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Midterm 1 Review #2

Math 2300 – Calculus II – University of Colorado

Fall 2010 – Review for Midterm I

This review is from instructor Anca Radulescu. I've included my own solutions up through the 'table of integrals' problems. (There may exist better, or more correct solutions.)

§ Substitution

Understand the concept of substitution (as a rewrite of the chain rule in terms of integrals), when to use it and how to carry it through (for both indefinite and definite integrals).

Some practice exercises: Using an appropriate substitution, show that the two integrals are identical:

  1. A\displaystyle{\int _{0}^{{1}}{f(Ax)dx}=\int _{0}^{{A}}{f(x)dx}}

    Solution: If w=Ax and dw=A\, dx then

    \displaystyle A\int _{0}^{1}f(Ax)\, dx \displaystyle=A\int _{{w=0}}^{{w=A}}f(w)\frac{dw}{A}
    \displaystyle=\int _{0}^{A}f(w)\, dw
    \displaystyle=\int _{0}^{A}f(x)\, dx

Gateway Exam Info

Date: 
Thursday, September 23, 2010 - 14:00 - 15:00

The gateway exam will be held on September 23 during class. No calculators are allowed on this exam.

Midterm 1 Info

Date: 
Wednesday, September 15, 2010 - 17:15 - 18:45

Our midterm will be held in HALE270 during 5:15-6:45PM on Wednesday, September 14.

Book Assignment 4

Due Date: 
Monday, September 20, 2010 - 16:00

Section 7.8: Exercises 18, 20, 22
Section 8.1: Problem 14 and Exercise 26

Improper Integral Comparisons

Section: 
7.8
Date: 
Monday, September 13, 2010 - 14:00

Math 2300 Section 005 – Calculus II – Fall 2010

Comparison of Improper Integrals Examples – September 13, 2010

Comparison of Improper Integrals

  1. \int _{1}^{\infty}\frac{1}{\sqrt{x^{3}+5}}\, dx

    Solution: This is example 1 from Section 7.8 in the text.

  2. Converge or Diverge?

    \int _{{10}}^{\infty}\frac{1}{\ln(t)}\, dt

    Solution: Notice that for t\ge 10 we have

    \frac{1}{\ln(t)}>\frac{1}{t}

    and so

    \int _{{10}}^{\infty}\frac{1}{\ln(t)}\, dt>\int _{{10}}^{\infty}\frac{1}{t}\, dt=\infty

    and so this integral diverges. Notice that starting the integral limit at t=10 makes no difference here. The simple fact is that we know

    \int _{1}^{\infty}\frac{1}{t}\, dt=\int _{1}^{{b}}\frac{1}{t}\, dt+\int _{b}^{\infty}\frac{1}{t}\, dt

    is infinite, while the integral

    \int _{1}^{b}\frac{1}{t}\, dt

    is finite for any b satisfying 1<b<\infty. It is really the integral on the far left, the one with the infinite limit, that causes the problem here.

Improper Integrals

Section: 
7.7
Date: 
Wednesday, September 8, 2010 - 14:00 - Thursday, September 9, 2010 - 15:00

Math 2300 Section 005 – Calculus II – Fall 2010

Improper Integrals Examples – September 8-10, 2010

  1. Limits of Integration Not Finite

    \int _{1}^{\infty}\frac{1}{x^{p}}\, dx

    Solution: We find that

    \displaystyle\int _{1}^{\infty}\frac{1}{x^{p}}\, dx \displaystyle=\lim _{{b\rightarrow\infty}}\int _{1}^{b}\frac{1}{x^{p}}\, dx
    \displaystyle(p\neq 1) \displaystyle=\lim _{{b\rightarrow\infty}}\left[\frac{1}{-p+1}x^{{-p+1}}\right]_{1}^{b}
    \displaystyle(p\neq 1) \displaystyle=\frac{1}{-p+1}\left(\lim _{{b\rightarrow\infty}}b^{{-p+1}}-1\right).

    This will converge if and only if p>1, meaning that it will diverge when p\le 1. We showed this in class in somewhat painful detail.

Midterm 1 Review

Math 2300 – Calculus II – University of Colorado

Fall 2010 – Review for Midterm I

One of the main areas that will be emphasized during this midterm, compared to previous calculus II exams that you may find, is abstract and conceptual understanding. From what we have done in class and on homework/webwork, you have all of the tools to solve these problems. But, they may initially look a bit unfamiliar. Some examples of such problems are given here to help prepare you for the sorts of questions that will be asked on the midterm.

Some Things to Make Sure You Know

  • Substitution of indefinite and definite integrals.

  • Integration by parts of indefinite and definite integrals. It is also always helpful conceptually to know how to derive the formula for integration by parts.

  • Know how to use the table of integrals. More importantly, know how to derive some of the “basic” formulas on and related to that table: \int e^{x}\cos x\, dx and \int\cos^{n}x\, dx are classic examples, and you did one of them as a homework problem.

  • Know how to use the method of partial fractions, including how to apply long division to simplify an integral.

  • Know which trig substitutions to use, and how to use completing the square to rewrite quadratics when needed.

  • Understand how the Left, Right, Mid and Trap approximation methods work and how they relate (when a function is increasing/decreasing, concave up/concave down). Know how Simp relates. Of course, you should be able to calculate any of these and produce the relevant formulae for their calculation (even if you are using a calculator). Know what increase in the factor of subdivisions results in one more decimal of accuracy for each method.

  • Know how to set up the calculations for improper integrals, be able to follow through with the corresponding limit calculations, and be able to explain why this weird limit process is needed.

Take Home Quiz - Improper Integrals

Due Date: 
Monday, September 13, 2010 - 16:00

MATH2300-005 – Fall 2010 – University of Colorado

Quiz - Due Monday, September 13, 2010

  1. Evaluate the integral

    \int _{1}^{4}\frac{dx}{(x-2)^{{2/3}}}\, dx
  2. Evaluate the integral

    \int _{{-\infty}}^{0}\frac{e^{x}\, dx}{3-2e^{x}}
  3. Replacement Question: While trying to escape a wildfire, a family of four prairie dogs comes to a bridge. They realize that the bridge is old and only two of them may cross at a single time. Making matters worse, it is night and they only have one headlamp. So, two will cross and then one will return with the headlamp, then two more will cross and one will return, etc. When two prairie dogs cross the bridge, they will always travel at the pace of the slower prairie dog. Daddy Prarie Dog can cross the bridge in 1 minute, Mommy Prairie Dog can cross in 2 minutes, Junior Prairie Dog crosses in 5 minutes and Baby Prairie Dog crosses in 10 minutes. If the fire will arrive at their location and burn the bridge, taking with it any possibility of escape, in 17 minutes and 30 seconds, is there a chance for the family to survive?

Worksheet 3

Solutions to worksheet 3 are posted.

Book Assignment 3

Due Date: 
Monday, September 13, 2010 - 16:00

Section 7.6: Problem 8
Section 7.7: Problems 28 and 50

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