If f"(3) DNE then the graph of f have a point of inflection at x = 3. y = L is a horizontal asymptote if lim f(x) = L and lim f(x) = L. y = mx + b is a slant asymptote if lim [f(x) – y] = 0. Let f'(c) = 0 and the second derivative exists on an open interval containing c. If f"(c) < 0, then f has a local minimum at (c, f (c)).
If f"(3) DNE then the graph of f have a point of inflection at x = 3. y = L is a horizontal asymptote if lim f(x) = L and lim f(x) = L. y = mx + b is a slant asymptote if lim [f(x) – y] = 0. Let f'(c) = 0 and the second derivative exists on an open interval containing c. If f"(c) < 0, then f has a local minimum at (c, f (c)).
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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Concept explainers
Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Question
![### Educational Content on Calculus Concepts
#### Statement Analysis
**True (T) False (F) Questions:**
1. **T F**
If \( f''(3) \) does not exist, then the graph of \( f \) has a point of inflection at \( x = 3 \).
2. **T F**
\( y = L \) is a horizontal asymptote if \( \lim_{{x \to \infty}} f(x) = L \) and \( \lim_{{x \to -\infty}} f(x) = L \).
3. **T F**
\( y = mx + b \) is a slant asymptote if \( \lim_{{x \to \infty}} [f(x) - y] = 0 \).
4. **T F**
Let \( f'(c) = 0 \) and the second derivative exists on an open interval containing \( c \). If \( f''(c) < 0 \), then \( f \) has a local minimum at \( (c, f(c)) \).
5. **T F**
\( f(x + \Delta x) = f(x) + \Delta y \).
#### Conceptual Explanation:
1. **Point of Inflection:**
- A point of inflection occurs where the graph of a function changes concavity. The existence of the second derivative is not a necessary condition for an inflection point.
2. **Horizontal Asymptote:**
- A line \( y = L \) is a horizontal asymptote if the function approaches \( L \) as \( x \) approaches positive or negative infinity.
3. **Slant Asymptote:**
- A slant (or oblique) asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator in rational functions.
4. **Local Extrema with Second Derivative Test:**
- If the first derivative is zero and the second derivative is negative at a point, the function has a local maximum there, not a minimum.
5. **Linear Approximation:**
- The equation \( f(x + \Delta x) = f(x) + \Delta y \) is a simplistic interpretation of linearity and does not hold true unless \( \Delta y \) is specific](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb40fa447-d3c2-41e2-8596-d1d31b5ac1d4%2Fa8b1b9f3-f615-48a6-b55b-62b267fb5181%2F1yyy18_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Content on Calculus Concepts
#### Statement Analysis
**True (T) False (F) Questions:**
1. **T F**
If \( f''(3) \) does not exist, then the graph of \( f \) has a point of inflection at \( x = 3 \).
2. **T F**
\( y = L \) is a horizontal asymptote if \( \lim_{{x \to \infty}} f(x) = L \) and \( \lim_{{x \to -\infty}} f(x) = L \).
3. **T F**
\( y = mx + b \) is a slant asymptote if \( \lim_{{x \to \infty}} [f(x) - y] = 0 \).
4. **T F**
Let \( f'(c) = 0 \) and the second derivative exists on an open interval containing \( c \). If \( f''(c) < 0 \), then \( f \) has a local minimum at \( (c, f(c)) \).
5. **T F**
\( f(x + \Delta x) = f(x) + \Delta y \).
#### Conceptual Explanation:
1. **Point of Inflection:**
- A point of inflection occurs where the graph of a function changes concavity. The existence of the second derivative is not a necessary condition for an inflection point.
2. **Horizontal Asymptote:**
- A line \( y = L \) is a horizontal asymptote if the function approaches \( L \) as \( x \) approaches positive or negative infinity.
3. **Slant Asymptote:**
- A slant (or oblique) asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator in rational functions.
4. **Local Extrema with Second Derivative Test:**
- If the first derivative is zero and the second derivative is negative at a point, the function has a local maximum there, not a minimum.
5. **Linear Approximation:**
- The equation \( f(x + \Delta x) = f(x) + \Delta y \) is a simplistic interpretation of linearity and does not hold true unless \( \Delta y \) is specific
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