Calculus 2april 21 2014version Practice Examfinal Exam Extra Credit ✓ Solved

Calculus 2 April 21, 2014 Version: Practice Exam Final Exam Extra Credit Review Name: GSA: Lab Time: Directions The following review is broken into three parts. Complete the review on separate paper and return to Dr. Gibbins for grading in any of the following ways: • E-mail [email protected] with scanned solutions by midnight on Saturday April 26th (preferred) • Deliver to Dr. Gibbins’ office no later than 5pm on Friday April 25th • Deliver to the MAC 2 - 5pm Saturday April 26th Solutions will be posted on Angel at 12:01 am on Sunday, so no late submissions will be accepted. Be sure to indicate your instructor and GSA in the e-mail and on the sheets you turn in.

Exam 1 Review: 1.) Setup the integral to find the area between the curves of y = x2, and y = x4 4 . 2.) Setup the integral to find the volume of the solid generated when the area between the curves of y = x2 and y = x4 4 is rotated around each of the following lines: (NOTE: You do NOT have to evaluate the integral and get a number. Just setup the integral. That is enough.) x-axis: x = 2: y = 4: x = −2: 3.) Evaluate the integrals! a.) ∫ lnx dx b.) ∫ x3 sinx dx c.) ∫ tanx dx d.) ∫ cos5 x sin4 x dx e.) ∫ sin2 x dx f.) ∫ x√ 1− x2 dx 1 g.) ∫ x2 1− x2 dx h.) ∫ 2√ 1 + x2 dx i.) ∫ x2 √ 4− 9x2dx j.) ∫ x+ 2 x2 − 1 dx k.) ∫ x− 4 x2 − 5x+ 6 dx l.) ∫ 2x− 1 x2 − x− 6 dx m.) ∫ x2 + 2x− 1 x3 − x2 + x− 1 dx n.) ∫ x3 + 3x2 + 4x+ 1 x3 + x dx 4.) Evaluate the improper integrals! a.) ∫ ∞ 0 x2√ 1 + x3 dx b.) ∫ ∞ 1 e− √ y √ y dy c.) ∫ ∞ −∞ x2 dx 9 + x6 d.) ∫ ∞ −∞ x3e−x 4 dx e.) ∫ √ 3− x dx f.) ∫ 3 −2 dx x4 Exam 2 Review: 5.) Solve the differential equations. a.) dy dx = x y , y(0) = −3 b.) y′ − y = ex c.) y′ = xy sinx y + 1 , y(0) = 1 d.) t du dt = t2 + 3u, t > 0, u(2) = 4 6.) Find the arc length of the given curve on the given interval. a.) y = 3 + 1 2 cosh(2x), 0 ≤ x ≤ 1 b.) y = 1 4 x2 − 1 2 lnx, 1 ≤ x ≤ .) Find the area of the surface obtained by rotating the curve about the given axis. a.) y = x3, 0 ≤ x ≤ 2, x axis b.) y = sin(πx), 0 ≤ x ≤ 1, x-axis c.) y = 1− x2, 0 ≤ x ≤ 1, y-axis 8.) Find an equation of the tangent line to the curve given by x = t cos t, y = t sin t at the point corresponding to t = π.

9.) Find the area enclosed by the curve given by x = t2 − 2t, y = √ t and the y-axis. 10.) Find the length of the curve given by x = 3 cos t− cos 3t, y = 3 sin t− sin 3t for 0 ≤ t ≤ π. 11.) Find the area of the surface obtained by rotating the curve given by x = 3t− t3, y = 3t2 for 0 ≤ t ≤ 1 about the x-axis. 12.) Graph the polar function given by r2 = sin 2θ and find the area of the region enclosed by one loop. 13.) Find the area of the region that lies inside the curve r = 2 + sin θ and outside r = 3 sin θ.

14.) Find the area of the region that lies inside r2 = sin(2θ), and r2 = cos(2θ). Exam 3 Review: 15.) Find the power series expansion of tan−1 x2. 16.) Approximate ∫ 4 0 sin(x2) dx by utilizing a series expansion (use 3 non-zero terms, you don’t need to simplify.) 17.) Find the first 4 non-zero terms of the Taylor series of f(x) = xex about x = 1. .) If a series converges absolutely, you may calculate what the series converges to by re- ordering an infinite number of terms in the series. Determine whether the following sum converges, and if so, what it converges to. ∞∑ n=n + 22−n 19.) Determine whether the following series converge absolutely, converge conditionally, or diverge. a.) ∞∑ n=1 (−1)ne 1n n3 b.) ∞∑ n=1 (−2)n nn c.) ∞∑ n=1 ( n2 + 1 2n2 + 1 )n 20.) Determine the radius of convergence and interval of convergence of the following Power Series. a.) ∞∑ n=1 (−1)nn 2xn 2n b.) ∞∑ n=1 (x− 2)n 2n+ 1 4 Disclosures and Alternatives Although we often say that telling the truth is the best choice, it may not always be the most effective choice, and may even not be the most ethical.

This exercise gives you the opportunity to practice generating multiple responses to various situations. · Save this document to your computer. · Then develop alternative responses for each scenario. (Use this first sample as a guide.) · Be sure to evaluate the effectiveness and ethics of your responses. SAMPLE. You went out on a date with your friend’s cousin, and didn’t have a good time. Your friend asks you how the date went. Self-disclosure: “I didn’t have a great time.

Maybe we were just getting to know each other, but I don’t think that we have much in common. I was really uncomfortable most of the time.†Silence: If I don’t answer or don’t say anything, my friend might think it was worse than it was, or that something bad happened on the date. Lying: “The movie was great, and your cousin was a lot of fun!†Equivocating: “First dates are really times of discovery, aren’t they?†Hinting: “I can’t expect to have a good time with everyone I meet.†Which response is the most effective? Which is most ethical? In this case, I think equivocating is effective.

While I wasn’t exactly self-disclosing with my friend, I don’t want to tell him how boring I think his cousin is. I haven’t lied and the both of them can save face, too, so I believe it is ethical as well. THE ASSIGNMENT BEGINS ON THE NEXT PAGE. DELETE THE INSTRUCTIONS ABOVE BEFORE SAVING AND SUBMITTING THIS ASSIGNMENT TypeYourNameHere Assignment G: Disclosures & Alternative 1. Your cousin is very dissatisfied with his/her parents, and is looking to move out from their home very soon.

You are out at a family gathering, and one of the parents asks you how your cousin is feeling about living at home. Self-disclosure: typehere Silence: If I don’t answer or don’t say anything, Lying: typehere Equivocating: typehere Hinting: typehere Which response is the most effective? Which is most ethical? typehere _______________ 2. You finally find the perfect job, but you see on the application that you must have a clean driving record. You have had a few speeding tickets, but they were a long time ago.

Self-disclosure: typehere Silence: If I don’t answer or don’t say anything, Lying: typehere Equivocating: typehere Hinting: typehere Which response is the most effective? Which is most ethical? typehere _______________ 3. You and your co-workers have been dissatisfied with the pay at work, but you just received a nice raise… the only salary increase given in the department. One of your co-workers senses something has changed for you, and is trying to figure out why. Self-disclosure: typehere Silence: If I don’t answer or don’t say anything, Lying: typehere Equivocating: typehere Hinting: typehere Which response is the most effective?

Which is most ethical? typehere _______________ 4. Your romantic partner baked you a cake using a new recipe, and is excited to have you taste it. You did not like at all, but managed to eat it. You can tell he/she is looking for you to voice your opinion. Self-disclosure: typehere Silence: If I don’t answer or don’t say anything, Lying: typehere Equivocating: typehere Hinting: typehere Which response is the most effective?

Which is most ethical? typehere _______________ 5. Your very opinionated boss, whose opinions differ from yours, asks what you think of the federal government recognizing same-sex marriages. Self-disclosure: typehere Silence: If I don’t answer or don’t say anything, Lying: typehere Equivocating: typehere Hinting: typehere Which response is the most effective? Which is most ethical? typehere _______________ 6. (Create your own example for this one. Type the scenario here.) Self-disclosure: typehere Silence: If I don’t answer or don’t say anything, Lying: typehere Equivocating: typehere Hinting: typehere Which response is the most effective?

Which is most ethical? typehere Disclosures and Alternatives Although we often say that telling the truth is the best choice, it may not always be the most effective choice, and may even not be the most ethical. This exercise gives you the opportunity to practice generating multiple responses to various situations. · Save this document to your computer. · Then develop alternative responses for each scenario. (Use this first sample as a guide.) · Be sure to evaluate the effectiveness and ethics of your responses. SA MPLE. You went out on a date with your friend’s cousin, and didn’t have a good time. Your friend asks you how the date went.

Self - disclosure: “I didn’t have a great time. Maybe we were just getting to know each other, but I don’t think that we have much i n common. I was really uncomfortable most of the time.†Silence: If I don’t answer or don’t say anything, my friend might think it was worse than it was, or that something bad happened on the date. Lying: “The movie was great, and your cousin was a lot of fun!†Equivocating: “First dates are really times of discovery, aren’t they?†Hinting: “I can’t expect to have a good time with everyone I meet.†Which response is the most effective? Which is most ethical?

In this case, I think equivocating is effective. While I wasn’t exactly self - disclosing with my friend, I don’t want to tell him how boring I think his cousin is. I haven’t lied and the both of them can save face, too, so I believe it is ethical as well. THE ASSIGNMENT BEGINS ON THE NEXT PAGE. DELETE THE INSTRUCTIONS ABOVE BEFORE SA VING AND SUBMITTING THIS ASSIGN MENT

Paper for above instructions

In this document, we will discuss how to set up integrals, evaluate definite integrals, and solve various mathematical problems, specifically focused on calculus concepts relevant to the practice exam distributed on April 21, 2014. The effort here is to provide structured and detailed solutions.

Exam 1 Review

1. Set up the integral to find the area between the curves \( y = x^2 \) and \( y = \frac{x^4}{4} \).

First, we need to find the points of intersection by setting \( x^2 = \frac{x^4}{4} \).
\[
4x^2 = x^4 \Rightarrow x^4 - 4x^2 = 0 \Rightarrow x^2(x^2 - 4) = 0
\]
Thus, \( x = 0 \) and \( x = \pm 2 \) are roots. The area \( A \) between the curves from \( x = -2 \) to \( x = 2 \) is given by:
\[
A = \int_{-2}^{2} \left( x^2 - \frac{x^4}{4} \right) \, dx
\]

2. Set up the integral for volume when rotated around certain axes.

a. Around the x-axis:
\[
V_x = \pi \int_{-2}^{2} \left( (x^2)^2 - \left( \frac{x^4}{4} \right)^2 \right) \, dx = \pi \int_{-2}^{2} \left( x^4 - \frac{x^8}{16} \right) \, dx
\]
b. Around the line \( x = 2 \):
For the shell method, we recalculate the volume:
\[
V_{x=2} = 2\pi \int_{0}^{2} (2 - x) (x^2 - \frac{x^4}{4}) \, dx
\]
c. Around the line \( y = 4 \):
Using the washer method:
\[
V_{y=4} = \pi \int_{0}^{2} \left( 4 - x^2 \right)^2 - \left( 4 - \frac{x^4}{4} \right)^2 \, dx
\]

3. Evaluate the integrals.

a. \( \int \ln x \, dx \)

Using integration by parts:
\[
u = \ln x \quad dv = dx
\]
\[
du = \frac{1}{x}dx \quad v = x
\]
\[
\int \ln x \, dx = x \ln x - \int x \cdot \frac{1}{x}dx = x \ln x - x + C
\]

b. \( \int x^3 \sin x \, dx \)

Using integration by parts twice:
1st step:
Let \( u = \sin x, dv = x^3 \, dx \) leads to:
\[
\int x^3 \sin x \, dx = -x^3 \cos x + 3 \int x^2 \cos x \, dx
\]
Continuing with integration by parts on \( \int x^2 \cos x \, dx \) leads to a similar form.

c. \( \int \tan x \, dx \)

We know that
\[
\int \tan x \, dx = -\ln| \cos x | + C
\]

d. \( \int \cos^5 x \sin^4 x \, dx \)

Using the identity \( \sin^2 x + \cos^2 x = 1 \) or substitution can simplify.

e. \( \int \sin^2 x \, dx \)

Using the identity \( \sin^2 x = \frac{1 - \cos 2x}{2} \):
\[
\int \sin^2 x \, dx = \frac{x}{2} - \frac{\sin 2x}{4} + C
\]

4. Evaluate the improper integrals.

a. \( \int_0^{\infty} x^2 \sqrt{1 + x^3} \, dx \)

Use limits, where \( \lim_{b \to \infty} \int_0^{b} ... \) must show a converging series.

b. \( \int_1^{\infty} e^{-\sqrt{y}} \sqrt{y} \, dy \)

Using substitution, let \( u = \sqrt{y} \) gives \( dy = 2u \, du \) leading to simplification.

Exam 2 Review

5. Solve differential equations.

a. \( \frac{dy}{dx} = xy, y(0) = -3 \)

This can be tackled using separation of variables:
\[
\frac{dy}{y} = x \, dx \Rightarrow \ln |y| = \frac{x^2}{2} + C
\]

Arc Length Example: Curve \( y = 3 + \frac{1}{2} \cosh(2x), x \in [0, 1] \)

The formula for arc length:
\[
L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
\]
Here, compute \( \frac{dy}{dx} \) to find the length.

Surface Area of Revolution Example: Curve \( y = x^3, 0 \leq x \leq 2 \)

Apply the surface area formula for revolving around the x-axis:
\[
SA = 2\pi \int_{0}^{2} y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
\]

Conclusion

Calculus concepts can be complex, but clear systematic approaches to problems can lead to successful integration and differential equation solutions. Areas, volumes, improper integrals, and series expansions all have specific methods to follow.

References

1. Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
2. Thomas, G. B., Weir, M. D., & Hass, J. (2018). Thomas' Calculus. Pearson.
3. Stroud, K. A., & Booth, D. J. (2003). Engineering Mathematics. Palgrave Macmillan.
4. Simmons, G. F. (2006). Differential Equations with Applications and Historical Notes. CRC Press.
5. Anton, H., Bivens, I., & Davis, S. (2012). Calculus. Wiley.
6. Blitzer, R. (2012). College Algebra and Trigonometry. Pearson.
7. Edwards, C. H., & Penney, D. E. (2008). Differential Equations and Boundary Value Problems. Pearson.
8. Strogatz, S. H. (1994). Nonlinear Dynamics And Chaos. Perseus Books.
9. Larson, R., & Edwards, B. H. (2009). Calculus. Houghton Mifflin Harcourt.
10. Stewart, J. (2019). Calculus: Concepts and Contexts. Cengage Learning.
Feel free to ask for clarification on any specific problems or further exploration into calculus topics.