Calculus II: Grading Policies, Cylindrical Shells, and Average Value of Functions

Grading Policies and Academic Procedures

• Grading Calculation Methods: The instructor primarily utilizes the average to determine grades. The median is not typically used for general grading (e.g., adding $5\%$ to reach a $70$). The instructor specifically compares the distribution of A's, B's, and C's against specific benchmarks.
• Grade Flooring Policy: The lowest grade assigned on a test is set at a floor of $50\%$. Even if a student earns a score significantly lower (e.g., $11.4\%$ or $36.4\%$), the value entered into the grade book for the average calculation is $50\%$. This policy exists because it is mathematically too difficult for a student to recover from a score in the $30\%$ range.
• Homework Weight and Impact: Homework accounts for $30\%$ of the total grade. This includes both written and online assignments. Diligent completion of homework is intended to improve test scores and acts as a "buffer" for test grades.
• Grade Minimum Scenario: If a student earns a $50\%$ on a test but maintains a $100\%$ average on both written and online homework, the minimum grade calculation would be: $(50 \times 0.70) + (100 \times 0.30) = 35 + 30 = 65$. Thus, a floor of $65\%$ is maintained until the final exam.
• Final Exam Policy: The final exam grade can replace the lowest test grade if the final exam score is higher.
• Office Hours and Support: Post-pandemic, student attendance at office hours has dropped significantly. The instructor cited a recent two-hour Friday session with zero attendees. Support options include:
  • Scheduled office hours.
  • Appointments at mutually agreeable times.
  • Assistance from tutors.
  • Reviewing test keys to understand specific errors.

Common Calculus II Errors and Test Feedback

• Integration Product Rule Misconception: A common error involves students attempting to integrate a product by integrating the terms individually. There is no product rule for integration equivalent to the derivative product rule. Sums or differences can be separated, but products cannot.
• Operator Confusion: Students frequently change operations without mathematical justification, such as turning a product into an addition problem mid-calculation.
• Simplification Errors: Many students lose points for failing to simplify fractional results. For example, leaving an answer as $\frac{3x^{7/2}}{7/2}$ rather than flipping the fraction to reach $\frac{6}{7}x^{7/2}$. If a student fails to simplify, they may lose $1$ or $2$ points, particularly because those who do attempt to simplify sometimes make arithmetic errors and are penalized accordingly.
• Homework vs. Test Performance: The instructor noted that errors seen on tests (like basic property failures) should have been caught during homework. Consistently getting homework right while failing tests suggests potential copying of homework answers, which leads to poor test performance.

Section 6.3: Cylindrical Shells Review

• Problem Statement: Find the volume of the solid generated by rotating the region bounded by $y = x^{3/2}$, $y = 8$, and $x = 0$ about the x-axis using the method of cylindrical shells.
• Setup and Graphing:
  • At $x = 0$, $y = 0$.
  • At $x = 1$, $y = 1$.
  • At $x = 4$, $y = \sqrt{4^3} = 2^3 = 8$. (Point: $(4, 8)$).
  • The region is bounded by the y-axis ($x = 0$) and the horizontal line $y = 8$.
• Shell Method Orientation: To rotate about the x-axis using shells, the side of the shell (the "can") must be parallel to the axis of rotation (horizontal). The thickness is $dy$.
• Solving for x: Since we integrate with respect to $y$, we must solve $y = x^{3/2}$ for $x$.
  • $x = y^{2/3}$.
• Identifying Radius and Height:
  • The radius ($r$) is the distance from the x-axis to the shell, which is simply $y$.
  • The height ($h$) of the shell is the horizontal distance, which is the function value $x = y^{2/3}$.
• Volume Formula: $V = 2\pi \int_0^8 (y)(y^{2/3}) \,dy$.
• Modified Rotation (About $y = 10$):
  • If the same area is rotated about the line $y = 10$:
  • The height remains $y^{2/3}$.
  • The radius changes to the distance between the line $y = 10$ and the shell at $y$, which is $10 - y$.
  • The integral becomes: $V = 2\pi \int_0^8 (10 - y)(y^{2/3}) \,dy$.
• Comparison with Disk/Washer Method: For this specific problem, the disk/washer method might be easier as it would not require solving for $x$ and would involve integrating from $0$ to $4$ along the x-axis.

Section 6.5: Average Value of a Function

• Concept Derivation: To find the average of a finite set of numbers, you sum them and divide by $n$. For a function with infinite points on an interval $[a, b]$, we use a limit of Riemann sums.
• Average Value Formula:
  • $f_{ave} = \frac{1}{b - a} \int_{a}^{b} f(x) \,dx$
• Geometric Interpretation: The average value $f_{ave}$ represents the height of a rectangle with width $(b - a)$ that has exactly the same area as the area under the curve $f(x)$ from $a$ to $b$.
  • $\text{Area} = f_{ave} \times (b - a)$.
• Worked Example: Find the average value of $h(u) = \frac{\ln(u)}{u}$ on the interval $[1, 5]$.
  • $h_{ave} = \frac{1}{5 - 1} \int_1^5 \frac{\ln(u)}{u} \,du$
  • Using u-substitution (letting $v = \ln(u)$):
    • $dv = \frac{1}{u} \,du$
    • When $u = 1$, $v = \ln(1) = 0$.
    • When $u = 5$, $v = \ln(5)$.
  • The integral becomes: $\frac{1}{4} \int_0^{\ln(5)} v \,dv = \frac{1}{4} [\frac{v^2}{2}]_0^{\ln(5)}$
  • Result: $\frac{1}{8}(\ln(5))^2$.

Properties of Logarithms

• Product Rule: $\log_b(AC) = \log_b(A) + \log_b(C)$.
• Quotient Rule: $\log_b(\frac{A}{C}) = \log_b(A) - \log_b(C)$.
• Power Rule: $\log_b(A^m) = m \log_b(A)$.

Historical Context of Logarithms and Calculation

• Invention: Logarithms were invented by John Napier to simplify complex astronomical calculations by converting multiplication into addition.
• Slide Rules: Before electronic calculators, engineers used slide rules. These devices used logarithmic scales on sliding rulers to perform approximate multiplications to a few significant digits.
• Electronic Transition: In $1971-1972$, the first electronic calculators (like those from HP) were released. They were marketed as "electronic slide rules."
• Economic Context: An early HP calculator cost approximately \$350 in $1971$. At the time, gasoline was roughly $29$ cents per gallon and minimum wage was approximately \$1.25 per hour. Buying a calculator required more than a month's worth of minimum wage earnings.

Questions & Discussion

• Passing with an A: A student asked what it takes to pass with an "A." The instructor noted it is highly background-dependent; some students are "cold" or "rusty" on Calculus I concepts, while others are better prepared.
• Time Constraints: A student advocated for more time on tests. The instructor agreed to allow the full class period for the next test because the integration techniques and area problems (like those in Chapter 7) are significantly more time-consuming.
• Substitution Logic: During a discussion on problem number 7, the instructor explained how to choose between substituting $u = \sec(x)$ or $u = \tan(x)$. Since the derivative of $\tan(x)$ is $\sec^2(x)$, and $\sec^2(x)$ was present in the integrand, $\tan(x)$ was the logical choice.
• Curriculum Note: The class is skipping Section 6.4; there is no homework assigned for that section.