AP Calculus BC Flashcards: First Derivative Test

What this deck covers

This deck focuses on First Derivative Test, 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 does $f'(x) = 0$ imply?

Answer: Possible extremum; critical point. Potential location for maximum, minimum, or inflection point.

Flashcard 2: What is the First Derivative Test used for?

Answer: To determine relative (local) extrema of a function. Identifies local maxima and minima by analyzing sign changes of $f'(x)$.

Flashcard 3: What is the derivative of $f(x) = x^5 - 5x^3$?

Answer: $f'(x) = 5x^4 - 15x^2$. Power rule applied to polynomial with multiple terms.

Flashcard 4: What test helps confirm if a critical point is an extremum?

Answer: The First Derivative Test. Distinguishes between maxima, minima, and inflection points.

Flashcard 5: Evaluate $f'(x)$ at $x = 2$ for $f(x) = x^3 - 3x^2$.

Answer: $f'(2) = 0$. Substitute $x = 2$ into $f'(x) = 3x^2 - 6x$.

Flashcard 6: Determine if there is an extremum at $x = 1$ for $f(x) = 2x^3 - 3x^2$.

Answer: No extremum at $x = 1$. $f'(x) = 6x^2 - 6x$; no sign change at $x = 1$.

Flashcard 7: Describe the behavior if $f'(x)$ changes from negative to positive.

Answer: There's a relative minimum. Function reaches a valley; slope changes from downward to upward.

Flashcard 8: What is a critical point?

Answer: Where $f'(x) = 0$ or $f'(x)$ is undefined. These are the only points where extrema can occur.

Flashcard 9: Evaluate $f'(x)$ at $x = 2$ for $f(x) = x^3 - 3x^2$.

Answer: $f'(2) = 0$. Substitute $x = 2$ into $f'(x) = 3x^2 - 6x$.

Flashcard 10: Describe the behavior if $f'(x)$ changes from positive to negative.

Answer: There's a relative maximum. Function reaches a peak; slope changes from upward to downward.

Flashcard 11: State the condition for a critical point.

Answer: $f'(x) = 0$ or $f'(x)$ is undefined. Necessary condition for potential extrema to exist.

Flashcard 12: What does $f'(x) > 0$ indicate about $f(x)$?

Answer: $f(x)$ is increasing. Positive derivative means function values are rising.

Flashcard 13: What must be true for a critical point to be a local extremum?

Answer: The derivative must change signs. Sign change of $f'(x)$ distinguishes extrema from inflection points.

Flashcard 14: What must be true for a critical point to be a local extremum?

Answer: The derivative must change signs. Sign change of $f'(x)$ distinguishes extrema from inflection points.

Flashcard 15: Find $f'(x)$: $f(x) = 2x^5 - 5x^4$.

Answer: $f'(x) = 10x^4 - 20x^3$. Apply power rule: bring down exponent, reduce power by 1.

Flashcard 16: Identify the relative extrema: $f(x) = x^3 - 3x^2$.

Answer: Relative maximum at $x = 0$; relative minimum at $x = 2$. $f'(x) = 3x^2 - 6x$; sign changes at $x = 0$ and $x = 2$.

Flashcard 17: Evaluate $f'(x)$ at $x = 0$ for $f(x) = \frac{1}{3}x^3 - x$.

Answer: $f'(0) = -1$. Substitute $x = 0$ into $f'(x) = x^2 - 1$.

Flashcard 18: Find the derivative of $f(x) = x^3 - 3x^2 + 2x$.

Answer: $f'(x) = 3x^2 - 6x + 2$. Apply power rule term by term to the polynomial.

Flashcard 19: What is the derivative of $f(x) = x^2 - 4x$?

Answer: $f'(x) = 2x - 4$. Basic power rule: derivative of $x^2$ is $2x$.

Flashcard 20: Describe the behavior if $f'(x)$ changes from negative to positive.

Answer: There's a relative minimum. Function reaches a valley; slope changes from downward to upward.

Flashcard 21: Determine extrema for $f(x) = x^2 - 4x + 3$ using the test.

Answer: Minimum at $x = 2$. $f'(x) = 2x - 4 = 0$ at $x = 2$; sign changes from $-$ to $+$.

Flashcard 22: State the condition for a critical point.

Answer: $f'(x) = 0$ or $f'(x)$ is undefined. Necessary condition for potential extrema to exist.

Flashcard 23: Determine if there is a maximum at $x = 0$ for $f(x) = -x^2$.

Answer: Yes, maximum at $x = 0$. $f'(x) = -2x$ changes from positive to negative at origin.

Flashcard 24: What is a relative extremum?

Answer: A point where $f(x)$ changes from increasing to decreasing or vice versa. Local maximum or minimum where function direction reverses.

Flashcard 25: Evaluate $f'(x)$ at $x = 0$ for $f(x) = \frac{1}{3}x^3 - x$.

Answer: $f'(0) = -1$. Substitute $x = 0$ into $f'(x) = x^2 - 1$.

Flashcard 26: What is the derivative of $f(x) = x^5 - 5x^3$?

Answer: $f'(x) = 5x^4 - 15x^2$. Power rule applied to polynomial with multiple terms.

Flashcard 27: Find $f'(x)$ for $f(x) = x^3 - 9x^2 + 27x$.

Answer: $f'(x) = 3x^2 - 18x + 27$. Apply power rule to each term in the polynomial.

Flashcard 28: Determine relative extrema for $f(x) = x^2$ using the test.

Answer: Minimum at $x = 0$. $f'(x) = 2x$; changes from $-$ to $+$ at $x = 0$.

Flashcard 29: Find $f'(x)$: $f(x) = 2x^5 - 5x^4$.

Answer: $f'(x) = 10x^4 - 20x^3$. Apply power rule: bring down exponent, reduce power by 1.

Flashcard 30: What is the next step after finding critical points in the test?

Answer: Evaluate $f'(x)$ around critical points. Check sign changes of $f'(x)$ on intervals around each critical point.

Flashcard 31: Which test determines where a function is increasing or decreasing?

Answer: The First Derivative Test. Sign of $f'(x)$ determines increasing/decreasing intervals.

Flashcard 32: Identify extrema for $f(x) = x^2 - 2x + 1$ using the test.

Answer: Minimum at $x = 1$. $f'(x) = 2x - 2 = 0$ at $x = 1$; sign changes from $-$ to $+$.

Flashcard 33: Determine relative extrema for $f(x) = x^3 - 3x + 1$ using the test.

Answer: No relative extrema. $f'(x) = 3x^2 - 3 = 0$ at $x = \pm 1$; no sign changes.

Flashcard 34: What indicates a relative minimum using the First Derivative Test?

Answer: Derivative changes from negative to positive. Sign change from $-$ to $+$ creates a valley in the graph.

Flashcard 35: Find $f'(x)$: $f(x) = \frac{x^3}{3} - x$.

Answer: $f'(x) = x^2 - 1$. Standard power rule application with fractional coefficient.

Flashcard 36: Determine if there is a maximum at $x = 0$ for $f(x) = -x^2$.

Answer: Yes, maximum at $x = 0$. $f'(x) = -2x$ changes from positive to negative at origin.

Flashcard 37: Describe the behavior if $f'(x)$ changes from positive to negative.

Answer: There's a relative maximum. Function reaches a peak; slope changes from upward to downward.

Flashcard 38: Identify the relative extrema: $f(x) = x^3 - 3x^2$.

Answer: Relative maximum at $x = 0$; relative minimum at $x = 2$. $f'(x) = 3x^2 - 6x$; sign changes at $x = 0$ and $x = 2$.

Flashcard 39: What indicates a relative maximum using the First Derivative Test?

Answer: Derivative changes from positive to negative. Sign change from $+$ to $-$ creates a peak in the graph.

Flashcard 40: What indicates a relative maximum using the First Derivative Test?

Answer: Derivative changes from positive to negative. Sign change from $+$ to $-$ creates a peak in the graph.

Flashcard 41: Which test determines where a function is increasing or decreasing?

Answer: The First Derivative Test. Sign of $f'(x)$ determines increasing/decreasing intervals.

Flashcard 42: What does $f'(x) > 0$ indicate about $f(x)$?

Answer: $f(x)$ is increasing. Positive derivative means function values are rising.

Flashcard 43: What is a relative extremum?

Answer: A point where $f(x)$ changes from increasing to decreasing or vice versa. Local maximum or minimum where function direction reverses.

Flashcard 44: Find the derivative: $f(x) = x^4 - 4x^3$.

Answer: $f'(x) = 4x^3 - 12x^2$. Apply power rule: multiply by exponent, reduce exponent by 1.

Flashcard 45: Find the derivative: $f(x) = x^4 - 4x^3$.

Answer: $f'(x) = 4x^3 - 12x^2$. Apply power rule: multiply by exponent, reduce exponent by 1.

Flashcard 46: Find $f'(x)$: $f(x) = \frac{x^3}{3} - x$.

Answer: $f'(x) = x^2 - 1$. Standard power rule application with fractional coefficient.

Flashcard 47: What does $f'(x) = 0$ imply?

Answer: Possible extremum; critical point. Potential location for maximum, minimum, or inflection point.

Flashcard 48: Find the derivative of $f(x) = x^3 - 3x^2 + 2x$.

Answer: $f'(x) = 3x^2 - 6x + 2$. Apply power rule term by term to the polynomial.

Flashcard 49: What is the next step after finding critical points in the test?

Answer: Evaluate $f'(x)$ around critical points. Check sign changes of $f'(x)$ on intervals around each critical point.

Flashcard 50: Identify extrema for $f(x) = x^2 - 2x + 1$ using the test.

Answer: Minimum at $x = 1$. $f'(x) = 2x - 2 = 0$ at $x = 1$; sign changes from $-$ to $+$.

Flashcard 51: What indicates a relative minimum using the First Derivative Test?

Answer: Derivative changes from negative to positive. Sign change from $-$ to $+$ creates a valley in the graph.

Flashcard 52: Determine if there is a minimum at $x = 0$ for $f(x) = x^2$.

Answer: Yes, minimum at $x = 0$. $f'(x) = 2x$ changes from negative to positive at origin.

Flashcard 53: What test helps confirm if a critical point is an extremum?

Answer: The First Derivative Test. Distinguishes between maxima, minima, and inflection points.

Flashcard 54: What does $f'(x) < 0$ indicate about $f(x)$?

Answer: $f(x)$ is decreasing. Negative derivative means function values are falling.

Flashcard 55: Determine relative extrema for $f(x) = x^3 - 3x + 1$ using the test.

Answer: No relative extrema. $f'(x) = 3x^2 - 3 = 0$ at $x = \pm 1$; no sign changes.

Flashcard 56: What is the derivative of $f(x) = \frac{1}{3}x^3 - x$?

Answer: $f'(x) = x^2 - 1$. Apply power rule to each term separately.

Flashcard 57: Determine if there is an extremum at $x = 1$ for $f(x) = 2x^3 - 3x^2$.

Answer: No extremum at $x = 1$. $f'(x) = 6x^2 - 6x$; no sign change at $x = 1$.

Flashcard 58: Determine relative extrema for $f(x) = x^2$ using the test.

Answer: Minimum at $x = 0$. $f'(x) = 2x$; changes from $-$ to $+$ at $x = 0$.

Flashcard 59: Determine extrema for $f(x) = x^2 - 4x + 3$ using the test.

Answer: Minimum at $x = 2$. $f'(x) = 2x - 4 = 0$ at $x = 2$; sign changes from $-$ to $+$.

Flashcard 60: Find $f'(x)$ for $f(x) = x^3 - 9x^2 + 27x$.

Answer: $f'(x) = 3x^2 - 18x + 27$. Apply power rule to each term in the polynomial.

Flashcard 61: Determine if there is a minimum at $x = 0$ for $f(x) = x^2$.

Answer: Yes, minimum at $x = 0$. $f'(x) = 2x$ changes from negative to positive at origin.

Flashcard 62: What is the derivative of $f(x) = x^2 - 4x$?

Answer: $f'(x) = 2x - 4$. Basic power rule: derivative of $x^2$ is $2x$.

Flashcard 63: What is the derivative of $f(x) = \frac{1}{3}x^3 - x$?

Answer: $f'(x) = x^2 - 1$. Apply power rule to each term separately.

Flashcard 64: What does $f'(x) < 0$ indicate about $f(x)$?

Answer: $f(x)$ is decreasing. Negative derivative means function values are falling.