Clac 2 test
.pdf
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Edmonds Community College *
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Course
152
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
2
Uploaded by BaronGerbil3993
**Calculus II Test**
**Instructions:**
- Answer all questions thoroughly and provide detailed explanations where required.
- You have 120 minutes to complete the entire test.
- Write your answers clearly and legibly.
**Multiple Choice:**
Choose the best answer for each question.
1. What is the convergence test commonly used to determine if an infinite series converges or
diverges when the terms alternate in sign?
a) Ratio test
b) Integral test
c) Alternating series test
d) Comparison test
2. Which of the following integrals represents the area bounded by the curve \(y = e^x\), the
x-axis, and the lines \(x = 0\) and \(x = 2\)?
a) \(\int_0^2 e^x \, dx\)
b) \(\int_0^2 e^{-x} \, dx\)
c) \(\int_0^2 e^{2x} \, dx\)
d) \(\int_0^2 e^{x^2} \, dx\)
3. What is the volume of the solid generated by revolving the region bounded by the curves \(y =
x^2\) and \(y = 4\) about the y-axis?
a) \(\frac{32}{3} \pi\)
b) \(8 \pi\)
c) \(\frac{16}{3} \pi\)
d) \(\frac{64}{3} \pi\)
4. What is the equation of the tangent line to the curve \(y = \ln(x)\) at the point (1, 0)?
a) \(y = x\)
b) \(y = x - 1\)
c) \(y = 2x\)
d) \(y = x + 1\)
5. Which of the following series converges by the integral test?
a) \(\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}\)
b) \(\sum_{n=1}^{\infty} \frac{1}{n^2}\)
c) \(\sum_{n=1}^{\infty} \frac{1}{n}\)
d) \(\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}\)
**Short Answer:**
Provide detailed answers to the following questions.
6. Explain the concept of convergence and divergence of infinite series. Discuss the importance
of convergence tests in determining the convergence of series.
7. Describe the method of partial fraction decomposition and explain how it is used to simplify
integrals of rational functions.
8. Discuss the application of improper integrals in calculus, including the types of improper
integrals and how they are evaluated.
9. Explain the concept of polar coordinates and polar curves. Describe how to convert between
polar and rectangular coordinates.
10. Define the concepts of arc length and surface area of curves and surfaces. Provide formulas
for calculating arc length and surface area and explain their derivations.
**Essay:**
Answer the following essay question in detail.
11. Discuss the concept of sequences and series in calculus. Explain the difference between
arithmetic and geometric sequences/series and provide examples to illustrate their properties.
**Bonus Question:**
This question is optional and will provide extra credit if answered correctly.
12. Explain the method of integration by parts and provide a step-by-step example of its
application to evaluate a definite integral.
**End of Test**
Remember to review your answers before submitting your test. Good luck!
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