Multiple Choice
Question 1
For the area of a circle A = pi r^2, what is dA/dt in terms of dr/dt?
A
2 pi r (dr/dt)
B
pi r^2 (dr/dt)
C
2 pi r
D
pi (dr/dt)
Solution: Differentiating A = pi r^2 with respect to t: dA/dt = 2 pi r (dr/dt).
Question 2
In a related-rates problem, the first step is usually to:
A
Write an equation relating the variables, then differentiate with respect to time
B
Substitute numbers before differentiating
C
Integrate the equation
D
Solve for a maximum
Solution: You relate the quantities with an equation, differentiate implicitly with respect to t, and only then substitute the known instantaneous values.
Question 3
The linearization of f at x = a is given by:
A
L(x) = f(a) + f'(a)(x - a)
B
L(x) = f'(a)(x - a)
C
L(x) = f(a)(x - a)
D
L(x) = f(a) + f(a)(x - a)
Solution: The linearization (tangent-line approximation) at x = a is L(x) = f(a) + f'(a)(x - a).
Question 4
L'Hopital's Rule can be applied directly to a limit only when it has which indeterminate form?
A
0/0 or infinity/infinity
B
0 * infinity
C
1/0
D
any limit
Solution: L'Hopital's Rule applies to indeterminate quotients of the form 0/0 or infinity/infinity.
Question 5
If g(x) measures the distance (miles) traveled after x hours, the units of g'(x) are:
A
miles per hour
B
hours per mile
C
miles
D
hours
Solution: The derivative gives distance per unit time: miles per hour (a speed).
Question 6
A particle moving along a line is momentarily at rest when:
A
v(t) = 0
B
a(t) = 0
C
s(t) = 0
D
v(t) is maximum
Solution: The particle is at rest exactly when its velocity is zero.
Question 7
When differentiating a related-rates equation, every variable is treated as a function of:
A
time t
B
x only
C
a constant
D
itself
Solution: In related rates, quantities change over time, so each variable is differentiated with respect to time t (implicit differentiation).
Question 8
The revenue from selling x units is R(x) = 50x - 0.1x^2 dollars. What is the marginal revenue at x = 100?
A
$30 per unit
B
$50 per unit
C
$40 per unit
D
$20 per unit
Solution: R'(x) = 50 - 0.2x. At x = 100, R'(100) = 50 - 20 = 30 dollars per unit.
Question 9
A population grows according to P(t) = 200 + 30t - t^2 (t in years). At what rate is the population changing at t = 5?
A
20 per year
B
30 per year
C
-10 per year
D
10 per year
Solution: P'(t) = 30 - 2t. At t = 5, P'(5) = 30 - 10 = 20 per year.
Question 10
The area of a circle is A = pi r^2. When r = 5, dA/dr equals:
A
10 pi
B
25 pi
C
5 pi
D
2 pi
Solution: dA/dr = 2 pi r. At r = 5, this is 10 pi (the circumference).
Question 11
For a sphere V = (4/3) pi r^3, what is dV/dt in terms of dr/dt?
A
4 pi r^2 (dr/dt)
B
(4/3) pi r^2 (dr/dt)
C
4 pi r (dr/dt)
D
pi r^2 (dr/dt)
Solution: Differentiating with respect to t: dV/dt = 4 pi r^2 (dr/dt).
Question 12
Given x^2 + y^2 = 100 with x and y functions of t, what equation relates dx/dt and dy/dt?
A
2x(dx/dt) + 2y(dy/dt) = 0
B
x(dx/dt) = y(dy/dt)
C
2x + 2y = 0
D
dx/dt + dy/dt = 0
Solution: Differentiating both sides with respect to t: 2x(dx/dt) + 2y(dy/dt) = 0.
Question 13
The volume of a cube with edge s is V = s^3. What is dV/dt in terms of ds/dt?
A
3s^2 (ds/dt)
B
s^2 (ds/dt)
C
3s (ds/dt)
D
s^3 (ds/dt)
Solution: Differentiating V = s^3 with respect to t: dV/dt = 3s^2 (ds/dt).
Question 14
The radius of a circle increases at 3 cm/s. How fast is the area increasing when r = 4 cm?
A
24 pi cm^2/s
B
12 pi cm^2/s
C
6 pi cm^2/s
D
16 pi cm^2/s
Solution: dA/dt = 2 pi r (dr/dt) = 2 pi (4)(3) = 24 pi cm^2/s.
Question 15
The area of a square increases at 8 cm^2/s. How fast is the side length increasing when the side is 4 cm?
A
1 cm/s
B
2 cm/s
C
0.5 cm/s
D
4 cm/s
Solution: A = s^2, so dA/dt = 2s(ds/dt). Then 8 = 2(4)(ds/dt) = 8(ds/dt), giving ds/dt = 1 cm/s.
Question 16
Use linearization of f(x) = sqrt(x) at a = 4 to estimate sqrt(4.1).
A
2.025
B
2.05
C
2.1
D
2.005
Solution: f(4) = 2, f'(x) = 1/(2 sqrt(x)), so f'(4) = 1/4. L(4.1) = 2 + (1/4)(0.1) = 2 + 0.025 = 2.025.
Question 17
The tangent line to f(x) = x^3 at x = 2 is used to approximate f(2.1). The approximation is:
A
9.2
B
9.261
C
8.4
D
12
Solution: f(2) = 8, f'(x) = 3x^2, f'(2) = 12. L(2.1) = 8 + 12(0.1) = 8 + 1.2 = 9.2. (The true value 9.261 is close.)
Question 18
Use the linearization of f(x) = e^x at a = 0 to approximate e^(0.02).
A
1.02
B
1.2
C
1.002
D
0.98
Solution: f(0) = 1, f'(0) = e^0 = 1. L(0.02) = 1 + 1(0.02) = 1.02.
Question 19
Use linearization of f(x) = ln(x) at a = 1 to estimate ln(1.1).
A
0.1
B
0.01
C
1.1
D
0.041
Solution: f(1) = 0, f'(x) = 1/x, f'(1) = 1. L(1.1) = 0 + 1(0.1) = 0.1. (True value ~0.0953.)
Question 20
Using L'Hopital's Rule, what is lim_{x->0} sin(x)/x?
A
1
B
0
C
infinity
D
does not exist
Solution: The form is 0/0. Differentiate top and bottom: lim_{x->0} cos(x)/1 = cos(0) = 1.