First Derivative Test - AP Calculus BC
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What test helps confirm if a critical point is an extremum?
What test helps confirm if a critical point is an extremum?
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The First Derivative Test. Distinguishes between maxima, minima, and inflection points.
The First Derivative Test. Distinguishes between maxima, minima, and inflection points.
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What test helps confirm if a critical point is an extremum?
What test helps confirm if a critical point is an extremum?
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The First Derivative Test. Distinguishes between maxima, minima, and inflection points.
The First Derivative Test. Distinguishes between maxima, minima, and inflection points.
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State the condition for a critical point.
State the condition for a critical point.
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$f'(x) = 0$ or $f'(x)$ is undefined. Necessary condition for potential extrema to exist.
$f'(x) = 0$ or $f'(x)$ is undefined. Necessary condition for potential extrema to exist.
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Which test determines where a function is increasing or decreasing?
Which test determines where a function is increasing or decreasing?
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The First Derivative Test. Sign of $f'(x)$ determines increasing/decreasing intervals.
The First Derivative Test. Sign of $f'(x)$ determines increasing/decreasing intervals.
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What indicates a relative minimum using the First Derivative Test?
What indicates a relative minimum using the First Derivative Test?
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Derivative changes from negative to positive. Sign change from $-$ to $+$ creates a valley in the graph.
Derivative changes from negative to positive. Sign change from $-$ to $+$ creates a valley in the graph.
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What indicates a relative maximum using the First Derivative Test?
What indicates a relative maximum using the First Derivative Test?
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Derivative changes from positive to negative. Sign change from $+$ to $-$ creates a peak in the graph.
Derivative changes from positive to negative. Sign change from $+$ to $-$ creates a peak in the graph.
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What is a relative extremum?
What is a relative extremum?
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A point where $f(x)$ changes from increasing to decreasing or vice versa. Local maximum or minimum where function direction reverses.
A point where $f(x)$ changes from increasing to decreasing or vice versa. Local maximum or minimum where function direction reverses.
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What is the derivative of $f(x) = \frac{1}{3}x^3 - x$?
What is the derivative of $f(x) = \frac{1}{3}x^3 - x$?
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$f'(x) = x^2 - 1$. Apply power rule to each term separately.
$f'(x) = x^2 - 1$. Apply power rule to each term separately.
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Identify the relative extrema: $f(x) = x^3 - 3x^2$.
Identify the relative extrema: $f(x) = x^3 - 3x^2$.
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Relative maximum at $x = 0$; relative minimum at $x = 2$. $f'(x) = 3x^2 - 6x$; sign changes at $x = 0$ and $x = 2$.
Relative maximum at $x = 0$; relative minimum at $x = 2$. $f'(x) = 3x^2 - 6x$; sign changes at $x = 0$ and $x = 2$.
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What does $f'(x) > 0$ indicate about $f(x)$?
What does $f'(x) > 0$ indicate about $f(x)$?
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$f(x)$ is increasing. Positive derivative means function values are rising.
$f(x)$ is increasing. Positive derivative means function values are rising.
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Describe the behavior if $f'(x)$ changes from negative to positive.
Describe the behavior if $f'(x)$ changes from negative to positive.
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There's a relative minimum. Function reaches a valley; slope changes from downward to upward.
There's a relative minimum. Function reaches a valley; slope changes from downward to upward.
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What is the derivative of $f(x) = x^5 - 5x^3$?
What is the derivative of $f(x) = x^5 - 5x^3$?
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$f'(x) = 5x^4 - 15x^2$. Power rule applied to polynomial with multiple terms.
$f'(x) = 5x^4 - 15x^2$. Power rule applied to polynomial with multiple terms.
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Find the derivative of $f(x) = x^3 - 3x^2 + 2x$.
Find the derivative of $f(x) = x^3 - 3x^2 + 2x$.
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$f'(x) = 3x^2 - 6x + 2$. Apply power rule term by term to the polynomial.
$f'(x) = 3x^2 - 6x + 2$. Apply power rule term by term to the polynomial.
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Find the derivative: $f(x) = x^4 - 4x^3$.
Find the derivative: $f(x) = x^4 - 4x^3$.
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$f'(x) = 4x^3 - 12x^2$. Apply power rule: multiply by exponent, reduce exponent by 1.
$f'(x) = 4x^3 - 12x^2$. Apply power rule: multiply by exponent, reduce exponent by 1.
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Determine if there is a minimum at $x = 0$ for $f(x) = x^2$.
Determine if there is a minimum at $x = 0$ for $f(x) = x^2$.
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Yes, minimum at $x = 0$. $f'(x) = 2x$ changes from negative to positive at origin.
Yes, minimum at $x = 0$. $f'(x) = 2x$ changes from negative to positive at origin.
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Find $f'(x)$ for $f(x) = x^3 - 9x^2 + 27x$.
Find $f'(x)$ for $f(x) = x^3 - 9x^2 + 27x$.
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$f'(x) = 3x^2 - 18x + 27$. Apply power rule to each term in the polynomial.
$f'(x) = 3x^2 - 18x + 27$. Apply power rule to each term in the polynomial.
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Evaluate $f'(x)$ at $x = 0$ for $f(x) = \frac{1}{3}x^3 - x$.
Evaluate $f'(x)$ at $x = 0$ for $f(x) = \frac{1}{3}x^3 - x$.
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$f'(0) = -1$. Substitute $x = 0$ into $f'(x) = x^2 - 1$.
$f'(0) = -1$. Substitute $x = 0$ into $f'(x) = x^2 - 1$.
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Determine relative extrema for $f(x) = x^3 - 3x + 1$ using the test.
Determine relative extrema for $f(x) = x^3 - 3x + 1$ using the test.
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No relative extrema. $f'(x) = 3x^2 - 3 = 0$ at $x = \pm 1$; no sign changes.
No relative extrema. $f'(x) = 3x^2 - 3 = 0$ at $x = \pm 1$; no sign changes.
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Describe the behavior if $f'(x)$ changes from positive to negative.
Describe the behavior if $f'(x)$ changes from positive to negative.
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There's a relative maximum. Function reaches a peak; slope changes from upward to downward.
There's a relative maximum. Function reaches a peak; slope changes from upward to downward.
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Determine if there is an extremum at $x = 1$ for $f(x) = 2x^3 - 3x^2$.
Determine if there is an extremum at $x = 1$ for $f(x) = 2x^3 - 3x^2$.
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No extremum at $x = 1$. $f'(x) = 6x^2 - 6x$; no sign change at $x = 1$.
No extremum at $x = 1$. $f'(x) = 6x^2 - 6x$; no sign change at $x = 1$.
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Determine extrema for $f(x) = x^2 - 4x + 3$ using the test.
Determine extrema for $f(x) = x^2 - 4x + 3$ using the test.
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Minimum at $x = 2$. $f'(x) = 2x - 4 = 0$ at $x = 2$; sign changes from $-$ to $+$.
Minimum at $x = 2$. $f'(x) = 2x - 4 = 0$ at $x = 2$; sign changes from $-$ to $+$.
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What is the next step after finding critical points in the test?
What is the next step after finding critical points in the test?
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Evaluate $f'(x)$ around critical points. Check sign changes of $f'(x)$ on intervals around each critical point.
Evaluate $f'(x)$ around critical points. Check sign changes of $f'(x)$ on intervals around each critical point.
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What must be true for a critical point to be a local extremum?
What must be true for a critical point to be a local extremum?
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The derivative must change signs. Sign change of $f'(x)$ distinguishes extrema from inflection points.
The derivative must change signs. Sign change of $f'(x)$ distinguishes extrema from inflection points.
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Identify extrema for $f(x) = x^2 - 2x + 1$ using the test.
Identify extrema for $f(x) = x^2 - 2x + 1$ using the test.
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Minimum at $x = 1$. $f'(x) = 2x - 2 = 0$ at $x = 1$; sign changes from $-$ to $+$.
Minimum at $x = 1$. $f'(x) = 2x - 2 = 0$ at $x = 1$; sign changes from $-$ to $+$.
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Determine if there is a maximum at $x = 0$ for $f(x) = -x^2$.
Determine if there is a maximum at $x = 0$ for $f(x) = -x^2$.
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Yes, maximum at $x = 0$. $f'(x) = -2x$ changes from positive to negative at origin.
Yes, maximum at $x = 0$. $f'(x) = -2x$ changes from positive to negative at origin.
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What is the derivative of $f(x) = x^2 - 4x$?
What is the derivative of $f(x) = x^2 - 4x$?
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$f'(x) = 2x - 4$. Basic power rule: derivative of $x^2$ is $2x$.
$f'(x) = 2x - 4$. Basic power rule: derivative of $x^2$ is $2x$.
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Find $f'(x)$: $f(x) = 2x^5 - 5x^4$.
Find $f'(x)$: $f(x) = 2x^5 - 5x^4$.
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$f'(x) = 10x^4 - 20x^3$. Apply power rule: bring down exponent, reduce power by 1.
$f'(x) = 10x^4 - 20x^3$. Apply power rule: bring down exponent, reduce power by 1.
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Evaluate $f'(x)$ at $x = 2$ for $f(x) = x^3 - 3x^2$.
Evaluate $f'(x)$ at $x = 2$ for $f(x) = x^3 - 3x^2$.
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$f'(2) = 0$. Substitute $x = 2$ into $f'(x) = 3x^2 - 6x$.
$f'(2) = 0$. Substitute $x = 2$ into $f'(x) = 3x^2 - 6x$.
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Find $f'(x)$: $f(x) = \frac{x^3}{3} - x$.
Find $f'(x)$: $f(x) = \frac{x^3}{3} - x$.
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$f'(x) = x^2 - 1$. Standard power rule application with fractional coefficient.
$f'(x) = x^2 - 1$. Standard power rule application with fractional coefficient.
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Determine relative extrema for $f(x) = x^2$ using the test.
Determine relative extrema for $f(x) = x^2$ using the test.
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Minimum at $x = 0$. $f'(x) = 2x$; changes from $-$ to $+$ at $x = 0$.
Minimum at $x = 0$. $f'(x) = 2x$; changes from $-$ to $+$ at $x = 0$.
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