MATH_352_Calculus_Winter2022_Exam_Solutions_2520.docx

calculus exam solutions 05

1. Quiz 3
MATH 338, Section 2
October 15, 2025
Name: ________________________________
Calculators are allowed for simple calculations only (arithmetic, log, etc). You must show all supporting work
on this exam paper. Answers without supporting work are worth zero points. There are 100 points possible.
1. (5 points) lim[x→1] (5x² + 2x - 7) / (x - 1)
Solution:
Solution:
Step 1: Direct substitution gives 0/0 (indeterminate form)
lim[x→1] (5x² + 2x - 7) / (x - 1)
Step 2: Factor the numerator
5x² + 2x - 7 = (x - 1)(5x + 7)
Step 3: Cancel common factor (x - 1)
lim[x→1] (x - 1)(5x + 7) / (x - 1) = lim[x→1] (5x + 7)
Step 4: Evaluate the limit
= 5(1) + 7 = 12
Answer: 12
The Limit Concept:
Limits are the foundation of calculus. We're asking "what value does the function approach?" rather than "what
value does it equal?" This distinction is crucial for understanding continuity and differentiability.

2. (5 points) Find the derivative of f(x) = (3x + 1)/(x² - 2)
Solution:
Solution:
Step 1: Identify the quotient rule formula
If f(x) = g(x)/h(x), then f'(x) = [g'(x)h(x) - g(x)h'(x)] / [h(x)]²
Step 2: Identify g(x) and h(x)
g(x) = 3x + 1 → g'(x) = 3
h(x) = x² - 2 → h'(x) = 2x
Step 3: Apply the quotient rule
f'(x) = [3(x² - 2) - (3x + 1)(2x)] / (x² - 2)²
Step 4: Expand the numerator
= [3x² - 6 - (6x² + 2x)] / (x² - 2)²
= [3x² - 6 - 6x² - 2x] / (x² - 2)²
= [-3x² - 2x - 6] / (x² - 2)²
Answer: f'(x) = (-3x² - 2x - 6)/(x² - 2)²

3. (4 points) Find the derivative of f(x) = e^(2x) + ln(x³)

2. Solution:
Solution:
Step 1: Differentiate each term separately
f(x) = e^(2x) + ln(x³)
f'(x) = d/dx[e^(2x)] + d/dx[ln(x³)]
Step 2: Find the derivative of e^(2x)
d/dx[e^(2x)] = e^(2x) · d/dx[2x] = e^(2x) · 2 = 2e^(2x)
Step 3: Find the derivative of ln(x³)
d/dx[ln(x³)] = 1/x³ · d/dx[x³] = 1/x³ · 3x² = 3x²/x³ = 3/x
Step 4: Combine the results
f'(x) = 2e^(2x) + 3/x
Answer: f'(x) = 2e^(2x) + 3/x
Why This Approach Works:
Derivatives measure the instantaneous rate of change. Each step follows from fundamental calculus rules that
were proven rigorously. Understanding the "why" behind each rule helps build intuition for more complex
problems.

4. (4 points) Find dy/dx for x² + y² = 25
Solution:
Solution:
Step 1: Differentiate both sides with respect to x
d/dx[x² + y²] = d/dx[25]
d/dx[x²] + d/dx[y²] = 0
Step 2: Apply the chain rule to d/dx[y²]
2x + 2y(dy/dx) = 0
Step 3: Solve for dy/dx
2y(dy/dx) = -2x
dy/dx = -2x/(2y)
dy/dx = -x/y
Answer: dy/dx = -x/y

5. (5 points) Evaluate: ∫(2x + 1)^3 dx
Solution:
Solution:
Step 1: Use substitution method
Let u = 2x + 1
Then du = 2dx, so dx = du/2
Step 2: Substitute into the integral
∫(2x + 1)^3 dx = ∫u³ · (du/2) = (1/2)∫u³ du
Step 3: Integrate using the power rule
= (1/2) · u /4 + C = u /8 + C
⁴ ⁴

3. Step 4: Substitute back u = 2x + 1
= (2x + 1) /8 + C
⁴
Answer: ∫f(x)dx = (2x + 1) /8 + C
⁴
Conceptual Understanding:
Integration is the reverse of differentiation (Fundamental Theorem of Calculus). We're essentially finding the
area under a curve or the original function from its rate of change. The constant C appears because derivatives
of constants are zero.

6. (5 points) lim[x→0] x² · sin(1/x)
Solution:
Solution:
Step 1: Recognize that sin(1/x) is bounded
-1 ≤ sin(1/x) ≤ 1 for all x ≠ 0
Step 2: Multiply all parts by x² (assuming x > 0 near 0)
-x² ≤ x² · sin(1/x) ≤ x²
Step 3: Take limits as x→0
lim[x→0] (-x²) ≤ lim[x→0] x² · sin(1/x) ≤ lim[x→0] x²
0 ≤ lim[x→0] x² · sin(1/x) ≤ 0
Step 4: By the Squeeze Theorem
lim[x→0] x² · sin(1/x) = 0
Answer: 0
The Limit Concept:
Limits are the foundation of calculus. We're asking "what value does the function approach?" rather than "what
value does it equal?" This distinction is crucial for understanding continuity and differentiability.

7. (5 points) A spherical balloon is being inflated. If the radius is increasing at a rate of 3 cm/min,
how fast is the volume increasing when the radius is 3 cm? (Volume of sphere: V = (4/3)πr³)
Solution:
Solution:
Step 1: Write the volume formula
V = (4/3)πr³
Step 2: Differentiate both sides with respect to time t
dV/dt = (4/3)π · 3r² · dr/dt = 4πr² · dr/dt
Step 3: Substitute the given values
dr/dt = 3 cm/min
r = 3 cm
Step 4: Calculate dV/dt
dV/dt = 4π(3)² · 3
= 4π · 9 · 3
= 108π cm³/min
Answer: The volume is increasing at 108π cm³/min

4. 8. (5 points) Evaluate: ∫(5x + 1)^3 dx
Solution:
Solution:
Step 1: Use substitution method
Let u = 5x + 1
Then du = 5dx, so dx = du/5
Step 2: Substitute into the integral
∫(5x + 1)^3 dx = ∫u³ · (du/5) = (1/5)∫u³ du
Step 3: Integrate using the power rule
= (1/5) · u /4 + C = u /20 + C
⁴ ⁴
Step 4: Substitute back u = 5x + 1
= (5x + 1) /20 + C
⁴
Answer: ∫f(x)dx = (5x + 1) /20 + C
⁴
Conceptual Understanding:
Integration is the reverse of differentiation (Fundamental Theorem of Calculus). We're essentially finding the
area under a curve or the original function from its rate of change. The constant C appears because derivatives
of constants are zero.

9. (5 points) lim[x→0] x² · sin(1/x)
Solution:
Solution:
Step 1: Recognize that sin(1/x) is bounded
-1 ≤ sin(1/x) ≤ 1 for all x ≠ 0
Step 2: Multiply all parts by x² (assuming x > 0 near 0)
-x² ≤ x² · sin(1/x) ≤ x²
Step 3: Take limits as x→0
lim[x→0] (-x²) ≤ lim[x→0] x² · sin(1/x) ≤ lim[x→0] x²
0 ≤ lim[x→0] x² · sin(1/x) ≤ 0
Step 4: By the Squeeze Theorem
lim[x→0] x² · sin(1/x) = 0
Answer: 0
The Limit Concept:
Limits are the foundation of calculus. We're asking "what value does the function approach?" rather than "what
value does it equal?" This distinction is crucial for understanding continuity and differentiability.

10. (4 points) Find the equation of the tangent line to y = x² - 3x + 2 at x = 1
Solution:
Solution:
Step 1: Find the point of tangency

5. y(1) = (1)² - 3(1) + 2 = 1 - 3 + 2 = 0
Point: (1, 0)
Step 2: Find the slope by taking the derivative
y = x² - 3x + 2
y' = 2x - 3
Step 3: Evaluate the derivative at x = 1
y'(1) = 2(1) - 3 = -1
Slope = -1
Step 4: Use point-slope form
y - y = m(x - x )
₁ ₁
y - 0 = -1(x - 1)
y = -x + 1
Answer: y = -x + 1
Why This Approach Works:
Derivatives measure the instantaneous rate of change. Each step follows from fundamental calculus rules that
were proven rigorously. Understanding the "why" behind each rule helps build intuition for more complex
problems.

11. (4 points) A company's cost function is C(x) = x³ - 12x² + 50x + 200. Find the marginal cost when
x = 8 units
Solution:
Solution:
Step 1: Find the marginal cost function
Marginal cost = C'(x)
C(x) = x³ - 12x² + 50x + 200
C'(x) = 3x² - 24x + 50
Step 2: Evaluate at x = 8
C'(8) = 3(8)² - 24(8) + 50
C'(8) = 3(64) - 192 + 50
C'(8) = 192 - 192 + 50 = 50
Step 3: Interpret the result
The marginal cost represents the cost of producing one additional unit
Answer: Marginal cost = $50 per unit

12. (5 points) Find the derivative: f(x) = 6x^4 + 6x² - 4x + 5
Solution:
Solution:
Step 1: Apply the power rule to each term
f'(x) = d/dx[6x^4] + d/dx[6x²] + d/dx[-4x] + d/dx[5]
Step 2: Use the power rule: d/dx[x^n] = nx^(n-1)
= 6 · 4x^3 + 6 · 2x^1 + (-4) · 1x^0 + 0

6. Step 3: Simplify
= 24x^3 + 12x - 4
Answer: f'(x) = 24x^3 + 12x - 4

13. (5 points) Find the value of k that makes f(x) continuous at x = 1, where: f(x) = { (x² - 1) / (x - 1)
if x ≠ 1 { k if x = 1
Solution:
Solution:
Step 1: For continuity at x = 1, we need:
lim[x→1] f(x) = f(1) = k
Step 2: Find the limit as x approaches 1
lim[x→1] (x² - 1) / (x - 1)
Step 3: Factor the numerator
x² - 1 = (x - 1)(x + 1)
Step 4: Simplify and evaluate
lim[x→1] (x - 1)(x + 1) / (x - 1) = lim[x→1] (x + 1) = 1 + 1 = 2
Step 5: Therefore, k = 2
Answer: k = 2
The Limit Concept:
Limits are the foundation of calculus. We're asking "what value does the function approach?" rather than "what
value does it equal?" This distinction is crucial for understanding continuity and differentiability.

14. (5 points) Find the derivative of f(x) = (3x + 1)/(x² - 2)
Solution:
Solution:
Step 1: Identify the quotient rule formula
If f(x) = g(x)/h(x), then f'(x) = [g'(x)h(x) - g(x)h'(x)] / [h(x)]²
Step 2: Identify g(x) and h(x)
g(x) = 3x + 1 → g'(x) = 3
h(x) = x² - 2 → h'(x) = 2x
Step 3: Apply the quotient rule
f'(x) = [3(x² - 2) - (3x + 1)(2x)] / (x² - 2)²
Step 4: Expand the numerator
= [3x² - 6 - (6x² + 2x)] / (x² - 2)²
= [3x² - 6 - 6x² - 2x] / (x² - 2)²
= [-3x² - 2x - 6] / (x² - 2)²
Answer: f'(x) = (-3x² - 2x - 6)/(x² - 2)²

15. (5 points) Find the derivative: f(x) = 9x^5 + 6x² - 3x + 5
Solution:

7. Solution:
Step 1: Apply the power rule to each term
f'(x) = d/dx[9x^5] + d/dx[6x²] + d/dx[-3x] + d/dx[5]
Step 2: Use the power rule: d/dx[x^n] = nx^(n-1)
= 9 · 5x^4 + 6 · 2x^1 + (-3) · 1x^0 + 0
Step 3: Simplify
= 45x^4 + 12x - 3
Answer: f'(x) = 45x^4 + 12x - 3