AP Calculus AB 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 AB.

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: Find the concavity of $f(x) = \text{ln}(x)$ at $x = 2$.

Answer: $f''(x) = -\frac{1}{x^2}$, $f''(2) < 0$, concave down. Natural logarithm has negative second derivative.

Flashcard 2: Determine concavity of $f(x) = x^3 + 6x^2 + 9x$ at $x = -2$.

Answer: $f''(x) = 6x + 12$, $f''(-2) = 0$. Concavity is undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 3: What is the concavity of $f(x) = x^2$ over its domain?

Answer: $f''(x) = 2 > 0$, so $f(x)$ is concave up everywhere. The second derivative is constant and positive.

Flashcard 4: At what point does concavity change for $f(x) = x^3$?

Answer: Concavity changes at $x = 0$, where $f''(x) = 0$. Inflection occurs where $f''(x) = 0$ and changes sign.

Flashcard 5: What is the concavity of $f(x) = x^3 + 3x^2$ at $x = -1$?

Answer: $f''(x) = 6x + 6$, $f''(-1) = 0$. Concavity is undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 6: Identify concavity of $f(x) = \text{arctan}(x)$ at $x = 0$.

Answer: $f''(x) = -\frac{2x}{(1+x^2)^2}$, $f''(0) = 0$. Undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 7: Identify an inflection point for $f(x) = x^3 - 3x^2 + x$.

Answer: Inflection at $x = 1$ where $f''(x)$ changes sign. Second derivative changes sign at this point.

Flashcard 8: Determine concavity of $f(x) = e^x$ over its domain.

Answer: $f''(x) = e^x > 0$, so $f(x)$ is concave up everywhere. The exponential function has constant positive concavity.

Flashcard 9: Determine concavity of $f(x) = x^4 - 4x^2$ at $x = 1$.

Answer: $f''(x) = 12x^2 - 8$, $f''(1) = 4 > 0$, concave up. Positive second derivative at the specified point.

Flashcard 10: What is the concavity of $f(x) = \text{ln}(x)$ over $x > 0$?

Answer: $f''(x) = -\frac{1}{x^2} < 0$, concave down for $x > 0$. Natural log is concave down on its domain.

Flashcard 11: Determine concavity of $f(x) = x^3 - x$ at $x = 1$.

Answer: $f''(x) = 6x$, $f''(1) = 6 > 0$, concave up. Positive second derivative indicates upward concavity.

Flashcard 12: Identify the concavity for $f(x) = \text{cos}(x)$ at $x = \frac{\text{π}}{2}$.

Answer: $f''(x) = -\text{cos}(x)$, $f''(\frac{\text{π}}{2}) = 0$, undetermined. When $f''(x) = 0$, concavity cannot be determined.

Flashcard 13: What is the concavity of $f(x) = 4x^4 - x^2$ at $x = 0$?

Answer: $f''(x) = 48x^2 - 2$, $f''(0) = -2 < 0$, concave down. Negative second derivative indicates downward concavity.

Flashcard 14: What is the concavity of $f(x) = -x^2$ over its domain?

Answer: $f''(x) = -2 < 0$, so $f(x)$ is concave down everywhere. Constant negative second derivative everywhere.

Flashcard 15: Identify concavity of $f(x) = x^2 + 4$ at $x = 0$.

Answer: $f''(x) = 2 > 0$, concave up. Constant positive second derivative for all quadratics.

Flashcard 16: What is the concavity of $f(x) = x^5$ at $x = 0$?

Answer: $f''(x) = 20x^3$, $f''(0) = 0$. Concavity is undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 17: Identify the concavity for $f(x) = \frac{1}{x}$ at $x = -1$.

Answer: $f''(x) = \frac{2}{x^3}$, $f''(-1) = -2 < 0$, concave down. Negative second derivative indicates downward concavity.

Flashcard 18: How does the sign of $f''(x)$ affect the graph of $f(x)$?

Answer: If $f''(x) > 0$, $f(x)$ is concave up; if $f''(x) < 0$, $f(x)$ is concave down. Sign of the second derivative directly determines concavity.

Flashcard 19: Identify concavity of $f(x) = \text{tan}(x)$ at $x = 0$.

Answer: $f''(x) = 2\text{tan}(x)\text{sec}^2(x)$, $f''(0) = 0$. Undetermined. Zero second derivative requires further analysis.

Flashcard 20: Determine concavity of $f(x) = \text{arcsin}(x)$ at $x = 0$.

Answer: $f''(x) = \frac{x}{(1-x^2)^{\frac{3}{2}}}$, $f''(0) = 0$. Undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 21: Determine concavity of $f(x) = e^{-x}$ over its domain.

Answer: $f''(x) = e^{-x} > 0$, so $f(x)$ is concave up everywhere. Exponential decay function maintains positive concavity.

Flashcard 22: Determine concavity of $f(x) = x^2 - 4x$ at $x = 3$.

Answer: $f''(x) = 2 > 0$, concave up. Constant positive second derivative for quadratic functions.

Flashcard 23: Identify concavity of $f(x) = \text{sin}(x)$ at $x = 0$.

Answer: $f''(x) = -\text{sin}(x)$, $f''(0) = 0$. Concavity is undetermined. Zero second derivative makes concavity indeterminate.

Flashcard 24: Identify concavity of $f(x) = x^3 - 3x$ at $x = -1$.

Answer: $f''(x) = 6x$, $f''(-1) = -6 < 0$, concave down. Negative second derivative indicates downward concavity.

Flashcard 25: Identify concavity of $f(x) = \frac{1}{x^2}$ at $x = 1$.

Answer: $f''(x) = \frac{6}{x^4}$, $f''(1) = 6 > 0$, concave up. Positive second derivative confirms upward concavity.

Flashcard 26: What is the concavity of $f(x) = x^4$ at $x = 1$?

Answer: $f''(x) = 12x^2$, $f''(1) = 12 > 0$, concave up. Positive second derivative indicates upward concavity.

Flashcard 27: Determine concavity of $f(x) = \frac{1}{x}$ at $x = 1$.

Answer: $f''(x) = \frac{2}{x^3}$, $f''(1) = 2 > 0$, concave up. Positive second derivative at a point indicates concave up.

Flashcard 28: Identify the concavity for $f(x) = x^3$ at $x = 0$.

Answer: $f''(x) = 6x$, so $f''(0) = 0$. Concavity is undetermined. When $f''(x) = 0$, the second derivative test is inconclusive.

Flashcard 29: What does it mean for a function to be concave down?

Answer: The graph is curved downwards, like a frown, and $f''(x) < 0$. Negative second derivative indicates downward curvature.

Flashcard 30: What does it mean for a function to be concave up?

Answer: The graph is curved upwards, like a cup, and $f''(x) > 0$. Positive second derivative indicates upward curvature.