MAT 251 Benchmark 2: Checkpoints 1-2; Attempt
review With a complete solution Updated RATED A+
NEW EDITION Straighterline
SECTION A: UNDERSTANDING STRAIGHTERLINE ASSESSMENTS
Benchmarks test mastery of course concepts. You have 3 attempts, and your
highest score counts. Checkpoints are quick knowledge checks on important course
concepts. All are open-book, and most have 1-3 attempts [citation:6].
SECTION B: RELATED RATES – VOLUME PROBLEMS
QUESTION 1
The volume of a spherical balloon is increasing at a constant rate of 3 m³/s. At
which of the following rates is the radius of the balloon increasing at the instant its
radius is 2 meters? [citation:1]
A) 3/(8π) m/s
B) 9/(8π) m/s
1
,C) 7/(16π) m/s
D) 3/(16π) m/s
Correct Answer: D
Solution:
Volume of a sphere: V = (4/3)πr³
Given dV/dt = 3 m³/s, r = 2 m
Differentiate implicitly:
dV/dt = 4πr²(dr/dt)
Solve for dr/dt:
dr/dt = (dV/dt) / (4πr²)
dr/dt = 3 / (4π × 2²)
dr/dt = 3 / (16π) m/s
SECTION C: DERIVATIVES – QUOTIENT RULE
2
,QUESTION 2
Find the derivative of y = (x² + 1) / (x² - 1). [citation:1]
A) -4x / (x² - 1)²
B) 4x³ / (x² - 1)²
C) 4x / (x² - 1)²
D) 4x⁴ / (x² - 1)²
Correct Answer: A
Solution:
Using the quotient rule: (u/v)' = (u'v - uv') / v²
u = x² + 1, u' = 2x
v = x² - 1, v' = 2x
y' = [2x(x² - 1) - (x² + 1)(2x)] / (x² - 1)²
y' = [2x³ - 2x - 2x³ - 2x] / (x² - 1)²
y' = -4x / (x² - 1)²
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, SECTION D: OPTIMIZATION – MAXIMUM DEFLECTION
QUESTION 3
The deflection of a hardwood beam of length L is given by:
D(x) = (9/4)x⁴ - 7Lx³ + 5L²x²
where x is the distance from the fixed end of the beam. Which of the following
values of x yields the maximum deflection? [citation:1]
A) 2L/3
B) 5L/3
C) 2L/3 and 5L/3
D) L
Correct Answer: C
Solution:
Find critical points by taking derivative and setting to zero:
D'(x) = 9x³ - 21Lx² + 10L²x
D'(x) = x(9x² - 21Lx + 10L²)
4
review With a complete solution Updated RATED A+
NEW EDITION Straighterline
SECTION A: UNDERSTANDING STRAIGHTERLINE ASSESSMENTS
Benchmarks test mastery of course concepts. You have 3 attempts, and your
highest score counts. Checkpoints are quick knowledge checks on important course
concepts. All are open-book, and most have 1-3 attempts [citation:6].
SECTION B: RELATED RATES – VOLUME PROBLEMS
QUESTION 1
The volume of a spherical balloon is increasing at a constant rate of 3 m³/s. At
which of the following rates is the radius of the balloon increasing at the instant its
radius is 2 meters? [citation:1]
A) 3/(8π) m/s
B) 9/(8π) m/s
1
,C) 7/(16π) m/s
D) 3/(16π) m/s
Correct Answer: D
Solution:
Volume of a sphere: V = (4/3)πr³
Given dV/dt = 3 m³/s, r = 2 m
Differentiate implicitly:
dV/dt = 4πr²(dr/dt)
Solve for dr/dt:
dr/dt = (dV/dt) / (4πr²)
dr/dt = 3 / (4π × 2²)
dr/dt = 3 / (16π) m/s
SECTION C: DERIVATIVES – QUOTIENT RULE
2
,QUESTION 2
Find the derivative of y = (x² + 1) / (x² - 1). [citation:1]
A) -4x / (x² - 1)²
B) 4x³ / (x² - 1)²
C) 4x / (x² - 1)²
D) 4x⁴ / (x² - 1)²
Correct Answer: A
Solution:
Using the quotient rule: (u/v)' = (u'v - uv') / v²
u = x² + 1, u' = 2x
v = x² - 1, v' = 2x
y' = [2x(x² - 1) - (x² + 1)(2x)] / (x² - 1)²
y' = [2x³ - 2x - 2x³ - 2x] / (x² - 1)²
y' = -4x / (x² - 1)²
3
, SECTION D: OPTIMIZATION – MAXIMUM DEFLECTION
QUESTION 3
The deflection of a hardwood beam of length L is given by:
D(x) = (9/4)x⁴ - 7Lx³ + 5L²x²
where x is the distance from the fixed end of the beam. Which of the following
values of x yields the maximum deflection? [citation:1]
A) 2L/3
B) 5L/3
C) 2L/3 and 5L/3
D) L
Correct Answer: C
Solution:
Find critical points by taking derivative and setting to zero:
D'(x) = 9x³ - 21Lx² + 10L²x
D'(x) = x(9x² - 21Lx + 10L²)
4
