W. Michael Kelley [W. Michael Kelley] - Calculus I

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About the Author

W. Michael Kelley is an award-winning math instructor from Calvert County, Maryland. He graduated from St. Marys College of Maryland as a mathematics major in 1994 with two goals in mind: to make mathematics easier for people, and to learn to communicate with raccoons and eventually advance through their complex social structure. Once the cold-and-flu medicine wore off, he decided just to stick with the first goal.

During his 7 years as a high school teacher, he received many accolades and was recognized as an outstanding teacher by numerous organizations, including the Maryland Council of Teachers of Mathematics. However, he most valued being voted the student bodys favorite teacher each year he taught calculus.

Kelley is the webmaster of the website calculus-help.com, which has been lauded by many, including the National Education Association and the National Council of Teachers of Mathematics. The site began as, and remains, a free resource of calculus exercises and tools. It posts a new problem every week to assist those fighting the beast that is calculus. If you enjoy this book, or have any feedback about it, pop Michael an email via the website.

Michael wrote the first edition of The Complete Idiots Guide to Calculus in 2001, and it quickly became one of the best-selling Idiots Guides, even though it was only his second book. Now, 15 years later, Mike has written 13 books, and he is still going strong.

APPENDIX
A

Solutions to Youve Got Problems

All of the answers to the problems that haunted you throughout the book are listed here, organized by chapter. The important steps are all shown, unless the skill needed to complete a problem was already discussed in a previous chapter. For example, once you learn how to do u-substitution in , I no longer focus on its details if problems in subsequent chapters require u-substitution as a component of their answers. If I didnt do that, this appendix would be a book unto itself!

Chapter 2

1. 6x + 9y = 11. Dont forget that 6x has to be positive to be in standard form. You may need to multiply everything by 1.

2. 2x 3y = 6. You can treat (0,2) as a point or use it as the y-intercept, so both forms work.

3. . Remember that .

4. . Calculate the slope of line j: . Substitute the slope and a point, such as (x,y) = (4,3), into slope-intercept form: . Solve: . Remember that b represents the y-intercept in slope-intercept form.

5. . When you square everything, you get 9x6y4, and the term with the negative exponent has to be moved to the denominator.

6. 7xy(x 3y2). The greatest common factor is 7xy, so divide it out of each term to get the factored form and write 7xy in front.

7. (2x + 7)(4x2 14x + 49). This is a sum of perfect cubes, with a = 2x and b = 7.

8. x = 0,4. Method one: Factor out 3x. Method two: First divide by 3 to get x2 + 4x = 0. Half of 4 is 2, whose square, 4, should be added to both sides. Take the square root of both sides, and dont forget the symbol. Then subtract 2 from both sides. Method three:a = 3, b = 12, and c = 0, because there is no constant term.

9. y = 3x2 + x 2. Set the equations x = 1 and equal to 0 to get x + 1 = 0 and . To eliminate fractions, multiply the second equation by 3 to get 3x 2 = 0. The quadratic equation you are seeking is y = (x + 1)(3x 2). Multiply and simplify.

Chapter 3

1. 4. f(43) = 7; g(7) = 64; h(64) = 4.

2. g(2) = 8, g(0) = 12, g(5) = 4. Because x = 2 and x = 0 are both less than or equal to 0, substitute them into 12 x2. Substitute 5 into .

3. 9. The graph of q(x) passes through point (1,5), so q(1) = 5. Next, p(5) = 4 (5) = 4 + 5 = 9.

4. Origin symmetry. Plug in x for x and y for y to get . Multiply both sides by 1, and youll get the original function.

5.

6. . After switching x and y, subtract 5 from both sides and then eliminate by multiplying each side of the equation by .

7. y = x2 3x + 3. Start by solving the x equation for