Multiple Choice
Question 1
Using L'Hopital's Rule, what is lim_{x->0} (e^x - 1)/x?
A
1
B
0
C
e
D
does not exist
Solution: The form is 0/0. Differentiate: lim_{x->0} e^x/1 = e^0 = 1.
Question 2
Evaluate lim_{x->infinity} x / e^x using L'Hopital's Rule.
A
0
B
1
C
infinity
D
e
Solution: The form is infinity/infinity. Differentiate: lim_{x->infinity} 1/e^x = 0.
Question 3
Evaluate lim_{x->2} (x^2 - 4)/(x - 2) using L'Hopital's Rule.
A
4
B
0
C
2
D
does not exist
Solution: Form 0/0: differentiate to get lim_{x->2} 2x/1 = 4. (Factoring gives the same result.)
Question 4
A tank's water depth is D(t) meters after t hours, with D'(4) = -0.5. This means at t = 4 hours the water is:
A
dropping at 0.5 meters per hour
B
rising at 0.5 meters per hour
C
0.5 meters deep
D
constant
Solution: A negative rate means the depth is decreasing; D'(4) = -0.5 means it drops 0.5 m per hour at that instant.
Question 5
A particle's velocity is v(t) = 3t^2 - 12. When is the particle at rest?
A
t = 2
B
t = 4
C
t = 0
D
t = 12
Solution: At rest when v(t) = 3t^2 - 12 = 0, so t^2 = 4 and t = 2 (taking t >= 0).
Question 6
The demand for a product is D(p) = 500 - 4p units at price p dollars. dD/dp = -4 means:
A
demand drops by 4 units for each $1 price increase
B
demand rises by 4 units per dollar
C
the price is $4
D
demand is 4 units
Solution: The derivative dD/dp = -4 gives the rate of change of demand with price: 4 fewer units sold per $1 increase.
Question 7
If the perimeter of a square is P = 4s, what is dP/dt in terms of ds/dt?
A
4 (ds/dt)
B
s (ds/dt)
C
4s (ds/dt)
D
16 (ds/dt)
Solution: Differentiating P = 4s with respect to t gives dP/dt = 4 (ds/dt).
Question 8
Using linearization of f(x) = x^(1/3) at a = 8, estimate (8.1)^(1/3).
A
2.0083
B
2.1
C
2.05
D
2.0333
Solution: f(8) = 2, f'(x) = (1/3)x^(-2/3), f'(8) = (1/3)(1/4) = 1/12. L(8.1) = 2 + (1/12)(0.1) ≈ 2.0083.
Question 9
A particle's position is s(t) = 2t^3 - 3t^2. What is its acceleration at t = 1?
A
6
B
0
C
12
D
3
Solution: v(t) = 6t^2 - 6t, a(t) = 12t - 6. At t = 1, a(1) = 12 - 6 = 6.
Question 10
Evaluate lim_{x->0} (sin(3x))/(x) using L'Hopital's Rule.
A
3
B
1
C
0
D
1/3
Solution: Form 0/0: differentiate to get lim_{x->0} 3cos(3x)/1 = 3cos(0) = 3.
Question 11
Let f(t) be the amount of a drug (mg) in the bloodstream after t hours. If f(2) = 40 and f'(2) = -8, estimate the amount at t = 2.5 hours.
A
36 mg
B
44 mg
C
32 mg
D
40 mg
Solution: Local linear approximation: f(2.5) ≈ f(2) + f'(2)(0.5) = 40 + (-8)(0.5) = 40 - 4 = 36 mg.
Question 12
Evaluate lim_{x->0} (cos(x) - 1)/x using L'Hopital's Rule.
A
0
B
1
C
-1
D
infinity
Solution: Form 0/0: differentiate to get lim_{x->0} (-sin x)/1 = -sin(0) = 0.
Question 13
A balloon's surface area A = 4 pi r^2 increases. When r = 3 and dr/dt = 2, how fast is the surface area increasing?
A
48 pi
B
24 pi
C
12 pi
D
36 pi
Solution: dA/dt = 8 pi r (dr/dt) = 8 pi (3)(2) = 48 pi.
Question 14
The differential dy for y = x^2 is:
A
2x dx
B
x^2 dx
C
2x
D
2 dx
Solution: By definition dy = f'(x) dx. For y = x^2, dy = 2x dx.
Question 15
A ball's height is h(t) = -4.9t^2 + 20t. What is its velocity at t = 1 second?
A
10.2 m/s
B
20 m/s
C
15.1 m/s
D
-9.8 m/s
Solution: h'(t) = -9.8t + 20. At t = 1, h'(1) = -9.8 + 20 = 10.2 m/s.
Question 16
If y = x^2 and x changes with time, then dy/dt equals:
A
2x (dx/dt)
B
2x
C
x^2 (dx/dt)
D
2 (dx/dt)
Solution: By the Chain Rule with respect to t: dy/dt = 2x (dx/dt).
Question 17
A car's fuel efficiency E(v) (miles per gallon) depends on speed v (mph). E'(50) = -0.4 means:
A
efficiency drops 0.4 mpg for each additional mph at 50 mph
B
efficiency is 0.4 mpg
C
the car goes 50 mph
D
efficiency rises with speed
Solution: E'(50) = -0.4 is the rate of change of efficiency with speed: a loss of 0.4 mpg per 1 mph increase at v = 50.
Question 18
The linear approximation of f(x) = cos(x) at a = 0 is:
A
L(x) = 1
B
L(x) = x
C
L(x) = 1 - x
D
L(x) = 1 + x
Solution: f(0) = 1, f'(x) = -sin(x), f'(0) = 0. So L(x) = 1 + 0*(x - 0) = 1.
Question 19
Before applying L'Hopital's Rule, you must first check that the limit:
A
has an indeterminate form (0/0 or infinity/infinity)
B
equals zero
C
is a polynomial
D
is continuous
Solution: L'Hopital's Rule requires confirming the quotient is indeterminate (0/0 or infinity/infinity) before differentiating numerator and denominator.
Question 20
If W(t) is a baby's weight (kg) at age t months and W'(6) = 0.5, this means at 6 months the baby is:
A
gaining 0.5 kg per month
B
0.5 kg
C
losing weight
D
6 months old only
Solution: W'(6) = 0.5 is the instantaneous growth rate: 0.5 kg gained per month at age 6 months.