Section 3.1 Quick Quiz
𝑓(2+ℎ)−𝑓(2)
1. The quotient
(if it is defined) equals
ℎ
(a) an average rate of change. (b) an instantaneous rate of change. (c) neither a nor b.
𝑓(2+ℎ)−𝑓(2)
2. The limit lim
ℎ→0
ℎ
(if it exists) equals
(a) an average rate of change. (b) an instantaneous rate of change. (c) neither a nor b.
3. If the line tangent to the curve y = f (x) at (4, 14) is y = 6x – 10, then lim
𝑓(4+ℎ)−𝑓(4)
ℎ
ℎ→0
(a) –10.
(b) 6.
equals
(c) 20.
4. The slope of the line tangent to the curve y = f (x) at (3, 4) (if it exists) equals
(a) lim
𝑓(𝑥)−𝑓(3)
𝑥→3
𝑥−3
(b) lim
.
𝑥→4
𝑓(𝑥)−𝑓(4)
𝑥−4
.
5. In the given figure, 𝑓′(−2) equals
(a) –1.
(b) 1.
(c) lim
𝑓(4+ℎ)−𝑓(4)
ℎ→0
ℎ
.
(c) 2.
6. In the given figure, f is
(a) not differentiable at x = 0. (b) not continuous at x = 0. (c) undefined at x = 0.
Figure for Exercises 5–6
7. If f (x) = x3 and f ′(x) = 3x2, then the slope of the curve y = f (x) at x = 2 is
(a) 8.
(b) 2.
(c) 12.
8. If a function g is not continuous at x = a, then g
(a) must be undefined at x = a. (b) is not differentiable at x = a. (c) has an asymptote at x = a.
9. If lim𝑓(𝑥) = 3 and lim
𝑥→5
(a) equals 3.
ℎ→5
𝑓(𝑥)−𝑓(5)
𝑥−5
= 4 then f (5)
(b) equals 4.
(c) cannot be determined.
10. If f (0) = 1 and f ′(0) = 6, then the equation of the line tangent to the curve y = f (x) at (0, 1) is
(a) y = 0.
(b) y = x + 6. (c) y = 6x + 1.
Quick Quiz 3.1 Answers:
1a 2b 3b 4a 5a 6a 7c 8b 9a 10c
Section 3.2 Quick Quiz
Answer the following multiple choice questions by circling the correct response.
1. The derivative of x4 with respect to x equals
(a) 0.
(c) 4x3.
(b) 4x.
2. The derivative of 104 with respect to x equals
(a) 0.
(b) 40.
(c) 4000.
3. The derivative of 5x4 with respect to x equals
(a) 20x3 .
(c) 5x3 .
(b) 20x.
4. The second derivative of 2x3 + 3x + 5 with respect to x equals
(a) 6x2 + 3 .
(b)12x + 3 .
(c)12x .
5. The slope of the line tangent to y = x(2x2 +1) at x = 1 is
(a) 5.
(b) 6.
(c) 7.
6. If 𝑓(𝑥) = 4√𝑥, then f ′(x) equals
(a)
1
(b)
√𝑥
2
(c)
√𝑥
4
√𝑥
𝑑
7. If f ′(x) exists, then 𝑑𝑥 [𝑥 − 𝑓(𝑥)]equals
(a) f ′(x) .
(b)1+ f ′(x) .
(c)1+ f (x) .
𝑑
8. If f ′ exists, then 𝑑𝑥 [𝑥 − 𝑓(𝑥)] evaluated at x = 0 equals
(a) − f ′(0) .
9.
𝑑3
𝑑𝑥 3
(b) 0.
(c)1− f ′(0) .
[3𝑥 3 ] equals
(a) 18. (b) 9x2. (c) 27x.
10. If 𝑓(𝑥) = (𝑥 2 − 1) − 2√𝑥 , then f ′(x) equals
(a) x3/2 .
(b) 2𝑥 − 1/√𝑥.
Quick Quiz 3.2 Answers:
1c 2a 3a 4c 5c 6b 7b 8c 9a 10b
(c)
3𝑥−1
2√𝑥
Section 3.3 Quick Quiz
1. The derivative of (x2 + 3x)(2x −1) with respect to x equals
(b) 6x2 +10x − 3.
(a) 2(2x + 3).
(c) 2x3 +5x2 − 3x.
𝑥3
2. The derivative of 𝑥 2 +1 with respect to x equals
(a)
3𝑥
.
2
(b)
−𝑥 4 −3𝑥 3
(𝑥 2 +1)2
.
(c)
𝑥 2 (𝑥 2 +3)
.
(𝑥 2 +1)2
3. The derivative of 6x−3 with respect to x equals
(a)18x2 .
(b) −18x−2 .
(c) −18x−4 .
2𝑥
4. The slope of the line tangent to 𝑦 = 𝑥−4 at x = 6 is
(a) –2.
(b) 6.
(c) 8.
5. If f (x) =𝑥 2 √𝑥, then f ′(4) equals
(b) 5⋅ 21/3 .
(a) 20.
6. If 𝑓(𝑥) =
(c) 2.
3𝑥−4
(a) x−2 .
𝑥2
, then f ′ (x) equals
(b)
3𝑥−4
𝑥6
(c) −6x−3 +12x−4 .
.
𝑑
7. If f ′(x) exists, then 𝑑𝑥 [𝑥 −2 + 𝑓(𝑥)] equals
(a) f ′(x) – 2x .
2
(b) f ′(x) – 𝑥 3 .
1
(c) f ′(x) + 𝑥 2 .
𝑑
8. If f ′ exists, then 𝑑𝑥 [𝑥𝑓(𝑥)]
(a) f ′(x) .
(b) f ′(x) + xf (x) .
(c) f (x) + xf ′(x) .
𝑑
9. If f ′ exists, then [𝑑𝑥 [𝑥/𝑓(𝑥)]
(a)1/ f ′(x) .
(b) (f (x) − xf ′(x)) / (𝑓(𝑥))2 . (c) (f (x) − x) / (𝑓(𝑥))2
10. If (𝑥) = (𝑥 2 − 1)√𝑥 , then f ′(x) equals
(a) √𝑥.
(b)
5𝑥 2 −1
√𝑥
.
Quick Quiz 3.3 Answers:
1b 2c 3c 4a 5a 6c 7b 8c 9b 10c
(c)
5𝑥 2 −1
2√𝑥
.
Section 3.4 Quick Quiz
Answer the following multiple choice questions by circling the correct response.
sin 𝑥
1. The value of lim 𝑥
𝑥→0
(a) 0.
(b) 1.
2. The value of lim
(c) undefined.
cos(𝑥)−1
𝑥
𝑥→0
(a) 0.
is
(b) 1.
(c) undefined.
3. If f (x) = cos x then f ′′(x) equals
(a) −sin x.
(c) −cos x.
(b) cos x.
d
4. 𝑑𝑥 (𝑥 + sin 𝑥) equals
(a) 1+ cos x.
d
tan 𝑥
5. 𝑑𝑥 (
𝑥
(b) 1− cos x.
(c) cos x.
) equals
(a) sec 2 (𝑥)
(b)
𝑥−sin 𝑥 cos 𝑥
𝑥 2 cos2 𝑥
.
(c)
1−sin 𝑥
𝑥 cos 𝑥
.
d
6. 𝑑𝑥 (𝑥 sec 𝑥) equals
(a) sec x + x sec x tan x.
(b) 1+ sec x tan x.
(c) sec x.
7. Let f (x) = sin x and g(x) = cos x . Then f ′(0) is
(a) g(0).
(b) −g(0).
(c) g′(0).
8. Let f (x) = sin x and g(x) = cos x . Then f ′′(0) is
(a) f ′(0).
(b) − f (0).
(c) g(0).
9. Let y = cot x. Then d 2 y / dx2 equals
(a) –cot x.
(b) –2csc x.
10. The value of the lim
𝑥→0
(c) 2csc2 x cot x.
sin(10 𝑥)
𝑥
(a) 10.
(b) 1.
Quick Quiz 3.4 Answers:
1b 2a 3c 4a 5b 6a 7a 8b 9c 10a
(c) 1/10.
Section 3.5 Quick Quiz
1. Let s = f (t) be the position of a rock above the ground t seconds after it is thrown vertically
upward. Then f ′(t) is the
(a) speed of the rock.
(b) velocity of the rock.
(c) acceleration of the rock.
2. Let s = f (t) be the position of a rock above the ground t seconds after it is thrown vertically
upward. Then f ′′(t) is the
(a) speed of the rock.
(b) velocity of the rock.
(c) acceleration of the rock.
3. Let s = f (t) be the position of a rock above the ground t seconds after it is thrown vertically
upward. To determine the time when the rock reaches its highest point, you should solve the
equation
(a) f (t) = 0 .
(b) f ′(t) = 0 .
(c) f ′′(t) = 0 .
4. An object begins moving along a straight line. Its displacement s relative to its starting point
after t seconds is given by s = 16t − t2. The object changes direction when
(a) t = 16.
(b) t = 8.
(c) t = 4.
5. A rock is dropped off a bridge and its distance s (in ft) from the bridge after t seconds is s = 16t2
. The acceleration of the rock at t = 1/2 is
(a) 8 ft/s2.
(b) 16 ft/s2.
(c) 32 ft/s2.
6. A rock is dropped off a bridge and its distance s (in ft) from the bridge after t seconds is s = 16t2
. The velocity of the rock at t = 2 is
(a) 16 ft/s.
Quick Quiz 3.5 Answers:
1b 2c 3b 4b 5c 6c
(b) 32 ft/s.
(c) 64 ft/s.
Section 3.6 Quick Quiz
𝑑𝑦
𝑑𝑦 𝑑𝑢
1. If 𝑦 = √𝑥 2 + 3𝑥 + 1, then 𝑑𝑥 = 𝑑𝑢 𝑑𝑥 , where
(b) u = x2 + 3x +1 .
(a) u = √𝑥 .
𝑑𝑦
(c) u = 2x + 3 .
𝑑𝑦 𝑑𝑢
2. If y = sin3 x, then 𝑑𝑥 = 𝑑𝑢 𝑑𝑥 , where
(a) u = x3
(b) u = 3sin2 x .
𝑑𝑦
(c) u = sin x .
𝑑𝑦 𝑑𝑢
3. If y = sin3 x, then 𝑑𝑥 = 𝑑𝑢 𝑑𝑥 , where
(a) y = u3 .
(b) y = sin u .
𝑑𝑦
𝑑𝑦 𝑑𝑢
3
4. If y = sin x , then =
, where
(c) neither a nor b is correct.
(a) u = x3 .
(c) u = sin x .
𝑑𝑥
𝑑𝑢 𝑑𝑥
(b) u = 3sin x2 .
𝑑
5. Let f (5) = 10 , f ′(5) = 3 , g(1) = 5 , and g′(1) = 7 . Then 𝑑𝑥 𝑓(𝑔(𝑥))|𝑥=1 equals
(a) 3.
(b) 15.
(c) 21.
6. Let h(x) = f ( f (x)) , f (2) = 3 , f ′(2) = 3 , and f ′(3) = 9 . Then h′(2) equals
(a) 9.
(b) 27.
(c) neither a nor b.
7. If f (x) = g(x2) , then f ′(x) equals
(a) 2xg′(x2).
(b) 2xg′(x).
(c) x2g′(x).
8. If f (x) = g(g(x)) , then f ′(x) equals
(a) g′(x)g′(x).
(b) g′(g(x)).
(c) g′(g(x))g′(x).
9. If g(x) = h(1000x) , then g′(x) equals
(a) h′(1000x).
(b) 1000h′(1000x).
(c) xh′(1000x).
10. If f (x) = g(20q(x)) , then f ′(x) equals
(a) 20q′(x)g′(20q(x)). (b) 20g′(20q(x)).
Quick Quiz 3.6 Answers:
1b 2c 3a 4a 5c 6b 7a 8c 9b 10a
(c) g′(20q(x)).
Section 3.7 Quick Quiz
1. For each value of x satisfying −1 < x < 1, how many tangent lines does the circle x2 + y2 = 1
have?
(a) 0 (b) 1 (c) 2
2. If xy = 1, then
𝑑𝑦
𝑑𝑦
(a) 𝑑𝑥 = 0.
𝑑𝑦
(b) 1 + 𝑥 𝑑𝑥 = 0.
(c) 𝑦 + 𝑥 𝑑𝑥 = 0.
3. If xy + y3 = 1, then
𝑑𝑦
(a) 𝑑𝑥 + 3𝑦 2 = 0.
𝑑𝑦
𝑑𝑦
𝑑𝑦
(b) 𝑦 + 𝑥 𝑑𝑥 + 3𝑦 2 𝑑𝑥 = 0.
𝑑𝑦
(c) 𝑦 + 𝑥 𝑑𝑥 + 3𝑦 2 = 0
4. If x = √𝑦 , then 𝑑𝑥 equals
(a) 2 √𝑦.
1
(b) √𝑦 .
(c) 2 𝑦.
√
𝑑𝑦
5. If sin y = x, then 𝑑𝑥 equals
(c) −sin y .
(a) cos y .
(b) sec y .
𝑑𝑦
6. If sin x = sin y, then 𝑑𝑥 equals
cos 𝑦
(a) cos 𝑥 .
cos 𝑥
(b) cos 𝑦.
(c) tan(xy) .
7. If y2 = x , then
(a)
𝑑2 𝑦
𝑑𝑥 2
= 2.
(b)
𝑑𝑦 𝑑2 𝑦
𝑑𝑥
𝑑𝑥 2
= 𝑦.
(c) 𝑦
𝑑2 𝑦
𝑑𝑥 2
𝑑𝑦 2
= −( )
𝑑𝑥
8. Suppose yx3 + sin (xy) = 1 and you want to evaluate dy/dx at (2, π). Then
(a) you should use explicit differentiation.
(b) you should use implicit differentiation.
(c) the problem doesn't make sense.
9. Consider the graph of x +8y2 = 10. How many lines tangent to this graph exist at points with x =
1?
(a) 1
(b) 2
(c) 3
10. A right circular cylinder has a fixed volume of 12 cubic inches, which meansπ r2h = 12 , where
r is the radius of the cylinder and h is its height. The rate of change of the radius with respect to the
height, dr/dh, is
(a) −r/(2h).
(b) −1/2.
(c) −r2.
Quick Quiz 3.7 Answers:
1c 2c 3b 4a 5b 6b 7c 8c 9b 10a
Section 3.8 Quick Quiz
Assume all derivatives (for example, x’and y’ ) are with respect to t.
1. If the radius of a circle decreases at a rate of 3 ft/s, what can you say about the rate of change of
the area A?
(a) A’(t) >0
(b) A’(t) = 0
(c) A’(t) ≤ 0
2. If the sides of a square increase in length at a rate of 3 in/s, then the rate of change
of the area
(a) cannot be determined (b) is negative.
(c) is always greater than 3 in/s.
3. The sides of a cube increase at a rate of R cm/s. When the sides have a length of 2 cm, the rate
of change of the volume is
(a) 10R.
(b) 12R.
(c) 16R.
4. The surface area of a sphere is given by S = 4πr2. The rate of change of the surface area as
the radius changes is
(a) dS/dt=8πr dr/dt
(b) dS/dt=4πr2 dr/dt
(c) dS/dt=4πr dr/dt
5. The volume of a sphere is given by V = (4/3)πr3. The rate of change of the volume as the radius
changes is
(a) dV/dt=12πr dr/dt
(b) dV/dt=2πr2 dr/dt
(c) dV/dt=4πr2 dr/dt
6. If the radius of a right circular cylinder increases and the height h of the cylinder changes in
such a way that the volume of the cylinder remains constant, then
(a) h’(t) >0
(c) h’(t) < 0
(b) h’(t) = 0
7. If the length of one leg of a right triangle increases and the length of the other leg decreases,
then the length of the hypotenuse
(a) increases.
(b) decreases.
(c) may increase or may decrease.
8. Suppose you walk along a circle centered at the origin in the xy-plane. Let dx/dt and dy/dt be
the rate of change 0f your x and y-coordinates. Then
a) dx/dt and dy/dt always have opposite signs
b) dx/dt and dy/dt always have the same sign
c) dx/dt and dy/dt have opposite signs in two quadrants.
9. Let x3 + tan y = 10. If x increases, then y
(a) always decreases.
(b) always increases.
(c) is constant.
10. The ideal gas law says that the pressure, volume, and temperature of an ideal gas are related
by pV/T = k, where k is a positive constant. If V’ (t ) =0 and T’ (t)> 0 , then
(a) P’(t) > 0.
Quick Quiz 3.8 Answers
1c 2a 3b 4a 5c 6c 7c 8c 9a 10a
(b) P’(t)< 0.
(c) P’(t)= 0.