AP Calculus BC Flashcards: Sketching Graphs Of Functions And Derivatives

What this deck covers

This deck focuses on Sketching Graphs Of Functions And Derivatives, 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 significance of $f'(x) = 0$?

Answer: Potential local max, min, or saddle point. Zero derivative indicates a horizontal tangent line at that point.

Flashcard 2: What does the second derivative of a function indicate?

Answer: The concavity of the function. $f''(x)$ describes whether the graph curves upward or downward.

Flashcard 3: What does a positive $f''(x)$ indicate about $f(x)$?

Answer: The graph of $f(x)$ is concave up. Positive second derivative means the graph curves upward like a bowl.

Flashcard 4: Find $f''(x)$ for $f(x) = x^4 - 4x^2 + 6$.

Answer: $f''(x) = 12x^2 - 8$. Differentiate twice using power rule: $(x^4)'' = 12x^2$, $(-4x^2)'' = -8$.

Flashcard 5: State the definition of an inflection point.

Answer: Where $f''(x)$ changes sign. Inflection points occur where concavity changes from up to down or vice versa.

Flashcard 6: What is the first derivative of $f(x)$ used to determine?

Answer: The slope of the tangent line to $f(x)$. $f'(x)$ represents the instantaneous rate of change at any point.

Flashcard 7: Identify the intervals of increase for $f(x) = x^2 - 4x + 3$.

Answer: Increasing on $(2, \text{infinity})$. Find where $f'(x) = 2x - 4 > 0$, so $x > 2$.

Flashcard 8: What is the relationship between $f(x)$ and $f'(x)$?

Answer: $f'(x)$ gives the slope of $f(x)$. The derivative measures how steeply the function rises or falls.

Flashcard 9: Identify the local minimum of $f(x) = x^2 - 4x + 4$.

Answer: Local minimum at $x = 2$. Complete the square: $f(x) = (x-2)^2$, minimum at vertex.

Flashcard 10: Compute $f''(x)$ for $f(x) = x^5 - 5x^3 + 10x$.

Answer: $f''(x) = 20x^3 - 30x$. Differentiate twice: $f'(x) = 5x^4 - 15x^2$, then $f''(x) = 20x^3 - 30x$.

Flashcard 11: What is the effect of $f'(x) = 0$ on $f(x)$?

Answer: Potential local extremum. Zero derivative creates a horizontal tangent, possibly an extremum.

Flashcard 12: What can be concluded if $f'(x) > 0$?

Answer: $f(x)$ is increasing on that interval. Positive first derivative means function values are getting larger.

Flashcard 13: What is the relationship between $f''(x)$ and concavity?

Answer: $f''(x) > 0$: concave up; $f''(x) < 0$: concave down. Second derivative test determines the direction of curvature.

Flashcard 14: What does $f''(x) > 0$ imply about $f(x)$?

Answer: Function is concave up. Positive second derivative indicates the function curves upward.

Flashcard 15: Calculate $f'(x)$ for $f(x) = x^3 - 3x + 1$.

Answer: $f'(x) = 3x^2 - 3$. Apply power rule: derivative of $x^3$ is $3x^2$, derivative of $-3x$ is $-3$.

Flashcard 16: Calculate $f''(x)$ for $f(x) = x^3 - 3x^2 + 1$.

Answer: $f''(x) = 6x - 6$. Differentiate twice: $f'(x) = 3x^2 - 6x$, then $f''(x) = 6x - 6$.

Flashcard 17: What does $f''(x) = 0$ suggest about $f(x)$?

Answer: Possible inflection point. Zero second derivative may indicate where concavity changes.

Flashcard 18: What occurs at a local minimum of $f(x)$?

Answer: $f'(x) = 0$ and $f''(x) > 0$. Both conditions ensure a true local minimum exists.

Flashcard 19: Find inflection points for $f(x) = x^3 - 3x + 1$.

Answer: Inflection point at $x = 0$. Set $f''(x) = 6x = 0$ to find where concavity changes.

Flashcard 20: What does $f'(x) < 0$ imply about $f(x)$?

Answer: $f(x)$ is decreasing on that interval. Negative first derivative means function values are getting smaller.

Flashcard 21: What does a negative $f''(x)$ indicate?

Answer: The graph is concave down. Negative second derivative means the graph curves downward.

Flashcard 22: Find the critical points for $f(x) = x^2 - 4x + 3$.

Answer: Critical points: $x = 2$. Set $f'(x) = 2x - 4 = 0$ to find $x = 2$.

Flashcard 23: What is the critical point of a function $f(x)$?

Answer: Where $f'(x) = 0$ or $f'(x)$ is undefined. Critical points occur where the derivative equals zero or doesn't exist.

Flashcard 24: Determine where $f(x) = -x^3 + 3x^2$ is decreasing.

Answer: Decreasing on $(0, 2)$. Find where $f'(x) = -3x^2 + 6x = -3x(x-2) < 0$.

Flashcard 25: When does $f(x)$ have a point of inflection?

Answer: When $f''(x)$ changes sign. Sign change in second derivative indicates concavity reversal.

Flashcard 26: What is the significance of $f'(x) = 0$?

Answer: Potential local max, min, or saddle point. Zero derivative indicates a horizontal tangent line at that point.

Flashcard 27: When does $f(x)$ have a point of inflection?

Answer: When $f''(x)$ changes sign. Sign change in second derivative indicates concavity reversal.

Flashcard 28: Determine where $f(x) = -x^3 + 3x^2$ is decreasing.

Answer: Decreasing on $(0, 2)$. Find where $f'(x) = -3x^2 + 6x = -3x(x-2) < 0$.

Flashcard 29: What does a negative $f''(x)$ indicate?

Answer: The graph is concave down. Negative second derivative means the graph curves downward.

Flashcard 30: What does $f'(x) = 0$ imply about $f(x)$?

Answer: Possible local max, min, or point of inflection. Zero derivative indicates a horizontal tangent line.