Sketching Graphs of Functions and Derivatives - AP Calculus BC
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What is the significance of $f'(x) = 0$?
What is the significance of $f'(x) = 0$?
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Potential local max, min, or saddle point. Zero derivative indicates a horizontal tangent line at that point.
Potential local max, min, or saddle point. Zero derivative indicates a horizontal tangent line at that point.
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What does the second derivative of a function indicate?
What does the second derivative of a function indicate?
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The concavity of the function. $f''(x)$ describes whether the graph curves upward or downward.
The concavity of the function. $f''(x)$ describes whether the graph curves upward or downward.
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What does a positive $f''(x)$ indicate about $f(x)$?
What does a positive $f''(x)$ indicate about $f(x)$?
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The graph of $f(x)$ is concave up. Positive second derivative means the graph curves upward like a bowl.
The graph of $f(x)$ is concave up. Positive second derivative means the graph curves upward like a bowl.
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Find $f''(x)$ for $f(x) = x^4 - 4x^2 + 6$.
Find $f''(x)$ for $f(x) = x^4 - 4x^2 + 6$.
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$f''(x) = 12x^2 - 8$. Differentiate twice using power rule: $(x^4)'' = 12x^2$, $(-4x^2)'' = -8$.
$f''(x) = 12x^2 - 8$. Differentiate twice using power rule: $(x^4)'' = 12x^2$, $(-4x^2)'' = -8$.
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State the definition of an inflection point.
State the definition of an inflection point.
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Where $f''(x)$ changes sign. Inflection points occur where concavity changes from up to down or vice versa.
Where $f''(x)$ changes sign. Inflection points occur where concavity changes from up to down or vice versa.
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What is the first derivative of $f(x)$ used to determine?
What is the first derivative of $f(x)$ used to determine?
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The slope of the tangent line to $f(x)$. $f'(x)$ represents the instantaneous rate of change at any point.
The slope of the tangent line to $f(x)$. $f'(x)$ represents the instantaneous rate of change at any point.
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Identify the intervals of increase for $f(x) = x^2 - 4x + 3$.
Identify the intervals of increase for $f(x) = x^2 - 4x + 3$.
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Increasing on $(2, \text{infinity})$. Find where $f'(x) = 2x - 4 > 0$, so $x > 2$.
Increasing on $(2, \text{infinity})$. Find where $f'(x) = 2x - 4 > 0$, so $x > 2$.
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What is the relationship between $f(x)$ and $f'(x)$?
What is the relationship between $f(x)$ and $f'(x)$?
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$f'(x)$ gives the slope of $f(x)$. The derivative measures how steeply the function rises or falls.
$f'(x)$ gives the slope of $f(x)$. The derivative measures how steeply the function rises or falls.
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Identify the local minimum of $f(x) = x^2 - 4x + 4$.
Identify the local minimum of $f(x) = x^2 - 4x + 4$.
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Local minimum at $x = 2$. Complete the square: $f(x) = (x-2)^2$, minimum at vertex.
Local minimum at $x = 2$. Complete the square: $f(x) = (x-2)^2$, minimum at vertex.
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Compute $f''(x)$ for $f(x) = x^5 - 5x^3 + 10x$.
Compute $f''(x)$ for $f(x) = x^5 - 5x^3 + 10x$.
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$f''(x) = 20x^3 - 30x$. Differentiate twice: $f'(x) = 5x^4 - 15x^2$, then $f''(x) = 20x^3 - 30x$.
$f''(x) = 20x^3 - 30x$. Differentiate twice: $f'(x) = 5x^4 - 15x^2$, then $f''(x) = 20x^3 - 30x$.
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What is the effect of $f'(x) = 0$ on $f(x)$?
What is the effect of $f'(x) = 0$ on $f(x)$?
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Potential local extremum. Zero derivative creates a horizontal tangent, possibly an extremum.
Potential local extremum. Zero derivative creates a horizontal tangent, possibly an extremum.
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What can be concluded if $f'(x) > 0$?
What can be concluded if $f'(x) > 0$?
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$f(x)$ is increasing on that interval. Positive first derivative means function values are getting larger.
$f(x)$ is increasing on that interval. Positive first derivative means function values are getting larger.
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What is the relationship between $f''(x)$ and concavity?
What is the relationship between $f''(x)$ and concavity?
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$f''(x) > 0$: concave up; $f''(x) < 0$: concave down. Second derivative test determines the direction of curvature.
$f''(x) > 0$: concave up; $f''(x) < 0$: concave down. Second derivative test determines the direction of curvature.
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What does $f''(x) > 0$ imply about $f(x)$?
What does $f''(x) > 0$ imply about $f(x)$?
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Function is concave up. Positive second derivative indicates the function curves upward.
Function is concave up. Positive second derivative indicates the function curves upward.
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Calculate $f'(x)$ for $f(x) = x^3 - 3x + 1$.
Calculate $f'(x)$ for $f(x) = x^3 - 3x + 1$.
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$f'(x) = 3x^2 - 3$. Apply power rule: derivative of $x^3$ is $3x^2$, derivative of $-3x$ is $-3$.
$f'(x) = 3x^2 - 3$. Apply power rule: derivative of $x^3$ is $3x^2$, derivative of $-3x$ is $-3$.
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Calculate $f''(x)$ for $f(x) = x^3 - 3x^2 + 1$.
Calculate $f''(x)$ for $f(x) = x^3 - 3x^2 + 1$.
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$f''(x) = 6x - 6$. Differentiate twice: $f'(x) = 3x^2 - 6x$, then $f''(x) = 6x - 6$.
$f''(x) = 6x - 6$. Differentiate twice: $f'(x) = 3x^2 - 6x$, then $f''(x) = 6x - 6$.
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What does $f''(x) = 0$ suggest about $f(x)$?
What does $f''(x) = 0$ suggest about $f(x)$?
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Possible inflection point. Zero second derivative may indicate where concavity changes.
Possible inflection point. Zero second derivative may indicate where concavity changes.
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What occurs at a local minimum of $f(x)$?
What occurs at a local minimum of $f(x)$?
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$f'(x) = 0$ and $f''(x) > 0$. Both conditions ensure a true local minimum exists.
$f'(x) = 0$ and $f''(x) > 0$. Both conditions ensure a true local minimum exists.
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Find inflection points for $f(x) = x^3 - 3x + 1$.
Find inflection points for $f(x) = x^3 - 3x + 1$.
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Inflection point at $x = 0$. Set $f''(x) = 6x = 0$ to find where concavity changes.
Inflection point at $x = 0$. Set $f''(x) = 6x = 0$ to find where concavity changes.
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What does $f'(x) < 0$ imply about $f(x)$?
What does $f'(x) < 0$ imply about $f(x)$?
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$f(x)$ is decreasing on that interval. Negative first derivative means function values are getting smaller.
$f(x)$ is decreasing on that interval. Negative first derivative means function values are getting smaller.
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What does a negative $f''(x)$ indicate?
What does a negative $f''(x)$ indicate?
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The graph is concave down. Negative second derivative means the graph curves downward.
The graph is concave down. Negative second derivative means the graph curves downward.
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Find the critical points for $f(x) = x^2 - 4x + 3$.
Find the critical points for $f(x) = x^2 - 4x + 3$.
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Critical points: $x = 2$. Set $f'(x) = 2x - 4 = 0$ to find $x = 2$.
Critical points: $x = 2$. Set $f'(x) = 2x - 4 = 0$ to find $x = 2$.
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What is the critical point of a function $f(x)$?
What is the critical point of a function $f(x)$?
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Where $f'(x) = 0$ or $f'(x)$ is undefined. Critical points occur where the derivative equals zero or doesn't exist.
Where $f'(x) = 0$ or $f'(x)$ is undefined. Critical points occur where the derivative equals zero or doesn't exist.
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Determine where $f(x) = -x^3 + 3x^2$ is decreasing.
Determine where $f(x) = -x^3 + 3x^2$ is decreasing.
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Decreasing on $(0, 2)$. Find where $f'(x) = -3x^2 + 6x = -3x(x-2) < 0$.
Decreasing on $(0, 2)$. Find where $f'(x) = -3x^2 + 6x = -3x(x-2) < 0$.
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When does $f(x)$ have a point of inflection?
When does $f(x)$ have a point of inflection?
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When $f''(x)$ changes sign. Sign change in second derivative indicates concavity reversal.
When $f''(x)$ changes sign. Sign change in second derivative indicates concavity reversal.
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What is the significance of $f'(x) = 0$?
What is the significance of $f'(x) = 0$?
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Potential local max, min, or saddle point. Zero derivative indicates a horizontal tangent line at that point.
Potential local max, min, or saddle point. Zero derivative indicates a horizontal tangent line at that point.
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When does $f(x)$ have a point of inflection?
When does $f(x)$ have a point of inflection?
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When $f''(x)$ changes sign. Sign change in second derivative indicates concavity reversal.
When $f''(x)$ changes sign. Sign change in second derivative indicates concavity reversal.
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Determine where $f(x) = -x^3 + 3x^2$ is decreasing.
Determine where $f(x) = -x^3 + 3x^2$ is decreasing.
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Decreasing on $(0, 2)$. Find where $f'(x) = -3x^2 + 6x = -3x(x-2) < 0$.
Decreasing on $(0, 2)$. Find where $f'(x) = -3x^2 + 6x = -3x(x-2) < 0$.
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What does a negative $f''(x)$ indicate?
What does a negative $f''(x)$ indicate?
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The graph is concave down. Negative second derivative means the graph curves downward.
The graph is concave down. Negative second derivative means the graph curves downward.
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What does $f'(x) = 0$ imply about $f(x)$?
What does $f'(x) = 0$ imply about $f(x)$?
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Possible local max, min, or point of inflection. Zero derivative indicates a horizontal tangent line.
Possible local max, min, or point of inflection. Zero derivative indicates a horizontal tangent line.
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