Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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## Text Transcription for Educational Purposes ### Problems 1-6: Function and Derivative Equations 1. **\( f(x) = \frac{2 + x - x^2}{x^2 - 2x + 1} \); \( f''(x) = \frac{2(7 - x)}{(x - 1)^4} \)** 2. **\( f(x) = \frac{-x^2}{(x - 1)^2} \); \( f''(x) = \frac{2(x + 1)}{(x - 1)^4} \)** 3. **\( f(x) = \frac{x^2 + 1}{x^2 - 2} \); \( f''(x) = \frac{6(3x^2 + 2)}{(x^2 - 2)^3} \)** 4. **\( f(x) = x\sqrt{a^2 - x^2} \); \( f''(x) = \frac{x(2x^2 - 3a^2)}{(a^2 - x^2)^{3/2}} \)** ### Problems 7-34: Determine Concavity and Points of Inflection In these problems, determine concavity and the x-values where points of inflection occur. Do not sketch the graphs. 7. \( y = -2x^2 + 4x \) 8. \( y = -74x^2 + 19x - 37 \) 9. \( y = 4x^3 + 12x^2 - 12x \) 10. \( y = x^3 - 6x^2 + 9x + 1 \) 11. \( y = ax^3 + bx^2 + cx + d \) 12. \( y = x^4 - 8x^2 - 6 \) 13. \( y = 2x^4 - 48x^2 + 7x + 3 \) 14. \( y = \frac{x^4}{4} + \frac{9x^2}{2} + 2x \) 15. \( y = 2x^{1/5} \) 16. \( y = \frac{a}{x^3} \
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Transcribed Image Text:## Text Transcription for Educational Purposes ### Problems 1-6: Function and Derivative Equations 1. **\( f(x) = \frac{2 + x - x^2}{x^2 - 2x + 1} \); \( f''(x) = \frac{2(7 - x)}{(x - 1)^4} \)** 2. **\( f(x) = \frac{-x^2}{(x - 1)^2} \); \( f''(x) = \frac{2(x + 1)}{(x - 1)^4} \)** 3. **\( f(x) = \frac{x^2 + 1}{x^2 - 2} \); \( f''(x) = \frac{6(3x^2 + 2)}{(x^2 - 2)^3} \)** 4. **\( f(x) = x\sqrt{a^2 - x^2} \); \( f''(x) = \frac{x(2x^2 - 3a^2)}{(a^2 - x^2)^{3/2}} \)** ### Problems 7-34: Determine Concavity and Points of Inflection In these problems, determine concavity and the x-values where points of inflection occur. Do not sketch the graphs. 7. \( y = -2x^2 + 4x \) 8. \( y = -74x^2 + 19x - 37 \) 9. \( y = 4x^3 + 12x^2 - 12x \) 10. \( y = x^3 - 6x^2 + 9x + 1 \) 11. \( y = ax^3 + bx^2 + cx + d \) 12. \( y = x^4 - 8x^2 - 6 \) 13. \( y = 2x^4 - 48x^2 + 7x + 3 \) 14. \( y = \frac{x^4}{4} + \frac{9x^2}{2} + 2x \) 15. \( y = 2x^{1/5} \) 16. \( y = \frac{a}{x^3} \
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