AP Calculus BC Flashcards: Determining Intervals On Increasing Decreasing Functions

What this deck covers

This deck focuses on Determining Intervals On Increasing Decreasing Functions, 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 relationship between critical points and intervals of increase/decrease?

Answer: Critical points separate intervals. They divide the domain into regions with consistent behavior.

Flashcard 2: What is the test to determine if a function is increasing on an interval?

Answer: The first derivative is positive. When $f'(x) > 0$, the function slopes upward.

Flashcard 3: How can you determine the critical points of $f(x) = x^3 - 6x^2$?

Answer: Solve $f'(x) = 0$: $x = 0, 4$. $f'(x) = 3x^2 - 12x = 3x(x-4)$, set equal to zero.

Flashcard 4: What is the condition for $f(x)$ to be constant on an interval?

Answer: $f'(x) = 0$ for all $x$ in that interval. Zero derivative means no rate of change.

Flashcard 5: What does a negative $f'(x)$ indicate about $f(x)$ on an interval?

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

Flashcard 6: For $f'(x) = 4x^3 - 12x$, find the critical points.

Answer: $x = 0, \pm \sqrt{3}$. $f'(x) = 4x(x^2 - 3)$, set equal to zero.

Flashcard 7: What does $f''(x) < 0$ indicate about $f(x)$?

Answer: $f(x)$ is concave down. Negative second derivative means curve bends downward.

Flashcard 8: State the critical point condition for a function $f(x)$.

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

Flashcard 9: Find intervals where $f(x) = x^2 - 8x + 15$ is decreasing.

Answer: $x \in (-\infty, 4)$. $f'(x) = 2x - 8$, negative when $x < 4$.

Flashcard 10: If $f'(x) < 0$ and $f''(x) > 0$, what can be said about $f(x)$?

Answer: $f(x)$ is decreasing and concave up. Function falls while curving upward like a valley.

Flashcard 11: What does $f'(x) = (x-1)^2$ imply about intervals of increase/decrease?

Answer: $f(x)$ is non-decreasing. $f'(x) = (x-1)^2 \geq 0$, never decreasing.

Flashcard 12: What does a positive $f'(x)$ indicate about $f(x)$ on an interval?

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

Flashcard 13: For $f'(x) = -4x$, on which interval is $f(x)$ decreasing?

Answer: $x \in (0, \infty)$. $f'(x) = -4x < 0$ when $x > 0$.

Flashcard 14: Find intervals where $f(x) = x^2 - 8x + 15$ is increasing.

Answer: $x \in (4, \infty)$. $f'(x) = 2x - 8$, positive when $x > 4$.

Flashcard 15: What is the second derivative of $f(x) = x^3 - 3x^2 + 2x$?

Answer: $f''(x) = 6x - 6$. Take derivative of $f'(x) = 3x^2 - 6x + 2$.

Flashcard 16: What does $f'(x) = 0$ but no sign change indicate at a critical point?

Answer: No local extremum. No sign change means no local maximum or minimum.

Flashcard 17: What does $f''(x) > 0$ indicate about $f(x)$?

Answer: $f(x)$ is concave up. Positive second derivative means curve bends upward.

Flashcard 18: What does $f'(x) = 0$ at $x = a$ indicate?

Answer: Possible local extremum. Critical points may be local maxima, minima, or inflection points.

Flashcard 19: What does it mean if $f'(x) > 0$ for all $x$ in $(a, b)$?

Answer: $f(x)$ is strictly increasing on $(a, b)$. Positive derivative throughout means consistently rising.

Flashcard 20: What is the significance of a sign change in $f'(x)$ at a critical point?

Answer: Indicates a local extremum. Sign changes indicate transitions between increasing/decreasing.

Flashcard 21: What is a necessary condition for a point to be a local extremum?

Answer: $f'(x) = 0$ or $f'(x)$ is undefined. Local extrema occur only at critical points.

Flashcard 22: Identify the intervals of increase for $f(x) = x^3 - 3x^2 + 4$.

Answer: $x \in (2, \infty)$. $f'(x) = 3x^2 - 6x = 3x(x-2)$, positive when $x > 2$.

Flashcard 23: What is the test to determine if a function is decreasing on an interval?

Answer: The first derivative is negative. When $f'(x) < 0$, the function slopes downward.

Flashcard 24: What is the derivative of $f(x) = x^3 - 3x^2$?

Answer: $f'(x) = 3x^2 - 6x$. Power rule: bring down exponent and reduce by 1.

Flashcard 25: What does it mean if $f'(x) < 0$ for all $x$ in $(a, b)$?

Answer: $f(x)$ is strictly decreasing on $(a, b)$. Negative derivative throughout means consistently falling.

Flashcard 26: For $f'(x) = 4x^3 - 12x$, find the critical points.

Answer: $x = 0, \pm \sqrt{3}$. $f'(x) = 4x(x^2 - 3)$, set equal to zero.

Flashcard 27: What does it mean if $f'(x) > 0$ for all $x$ in $(a, b)$?

Answer: $f(x)$ is strictly increasing on $(a, b)$. Positive derivative throughout means consistently rising.

Flashcard 28: State the critical point condition for a function $f(x)$.

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

Flashcard 29: What does it mean if $f'(x) < 0$ for all $x$ in $(a, b)$?

Answer: $f(x)$ is strictly decreasing on $(a, b)$. Negative derivative throughout means consistently falling.

Flashcard 30: What is the derivative of $f(x) = x^3 - 3x^2$?

Answer: $f'(x) = 3x^2 - 6x$. Power rule: bring down exponent and reduce by 1.