Section 3.1 Quick Quiz
1. The quotient(if it is defined) equals
(a) an average rate of change. (b) an instantaneous rate of change. (c) neither a nor b.
2. The limit (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 equals
(a) –10. (b) 6. (c) 20.
4. The slope of the line tangent to the curve y = f (x) at (3, 4) (if it exists) equals
(a).(b).(c).
5. In the given figure, equals
(a) –1. (b) 1. (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) = x3and 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. Ifandthen f (5)
(a) equals 3.(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. (b) 4x. (c) 4x3.
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 .(b) 20x. (c) 5x3 .
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, then f ′(x) equals
(a) (b) (c)
7. If f ′(x) exists, then equals
(a) f ′(x) .(b)1+ f ′(x) .(c)1+ f (x) .
8. If f ′ exists, then evaluated at x = 0 equals
(a) − f ′(0) . (b) 0. (c)1− f ′(0) .
9. equals
(a) 18. (b) 9x2. (c) 27x.
10. If , then f ′(x) equals
(a) x3/2 .(b).(c)
Quick Quiz 3.2 Answers:
1c 2a 3a 4c 5c 6b 7b 8c 9a 10b
Section 3.3 Quick Quiz
1. The derivative of (x2+ 3x)(2x −1) with respect to x equals
(a) 2(2x + 3). (b) 6x2+10x − 3. (c) 2x3+5x2− 3x.
2. The derivative ofwith respect to x equals
(a). (b) .(c) .
3. The derivative of 6x−3 with respect to x equals
(a)18x2 .(b) −18x−2 .(c) −18x−4 .
4. The slope of the line tangent to at x = 6 is
(a) –2. (b) 6. (c) 8.
5. If f (x) =, then f ′(4) equals
(a) 20. (b) 5⋅21/3 .(c) 2.
6. If , then f′ (x)equals
(a) x−2 .(b). (c) −6x−3+12x−4 .
7. If f ′(x) exists, then equals
(a) f ′(x) – 2x . (b) f ′(x) – . (c) f ′(x) +.
8. If f ′ exists, then
(a) f ′(x) . (b) f ′(x) + xf(x) .(c) f (x) + xf′(x) .
9. If f ′ exists, then [
(a)1/ f ′(x) .(b) (f (x) − xf′(x)) / .(c)(f (x) − x) /
10. If , then f ′(x) equals
(a) . (b). (c).
Quick Quiz 3.3 Answers:
1b 2c 3c 4a 5a 6c 7b 8c 9b 10c
Section 3.4 Quick Quiz
Answer the following multiple choice questions by circling the correct response.
1. The value of
(a) 0. (b) 1. (c) undefined.
2. The value ofis
(a) 0. (b) 1. (c) undefined.
3. If f (x) = cos x then f ′′(x) equals
(a) −sin x. (b) cos x. (c) −cos x.
4. equals
(a) 1+ cos x. (b) 1− cos x. (c) cos x.
5.equals
(a) (b) . (c).
6. 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 2y / dx2 equals
(a) –cot x. (b) –2csc x. (c) 2csc2x cot x.
10. The value of the
(a) 10. (b) 1. (c) 1/10.
Quick Quiz 3.4 Answers:
1b 2a 3c 4a 5b 6a 7a 8b 9c 10a
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. (b) 32 ft/s. (c) 64 ft/s.
Quick Quiz 3.5 Answers:
1b 2c 3b 4b 5c 6c
Section 3.6 Quick Quiz
1. If , then, where
(a) u = . (b) u = x2+ 3x +1 .(c) u = 2x + 3 .
2. If y = sin3x, then, where
(a) u = x3(b) u = 3sin2x .(c) u = sin x .
3. If y = sin3x, then, where
(a) y = u3 .(b) y = sin u .(c) neither a nor b is correct.
4. If y = sin x3, then, where
(a) u = x3 .(b) u = 3sin x2 .(c) u = sin x .
5. Let f (5) = 10 ,f ′(5) = 3 , g(1) = 5 , and g′(1) = 7 . Then 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)). (c) g′(20q(x)).
Quick Quiz 3.6 Answers:
1b 2c 3a 4a 5c 6b 7a 8c 9b 10a
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
(a). (b). (c) .
3. If xy+ y3 = 1, then
(a) .(b) . (c)
4. If x = , thenequals
(a) 2 . (b) . (c).
5. If sin y = x, thenequals
(a) cos y .(b) sec y .(c) −sin y .
6. If sin x = sin y, thenequals
(a). (b) . (c) tan(xy) .
7. If y2 = x, then
(a). (b). (c)
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 theradius 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
Assumeallderivatives(forexample,x’and y’)arewith respectto t.
1. Ifthe radiusofacircle decreasesatarate of3 ft/s,whatcanyousayaboutthe rate ofchangeofthe area A?
(a) A’(t) >0(b) A’(t) = 0 (c) A’(t) ≤ 0
2. Ifthe sidesofasquareincreaseinlengthatarate of3 in/s,thenthe rate ofchangeofthe area
(a) cannotbe determined (b)is negative. (c) isalways greaterthan3 in/s.
3.The sidesofacubeincreaseatarate ofRcm/s.Whenthe sideshavealengthof2 cm,therate ofchangeof the volumeis
(a) 10R. (b)12R. (c) 16R.
4. The surfacearea ofasphereisgivenbyS =4πr2. The rate ofchangeofthe surfacearea asthe radius changesis
(a) dS/dt=8πr dr/dt (b) dS/dt=4πr2dr/dt (c) dS/dt=4πr dr/dt
5.The volumeofasphereisgivenbyV=(4/3)πr3.The rate ofchangeofthevolumeastheradiuschangesis
(a) dV/dt=12πr dr/dt (b) dV/dt=2πr2dr/dt (c) dV/dt=4πr2dr/dt
6. Ifthe radiusofarightcircularcylinderincreasesandthe heighth ofthe cylinderchangesinsuchawaythat the volumeofthecylinderremainsconstant,then
(a) h’(t) >0(b) h’(t) = 0 (c) h’(t) < 0
7. Ifthe lengthofonelegofarighttriangleincreasesandthe lengthofthe otherlegdecreases,thenthe length ofthehypotenuse
(a) increases. (b)decreases. (c) mayincreaseormaydecrease.
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 signsb) 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+tany= 10.Ifxincreases,theny
(a) alwaysdecreases. (b)alwaysincreases. (c) isconstant.
10. The ideal gaslawsaysthatthepressure,volume,andtemperatureofanidealgasare relatedbypV/T=k, wherekisa positiveconstant.IfV’(t)=0andT’ (t)0, then
(a)P’(t)0. (b)P’(t)0.(c)P’(t)=0.
Quick Quiz 3.8 Answers
1c 2a 3b 4a 5c 6c 7c 8c 9a 10a