Part B- Pre-Test 2 for Cal 1 (2.4, 2.5, 2.6)
Test 2 will be on Oct 14th, chapter 2 (except 2.6)
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Given y = f(u) and u = g(x), find dy/dx = f′(g(x))g′(x).
1
1) y = , u = 6x - 7
u2
A) - 12
6x - 7
B) - 1)
6
6x - 7
C) - 12
(6x - 7)3
D)
12x
6x - 7
2) y = tan u, u = -14x + 18
2)
A) -14 sec 2 (-14x + 18)
B) -14 sec (-14x + 18) tan (-14x + 18)
C) sec 2 (-14x + 18)
D) - sec 2 (-14x + 18)
Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.
10
4
3) y = 6x2 - - x
x
9
4
dy
4
4
A) y = u10; u = 6x2 - - x; = 10 6x2 - - x 12x + - 1
x
dx
x
x2
3)
4
dy
4
B) y = 6u2 - - u; u = x10; = 12x20 - - x10
u
dx
x10
9
dy
4
4
= 10 6x2 - - x
C) y = u10; u = 6x2 - - x; dx
x
x
9
4
dy
4
D) y = u10; u = 6x2 - - x; = 10 12x + - 1
x
dx
x2
4) y = tan π - 3
x
4)
3 dy
3
3
A) y = tan u; u = π - ; = sec 2 π - x dx x2
x
3 dy
3
B) y = tan u; u = π - ; = sec 2
x dx
x2
3 dy
3
3
3
C) y = tan u; u = π - ; = sec π - tan π - x dx x2
x
x
3 dy
3
D) y = tan u; u = π - ; = sec 2 π - x dx
x
Find the derivative of the function.
5) q = 15r - r7
A)
1
2 15 - 7r6
5)
B)
15 - 7r6
C)
2 15r - r7
1
1
2 15r - r7
D)
-7r6
15r - r7
6) r = (sec θ + tan θ)-5
6)
A) -5(sec θ tan θ + sec 2 θ)-6
C)
B) -5(sec θ + tan θ)-6 (tan2 θ + sec θ tan θ)
-5 sec θ
D) -5(sec θ + tan θ)-6
(sec θ + tan θ)5
Find dy/dt.
7) y = cos6 (πt - 9)
7)
A) 6 cos5 (πt - 9)
B) -6 cos5 (πt - 9) sin(πt - 9)
C) -6π cos5 (πt - 9) sin(πt - 9)
D) -6π sin5 (πt - 9)
Find y′′ .
8) y = ( x - 5)-5
8)
A) 20( x - 5)-7
C)
5
5
B) - ( x - 5)-7
- 5
2x
x
5
5
( x - 5)-7 - + 7
4x
x
D) - 5
2 x
( x - 5)-6
1
9) y = tan(6x - 4)
4
A)
9)
3
sec 2 (6x - 4)
2
C) 18 sec 2 (6x - 4) tan(6x - 4)
B)
1
sec 2 (6x - 4) tan(6x - 4)
2
D)
1
sec(6x - 4)
2
Find the value of (f ∘ g)′ at the given value of x.
1
1
10) f(u) = + 12, u = g(x) = , x = 7
u
2
x - 12
A) -26
11) f(u) = 10)
B) 14
C) 7
D) 49
1
- u, u = g(x) = πx, x = 6
cos3 u
A) 6π
11)
C) -π
B) 3 - π
D) -6π
Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values
of x. Find the derivative with respect to x of the given combination at the given value of x.
x f(x) g(x) f ′(x) g ′(x)
12) 3 1 16
12)
6
5
4 3
3
5
-5
f(x) + g(x), x = 3
1
A) - 2 17
B)
11
17
C)
2
1
2 17
D)
11
2 17
Use implicit differentiation to find dy/dx.
13) xy + x = 2
1 + x
A) - y
13)
1 + x
B)
y
1 + y
C) - x
1 + y
D)
x
Find dy/dt.
14) y = t5 (t4 + 8)5
14)
A) t5 (t4 + 8)4 (25t3 + 40)
C) t4 (t4 + 8)4 (25t4 + 40)
B) 5t4 (t4 + 8)4 (20t4 + 8)
D) 100t18(t4 + 8)4
Use implicit differentiation to find dy/dx.
1
15) y cos = 8x + 8y
y
8 - y sin 8y
A)
sin C)
15)
B)
1
1
+ y cos - 8y
y
y
cos 8y2
1
sin - 8y2
y
1
y
1
- 8
y
8
D)
sin 1
1
+ y cos - 8
y
y
Use implicit differentiation to find dy/dx and d2 y/dx 2 .
16) 2y - x + xy = 8
16)
A)
dy y + 1 d2 y
2y + 2
= ; = dx
x + 2 dx2 (x + 2)2
B)
dy
1 + y d2 y
2y - 2
= - ; = dx
x + 2 dx2 (x + 2)2
C)
dy 1 - y d2 y
2y - 2
= ; = dx 2 + x dx2 (2 + x)2
D)
dy
1 + y d2 y
y + 1
= - ; = dx
x + 2 dx2 (2 + x)2
17) x3/5 + y3/5 = 2
dy x2/5 d2 y
2x3/5 + 2y3/5
A)
= ; = - dx y2/5 dx2
5x1/5y7/5
C)
17)
dy
y2/5 d2 y 2y1/5 - 2x
= - ; = dx
x2/5 dx2 5x7/5y3/5
B)
dy y2/5 d2 y
2x3/5 + 2y3/5
= ; = - dx x2/5 dx2
5x7/5y1/5
D)
dy
y2/5 d2 y 2x3/5 + 2y3/5
= - ; = dx
x2/5 dx2
5x7/5y1/5
At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve,
as requested.
18) x4 y4 = 16, tangent at (2, 1)
A) y = -8x + 1
18)
1
B) y = x
2
C) y = 8x - 1
3
1
D) y = - x + 2
2
19) x3 y3 = 8, slope at (2, 1)
1
A) - 2
19)
1
B) - 4
C) 4
D) 2
Solve the problem.
20) Suppose that the radius r and the circumference C = 2πr of a circle are differentiable functions of t.
Write an equation that relates dC/dt to dr/dt.
dr
dC
dr
dC dr
dr
dC
dC
= 2π
B)
= 2πr
C)
= D)
= 2π
A)
dt
dt
dt
dt
dt
dt
dt
dt
20)
21) If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse,
c2 = a 2 + b2 . How is dc/dt related to da/dt and db/dt?
21)
A)
da
db
dc
= a 2
+ b2
dt
dt
dt
B)
dc
da
db
= a + b
dt
dt
dt
C)
dc
da
db
= 2a + 2b
dt
dt
dt
D)
dc 1 da
db
= a
+ b
dt c dt
dt
Provide an appropriate response.
22) If x3 + y 3 = 9 and dx/dt = -3, then what is dy/dt when x = 1 and y = 2?
3
3
4
A)
B) - C)
4
4
3
22)
4
D) - 3
Solve the problem.
23) Water is falling on a surface, wetting a circular area that is expanding at a rate of 8 mm2 /s. How
fast is the radius of the wetted area expanding when the radius is 172 mm? (Round your answer to
four decimal places.)
A) 0.0465 mm/s
B) 0.0148 mm/s
C) 0.0074 mm/s
23)
D) 135.0884 mm/s
Solve the problem. Round your answer, if appropriate.
24) One airplane is approaching an airport from the north at 144 km/hr. A second airplane approaches
from the east at 190 km/hr. Find the rate at which the distance between the planes changes when
the southbound plane is 33 km away from the airport and the westbound plane is 17 km from the
airport.
A) -107 km/hr
B) -215 km/hr
C) -430 km/hr
D) -322 km/hr
25) Water is being drained from a container which has the shape of an inverted right circular cone. The
container has a radius of 6.00 inches at the top and a height of 8.00 inches. At the instant when the
water in the container is 6.00 inches deep, the surface level is falling at a rate of 0.9 in./sec. Find the
rate at which water is being drained from the container.
A) 54.7 in.3 /s
B) 55.1 in.3 s
C) 57.3 in.3 /s
4
24)
D) 70.0 in.3 /s
25)
26) A man 6 ft tall walks at a rate of 7 ft/sec away from a lamppost that is 20 ft high. At what rate is the
length of his shadow changing when he is 35 ft away from the lamppost? (Do not round your
answer)
21
21
245
ft/sec
C)
ft/sec
D)
ft/sec
A) 3 ft/sec
B)
26
13
6
26)
27) Boyleʹs law states that if the temperature of a gas remains constant, then PV = c, where
P = pressure, V = volume, and c is a constant. Given a quantity of gas at constant temperature, if V
is decreasing at a rate of 14 in. 3 /sec, at what rate is P increasing when P = 70 lb/in.2 and V = 90
27)
in.3 ? (Do not round your answer.)
98
lb/in.2 per sec
A)
9
B)
C) 450 lb/in.2 per sec
49
lb/in.2 per sec
81
D) 18 lb/in.2 per sec
28) The radius of a right circular cylinder is increasing at the rate of 2 in./sec, while the height is
decreasing at the rate of 8 in./sec. At what rate is the volume of the cylinder changing when the
radius is 18 in. and the height is 5 in.?
A) 124 in.3 /sec
B) -2412π in.3 /sec
C) -2232π in.3 /sec
D) -2412 in.3 /sec
Use implicit differentiation to find dy/dx.
x + y
29)
= x2 + y 2
x - y
A)
x(x - y)2 - y
x - y(x - y)2
B)
28)
29)
x(x - y)2 + y
x + y(x - y)2
C)
5
x(x - y)2 - y
x + y(x - y)2
D)
x(x - y)2 + y
x - y(x - y)2
Answer Key
Testname: PART B‐CAL1‐PRE‐TEST 2
1) C
2) A
3) A
4) A
5) B
6) C
7) C
8) C
9) C
10) B
11) C
12) D
13) C
14) C
15) A
16) C
17) D
18) D
19) A
20) A
21) D
22) C
23) C
24) B
25) C
26) A
27) A
28) C
29) D
6