AP Calculus BC Flashcards: Concavity Of Functions Over Their Domains

What this deck covers

This deck focuses on Concavity Of Functions Over Their Domains, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the second derivative test used for?

Answer: To determine concavity and points of inflection. The second derivative reveals how the slope is changing.

Flashcard 2: How do you determine concavity using the second derivative?

Answer: Concave up if $f''(x) > 0$; concave down if $f''(x) < 0$. Positive second derivative means upward curve, negative means downward.

Flashcard 3: What is the sign of $f''(x)$ when the function is concave down?

Answer: $f''(x) < 0$. Negative second derivative indicates the graph curves downward.

Flashcard 4: Find the concavity for $f(x) = x^2$ at $x = 0$.

Answer: Concave up. $f''(x) = 2 > 0$, so the parabola opens upward.

Flashcard 5: Find the concavity for $f(x) = -x^2$ at $x = 0$.

Answer: Concave down. $f''(x) = -2 < 0$, so the parabola opens downward.

Flashcard 6: State the general condition for a function to be concave up.

Answer: $f''(x) > 0$ over the interval. When the second derivative is positive everywhere.

Flashcard 7: What does $f''(x) < 0$ throughout an interval imply?

Answer: Function is concave down on that interval. The graph curves downward throughout the interval.

Flashcard 8: Determine concavity for $f(x) = 3x^5 - 10x^3$ at $x = 2$.

Answer: Concave up. $f''(x) = 60x^3 - 60x$ and $f''(2) = 360 > 0$.

Flashcard 9: Identify the concavity of $f(x) = x^4$ for $x > 0$.

Answer: Concave up. $f''(x) = 12x^2 > 0$ for all $x ≠ 0$.

Flashcard 10: What does $f''(x) < 0$ indicate about a function's graph?

Answer: The graph is concave down. The curve bends downward like an upside-down bowl.

Flashcard 11: Determine the concavity of $f(x) = \text{cos}(x)$ at $x = 0$.

Answer: Concave down. $f''(x) = -\cos(x)$ and $f''(0) = -1 < 0$.

Flashcard 12: For $f(x) = x^3$, what does $f''(0)$ tell us?

Answer: Possible inflection point. $f''(0) = 0$ and sign changes from negative to positive.

Flashcard 13: Evaluate concavity for $f(x) = x^4 - 4x^3$ at $x = 3$.

Answer: Concave up. $f''(x) = 12x^2 - 24x$ and $f''(3) = 36 > 0$.

Flashcard 14: Determine concavity for $f(x) = e^x$ for all $x$.

Answer: Concave up. $f''(x) = e^x > 0$ for all real $x$.

Flashcard 15: What does $f''(x) > 0$ throughout an interval imply?

Answer: Function is concave up on that interval. The graph curves upward throughout the interval.

Flashcard 16: Analyze concavity of $f(x) = e^{-x}$ at $x = 0$.

Answer: Concave up. $f''(x) = e^{-x} > 0$ for all $x$.

Flashcard 17: Is the function $f(x) = \text{ln}(x)$ concave up or down for $x > 0$?

Answer: Concave down. $f''(x) = -\frac{1}{x^2} < 0$ for all $x > 0$.

Flashcard 18: Determine the concavity for $f(x) = \frac{1}{x^2}$ for $x > 0$.

Answer: Concave up. $f''(x) = \frac{6}{x^4} > 0$ for all $x ≠ 0$.

Flashcard 19: Find $f''(x)$ for $f(x) = x^3 - 3x^2$ and determine concavity at $x=2$.

Answer: Concave up. $f''(x) = 6x - 6$ and $f''(2) = 6 > 0$.

Flashcard 20: Determine the concavity of $f(x) = \frac{1}{3}x^3 + x$ at $x = 1$.

Answer: Concave up. $f''(x) = 2x$ and $f''(1) = 2 > 0$.

Flashcard 21: Identify the inflection point condition.

Answer: $f''(x) = 0$ or $f''(x)$ changes sign. Where the second derivative is zero and changes sign.

Flashcard 22: What is the sign of $f''(x)$ when the function is concave up?

Answer: $f''(x) > 0$. Positive second derivative indicates the graph curves upward.

Flashcard 23: Evaluate $f''(x)$ for $f(x) = \text{ln}(x)$. What does it imply for $x > 0$?

Answer: Concave down. $f''(x) = -\frac{1}{x^2} < 0$ for all $x > 0$.

Flashcard 24: Find $f''(x)$ for $f(x) = x^5$. What is the concavity at $x=1$?

Answer: Concave up. $f''(x) = 20x^3$ and $f''(1) = 20 > 0$.

Flashcard 25: For $f(x) = \text{sin}(x)$, identify concavity at $x = \frac{\text{pi}}{2}$.

Answer: Concave down. $f''(x) = -\sin(x)$ and $f''(\frac{\pi}{2}) = -1 < 0$.

Flashcard 26: What indicates a point of inflection in terms of $f''(x)$?

Answer: $f''(x)$ changes sign at that point. The concavity switches direction at inflection points.

Flashcard 27: Determine if $f(x) = 7x^2 - 3x$ is concave up or down at $x=0$.

Answer: Concave up. $f''(x) = 14 > 0$, so always concave up.

Flashcard 28: What is the sign of $f''(x)$ when the function is concave up?

Answer: $f''(x) > 0$. Positive second derivative indicates the graph curves upward.

Flashcard 29: What is the sign of $f''(x)$ when the function is concave down?

Answer: $f''(x) < 0$. Negative second derivative indicates the graph curves downward.

Flashcard 30: Identify the inflection point condition.

Answer: $f''(x) = 0$ or $f''(x)$ changes sign. Where the second derivative is zero and changes sign.