) = (10/x^{3}) + (144/x^{4})

C) (d^{2}y/dx^{2}) = – (10/x) – (144/x^{2})

D) (d^{2}y/dx^{2}) = – (10/x^{3}) – (144/x^{4})

34) r = ((1 + 4θ/4θ))(4 – θ)

A) (d^{2}r/dθ^{2}) = – (1/θ^{2}) – 1

B) (d^{2}r/dθ^{2}) = – (2/θ^{3}) – 1

C) (d^{2}r/dθ^{2}) = (1/θ) – θ

D) (d^{2}r/dθ^{2}) = (2/θ^{3})

35) y = (3 – 2x^{3} + x^{5}/x^{9})

A) (d^{2}y/dx^{2}) = (270/x^{11}) – (84/x^{8}) + (20/x^{6})

B) (d^{2}y/dx^{2}) = (270/x^{12}) + (84/x^{5}) – (20/x^{3})

C) (d^{2}y/dx^{2}) = (27/x^{11}) + (12/x^{6}) – (4/x^{4})

D) (d^{2}y/dx^{2}) = – (27/x^{10}) + (12/x^{7}) – (4/x^{5})

36) p = ((q + 2/q)) ((q + 5/q^{2}))

A) (d^{2}p/dq^{2}) = – (2/q^{3}) – (42/q^{4}) – (120/q^{5})

B) (d^{2}p/dq^{2}) = (2/q) + (42/q^{2}) + (120/q^{3})

C) (d^{2}p/dq^{2}) = (2/q^{3}) + (42/q^{4}) + (120/q^{5})

D) (d^{2}p/dq^{2}) = – (1/q^{2}) – (14/q^{3}) – (30/q^{4})

Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative.

37) u(1) = 3, u ‘(1) = -5, v(1) = 6, v ‘(1) = -4.

(d/dx) (uv) at x = 1

A) 42

B) -39

C) 18

D) -42

38) u(1) = 3, u ‘(1) = -6, v(1) = 7, v ‘(1) = -3.

(d/dx) ((u/v)) at x = 1

A) – (33/7)

B) – (11/3)

C) – (51/49)

D) – (33/49)

39) u(1) = 4, u ‘(1) = -5, v(1) = 6, v ‘(1) = -2.

(d/dx) ((v/u)) at x = 1

A) – (11/8)

B) (11/8)

C) – (19/8)

D) (11/2)

40) u(2) = 9, u ‘(2) = 2, v(2) = -1, v ‘(2) = -5.

(d/dx) (uv) at x = 2

A) 47

B) 23

C) -47

D) -43

41) u(2) = 6, u ‘(2) = 3, v(2) = -1, v ‘(2) = -5.

(d/dx) ((u/v)) at x = 2

A) – 33

B) – 27

C) (27/25)

D) 27

42) u(2) = 6, u ‘(2) = 4, v(2) = -2, v ‘(2) = -5.

(d/dx) ((v/u)) at x = 2

A) – (19/18)

B) – (11/18)

C) (11/18)

D) – (11/3)

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