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)(5x3 – x2 + 1)
A) 80x3 – 87x2 + 10x + 4
B) 60x3 + 87x2 – 29x + 4
C) 20x3 + 29x2 – 87x + 4
D) 80x3 – 29x2 + 87x + 4
3) y = (x2 – 2x + 2)(4x3 – x2 + 5)
A) 20x4 – 32x3 + 30x2 + 6x – 10
B) 4x4 – 32x3 + 30x2 + 6x – 10
C) 20x4 – 36x3 + 30x2 + 6x – 10
D) 4x4 – 36x3 + 30x2 + 6x – 10
4) y = (4x3 + 3)(4x7 – 4)
A) 160x9 + 84x6 – 48x2
B) 160x9 + 84x6 – 48x
C) 16x9 + 84x6 – 48x
D) 16x9 + 84x6 – 48x2
5) y = (x + (1/x)) (x – (1/x))
A) 2x + (1/x3)
B) 2x + (1/x2)
C) 2x – (1/x2)
D) 2x + (2/x3)
6) y = ((4/x) + x) ((4/x) – x)
A) (32/x3) + 2x
B) – (16/x3) – 2x
C) – (32/x) + 2x
D) – (32/x3) – 2x
7) y = ((1/x2) + 3) (x2 – (1/x2) + 3)
A) (4/x3) + 6x
B) – (1/x5) + 6x
C) – (4/x5) – 6x
D) (4/x5) + 6x
8) y = ((1/x) + 1) (x – (1/x) + 1)
A) – (2/x3) – 1
B) – (1/x3) – 1
C) (1/x3) + 1
D) (2/x3) + 1
Find the derivative of the function.
9) y = (x2 – 3x + 2/x7 – 2)
A) y ‘ = (-5x8 + 18x7 – 13x6 – 4x + 6/(x7 – 2)2)
B) y ‘ = (-5x8 + 18x7 – 14x6 – 3x + 6/(x7 – 2)2)
C) y ‘ = (-5x8 + 19x7 – 14x6 – 4x + 6/(x7 – 2)2)
D) y ‘ = (-5x8 + 18x7 – 14x6 – 4x + 6/(x7 – 2)2)
10) y = (x3/x – 1)
A) y ‘ = (-2x3 + 3x2/(x – 1)2)
B) y ‘ = (2x3 – 3x2/(x – 1)2)
C) y ‘ = (2x3 + 3x2/(x – 1)2)
D) y ‘ = (-2x3 – 3x2/(x – 1)2)
11) g(x) = (x2 + 5/x2 + 6x)
A) g ‘(x) = (2x3 – 5x2 – 30x/x2(x + 6)2)
B) g ‘(x) = (4x3 + 18x2 + 10x + 30/x2(x + 6)2)
C) g ‘(x) = (x4 + 6x3 + 5x2 + 30x/x2(x + 6)2)
D) g ‘(x) = (6x2 – 10x – 30/x2(x + 6)2)
12) y = (x2 + 8x + 3/√(x))
A) y ‘ = (3x2 + 8x – 3/x)
B) y ‘ = (2x + 8/x)
C) y ‘ = (3x2 + 8x – 3/2x3/2)
D) y ‘ = (2x + 8/2x3/2)
13) y = (x2 + 2x – 2/x2 – 2x + 2)
A) y ‘ = (4x2 + 8x/(x2 – 2x + 2)2)
B) y ‘ = (-4x2 + 8x/(x2 – 2x + 2)2)
C) y ‘ = (-4x2 – 8x/(x2 – 2x + 2)2)
D) y ‘ = (4x2 – 8x/(x2 – 2x + 2)2)
14) f(t) = (6 – t)(6 + t3)-1
A) f ‘(t) = (2t3 – 18t2 – 6/6 + t3)
B) f ‘(t) = (2t3 – 18t2 – 6/(6 + t3)2)
C) f ‘(t) = (- 2t3 + 18t2 – 6/(6 + t3)2)
D) f ‘(t) = (- 4t3 + 18t2 – 6/(6 + t3)2)
15) r = (√(θ) – 5/√(θ) + 5)
A) r ‘ = (10/(θ + 5)√(θ2 – 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 ‘ = (10x2 – 40/(x – 4)2(x – 1)2)
B) y ‘ = (10x – 40/(x – 4)2(x – 1)2)
C) y ‘ = (-x2 + 8/(x – 4)2(x – 1)2)
D) y ‘ = (-10x2 + 40/(x – 4)2(x – 1)2)
17) y = (-9x3 + 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) = 5x2 + 4x – 1 – (3/x3)
A) f'(x) = 5x + 4 – (9/x4)
B) f'(x) = 10x + 4 + (9/x4)
C) f'(x) = 5x2 + 4 + (9/x4)
D) f'(x) = 10x2 + 4 – (3/x2)
Provide an appropriate response.
19) Find an equation for the tangent to the curve y = (10x/x2 + 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/x2 + 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 = πr2 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 ft2/ft
B) 4π ft2/ft
C) 8π ft2/ft
D) 16π ft2/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 + 0.02t2). 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 + 0.02t2). 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/m2)
C) (da/dm) = qEm
D) (da/dm) = – (qE/m2)
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 (d2a/dm2).
A) (d2a/dm2) = (qE/m3)
B) (d2a/dm2) = (2qE/m3)
C) (d2a/dm2) = (qE/2m)
D) (d2a/dm2) = – (qE/m2)
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)πt2n),
where n, the volume density of the gas, is a constant. Find (dL/dt).
A) (dL/dt) = – (2/√(2)πt3n)
B) (dL/dt) = – (1/√(2)πtn)
C) (dL/dt) = (1/2√(2)πt3n)
D) (dL/dt) = (2/√(2)πt3n)
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)πt2n),
where n, the volume density of the gas, is a constant. Find (d2L/dt2).
A) (d2L/dt2) = – (2/√(2)πt2n)
B) (d2L/dt2) = (6/√(2)πt4n)
C) (d2L/dt2) = – (6/√(2)πt4n)
D) (d2L/dt2) = – (2/√(2)πt3n)
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 = (Qh – Q C/Qh),
where Qh is the heat absorbed in one cycle and Q
C, the heat released into a reservoir in one cycle, is a constant. Find (d2e/dQh2).
A) (d2e/dQh2) = (- Q C/2Qh2)
B) (d2e/dQh2) = (Q C/Qh2)
C) (d2e/dQh2) = (Q C/Qh3)
D) (d2e/dQh2) = (- 2Q C/Qh3)
Find the second derivative of the function.
31) y = (x4 + 7/x2)
A) (d2y/dx2) = 2 + (42/x4)
B) (d2y/dx2) = 2 – (42/x4)
C) (d2y/dx2) = 2x – (14/x3)
D) (d2y/dx2) = 1 + (42/x4)
32) s = (t7 + 3t + 5/t2)
A) (d2s/dt2) = 20t3 + (6/t3) + (30/t4)
B) (d2s/dt2) = 5t3 – (3/t3) – (10/t4)
C) (d2s/dt2) = 20t5 + (6/t) + (30/t2)
D) (d2s/dt2) = 5t4 – (3/t2) – (10/t3)
33) y = ((x – 8)(x2 + 3x)/x3)
A) (d2y/dx2) = (5/x2) + (48/x3)
B) (d2y/dx2) = (10/x3) + (144/x4)
C) (d2y/dx2) = – (10/x) – (144/x2)
D) (d2y/dx2) = – (10/x3) – (144/x4)
34) r = ((1 + 4θ/4θ))(4 – θ)
A) (d2r/dθ2) = – (1/θ2) – 1
B) (d2r/dθ2) = – (2/θ3) – 1
C) (d2r/dθ2) = (1/θ) – θ
D) (d2r/dθ2) = (2/θ3)
35) y = (3 – 2x3 + x5/x9)
A) (d2y/dx2) = (270/x11) – (84/x8) + (20/x6)
B) (d2y/dx2) = (270/x12) + (84/x5) – (20/x3)
C) (d2y/dx2) = (27/x11) + (12/x6) – (4/x4)
D) (d2y/dx2) = – (27/x10) + (12/x7) – (4/x5)
36) p = ((q + 2/q)) ((q + 5/q2))
A) (d2p/dq2) = – (2/q3) – (42/q4) – (120/q5)
B) (d2p/dq2) = (2/q) + (42/q2) + (120/q3)
C) (d2p/dq2) = (2/q3) + (42/q4) + (120/q5)
D) (d2p/dq2) = – (1/q2) – (14/q3) – (30/q4)
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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