Chris Monahan [Chris Monahan] - Calculus II

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About the Author

Chris Monahan has taught high school and college mathematics for more than 30 years. He has a Master of Arts in teaching from Colgate University and a Bachelor of Science from Manhattan College. President of the Association of Mathematics Teachers of New York State in 2009 and 2010, Chris has spoken at annual conferences around the country on the use of technology in the classroom, led workshops, and served on a number of committees for the New York State Education Department (NYSED). He currently consults for the NYSED, where he writes and assesses materials for the Common Core assessments.

APPENDIX
A

Solutions to Youve Got Problems

All the answers to the problems that you found in the Youve Got Problems sidebars throughout the book are listed here, organized by chapter.

Chapter 1

1.Using the relationship that 180 corresponds to radians, set up the proportion to arrive at .

2.You know that sin(2A) = 2sin(A)cos( A ). Use the Pythagorean identity sin2(A) + cos2(A) = 1 to solve for the value of becomes so and . Therefore, .

3.Separate the logarithm of a quotient into the difference of logarithms (change the square root to exponential form, too) . Use the rule for logarithms of powers to get .

4. and , so the coordinates are .

5.The common ratio for the series is . The first term of the series is 12, so the sum of the infinite geometric series is .

6.Factor the denominator to (x + 5)(x 1), and rewrite the fraction as . Multiply by the common denominator: x 19 = A(x 1) + B(x + 5).

Set x = 1: 18 = 6B so B = 3.

Set x = 5: 24 = 6A so A = 4.

Therefore, .

Chapter 2

1.Youll get the indeterminate form when you substitute x = 2. Factor and reduce the fractional expression and evaluate the limit to get .

2.Use the product rule for the first term in the function, ln(x) + 1 1 = ln(x).

3.Find the first derivative by using the chain rule and the trigonometric identity for the sine of the double angle, k'(x) = 2sin(3x)cos(3x)(3) = 3sin(6x). The second derivative is also found using the chain rule, k''(x) = 3cos(6x)(6) = 18cos(6x).

4.Evaluate the function to get the point through which the line passes, w (2) = 5. Find the derivative of w ( z ) using the quotient rule, , and evaluate it at z = 2, w '(2) = 11. The equation of the line is w 5 = 11( z 2).

5.Find the value of the first derivative. becomes , so . The derivative of this statement is . Substitute what was found for to get . Simplification beyond this is not necessary (and just plain tedious).

6.f'(c) = 3c2 and , so 3c2 = 31 implies that (but not because that is not in the interval [1,5]).

7.f'(