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54. | +. -1 | 0.5

54. | +. -1 | 0.5

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I only need 54. Calculus 1
### Functions and Sign Charts #### Overview The following functions are given, and corresponding sign charts are used to determine the behavior of these functions: 49. **\( f(x) = e^{1+2x-x^2} \)** 50. **\( f(x) = \frac{2^x}{1-2^x} \)** 51. **\( f(x) = \sin\left(x - \frac{\pi}{4}\right) \)** 52. **\( f(x) = 3 \cos\left(\frac{\pi}{2} x\right) + 5 \)** #### Sign Charts For each set of sign charts in Exercises 53–62, sketch a possible graph of \( f \). **53. Sign Chart** - **\( f'(x) \) (Red Line):** - Positive when \( x < 2 \) and when \( x > 2 \). - Negative between \( x = 2 \). - **\( f''(x) \) (Green Line):** - Positive when \( x < 2 \). - Negative when \( x = 2 \) and for \( x > 2 \). **54. Sign Chart** - **\( f'(x) \) (Red Line):** - Negative when \( x < -1 \) and when \( x > 2 \). - Positive between \( x = -1 \) and \( x = 2 \). - **\( f''(x) \) (Green Line):** - Positive when \( x < 0.5 \). - Negative when \( x > 0.5 \). **55. Sign Chart** - **\( f'(x) \) (Red Line):** - Positive when \( x < -2 \) and when \( x > -2 \). - Negative between \( x = -2 \). - **\( f''(x) \) (Green Line):** - Negative when \( x < -2 \). - Positive when \( x > -2 \). #### Explanation - **\( f'(x) \)** indicates the intervals of increase and decrease in the function. A positive \( f'(x) \) means the function is increasing, while a negative \( f'(x) \) means it is decreasing. -
Transcribed Image Text:### Functions and Sign Charts #### Overview The following functions are given, and corresponding sign charts are used to determine the behavior of these functions: 49. **\( f(x) = e^{1+2x-x^2} \)** 50. **\( f(x) = \frac{2^x}{1-2^x} \)** 51. **\( f(x) = \sin\left(x - \frac{\pi}{4}\right) \)** 52. **\( f(x) = 3 \cos\left(\frac{\pi}{2} x\right) + 5 \)** #### Sign Charts For each set of sign charts in Exercises 53–62, sketch a possible graph of \( f \). **53. Sign Chart** - **\( f'(x) \) (Red Line):** - Positive when \( x < 2 \) and when \( x > 2 \). - Negative between \( x = 2 \). - **\( f''(x) \) (Green Line):** - Positive when \( x < 2 \). - Negative when \( x = 2 \) and for \( x > 2 \). **54. Sign Chart** - **\( f'(x) \) (Red Line):** - Negative when \( x < -1 \) and when \( x > 2 \). - Positive between \( x = -1 \) and \( x = 2 \). - **\( f''(x) \) (Green Line):** - Positive when \( x < 0.5 \). - Negative when \( x > 0.5 \). **55. Sign Chart** - **\( f'(x) \) (Red Line):** - Positive when \( x < -2 \) and when \( x > -2 \). - Negative between \( x = -2 \). - **\( f''(x) \) (Green Line):** - Negative when \( x < -2 \). - Positive when \( x > -2 \). #### Explanation - **\( f'(x) \)** indicates the intervals of increase and decrease in the function. A positive \( f'(x) \) means the function is increasing, while a negative \( f'(x) \) means it is decreasing. -
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