Determining Intervals on Increasing, Decreasing Functions - AP Calculus AB
Card 1 of 30
What does it mean if $f'(x) = 0$ at some point?
What does it mean if $f'(x) = 0$ at some point?
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The function may have a local maximum, minimum, or saddle point. Critical points are candidates for local extrema or points of inflection.
The function may have a local maximum, minimum, or saddle point. Critical points are candidates for local extrema or points of inflection.
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State the derivative of $f(x) = 2x^3 - 3x^2 + 5x - 4$.
State the derivative of $f(x) = 2x^3 - 3x^2 + 5x - 4$.
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$f'(x) = 6x^2 - 6x + 5$. Apply power rule: derivative of $ax^n$ is $nax^{n-1}$.
$f'(x) = 6x^2 - 6x + 5$. Apply power rule: derivative of $ax^n$ is $nax^{n-1}$.
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What is the derivative of $f(x) = x^4$?
What is the derivative of $f(x) = x^4$?
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$f'(x) = 4x^3$. Apply power rule: $(x^4)' = 4x^3$.
$f'(x) = 4x^3$. Apply power rule: $(x^4)' = 4x^3$.
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What does the sign of $f'(x)$ indicate?
What does the sign of $f'(x)$ indicate?
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The sign indicates whether the function is increasing or decreasing. Positive derivative means increasing, negative means decreasing.
The sign indicates whether the function is increasing or decreasing. Positive derivative means increasing, negative means decreasing.
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Find $f'(x)$ for $f(x) = 4x^3 - 9x^2 + 6x - 1$.
Find $f'(x)$ for $f(x) = 4x^3 - 9x^2 + 6x - 1$.
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$f'(x) = 12x^2 - 18x + 6$. Apply power rule to each term of the polynomial.
$f'(x) = 12x^2 - 18x + 6$. Apply power rule to each term of the polynomial.
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Evaluate $f'(x)$ for $f(x) = \frac{1}{3}x^3 - 2x^2 + x$.
Evaluate $f'(x)$ for $f(x) = \frac{1}{3}x^3 - 2x^2 + x$.
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$f'(x) = x^2 - 4x + 1$. Apply power rule to each term: $\frac{d}{dx}[\frac{1}{3}x^3] = x^2$, etc.
$f'(x) = x^2 - 4x + 1$. Apply power rule to each term: $\frac{d}{dx}[\frac{1}{3}x^3] = x^2$, etc.
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What is the derivative of a constant function?
What is the derivative of a constant function?
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The derivative is 0. Constant functions have zero slope everywhere.
The derivative is 0. Constant functions have zero slope everywhere.
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Compute the derivative of $f(x) = x^2 + 4x - 5$.
Compute the derivative of $f(x) = x^2 + 4x - 5$.
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$f'(x) = 2x + 4$. Use power rule: $(x^2)' = 2x$ and $(4x)' = 4$.
$f'(x) = 2x + 4$. Use power rule: $(x^2)' = 2x$ and $(4x)' = 4$.
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Explain the significance of the second derivative's sign.
Explain the significance of the second derivative's sign.
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Indicates concavity and potential inflection points. Second derivative determines concavity: positive means concave up, negative means concave down.
Indicates concavity and potential inflection points. Second derivative determines concavity: positive means concave up, negative means concave down.
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What is the first derivative test for determining intervals of decrease?
What is the first derivative test for determining intervals of decrease?
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If $f'(x) < 0$, the function is decreasing on that interval. When the slope is negative, the function is falling.
If $f'(x) < 0$, the function is decreasing on that interval. When the slope is negative, the function is falling.
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What is the first derivative test for determining intervals of increase?
What is the first derivative test for determining intervals of increase?
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If $f'(x) > 0$, the function is increasing on that interval. This is the fundamental relationship between derivative sign and function behavior.
If $f'(x) > 0$, the function is increasing on that interval. This is the fundamental relationship between derivative sign and function behavior.
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Find $f'(x)$ given $f(x) = 5x^3 - 2x + 7$.
Find $f'(x)$ given $f(x) = 5x^3 - 2x + 7$.
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$f'(x) = 15x^2 - 2$. Apply power rule: $(5x^3)' = 15x^2$ and $(-2x)' = -2$.
$f'(x) = 15x^2 - 2$. Apply power rule: $(5x^3)' = 15x^2$ and $(-2x)' = -2$.
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What is the derivative of $f(x) = \frac{1}{2}x^2 - 3x$?
What is the derivative of $f(x) = \frac{1}{2}x^2 - 3x$?
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$f'(x) = x - 3$. Apply power rule: $(\frac{1}{2}x^2)' = x$ and $(-3x)' = -3$.
$f'(x) = x - 3$. Apply power rule: $(\frac{1}{2}x^2)' = x$ and $(-3x)' = -3$.
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What is the significance of $f'(x)$ being zero at a point?
What is the significance of $f'(x)$ being zero at a point?
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Possible local maximum, minimum, or inflection point. Zero derivative indicates horizontal tangent line and potential extremum.
Possible local maximum, minimum, or inflection point. Zero derivative indicates horizontal tangent line and potential extremum.
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How is the first derivative used to find intervals of monotonicity?
How is the first derivative used to find intervals of monotonicity?
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By evaluating $f'(x)$ to determine where it is positive or negative. Sign of $f'(x)$ determines whether function increases or decreases.
By evaluating $f'(x)$ to determine where it is positive or negative. Sign of $f'(x)$ determines whether function increases or decreases.
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What is the purpose of finding the critical points?
What is the purpose of finding the critical points?
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To determine potential intervals of increase or decrease. Critical points divide the domain into intervals for monotonicity testing.
To determine potential intervals of increase or decrease. Critical points divide the domain into intervals for monotonicity testing.
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Find the critical points of $f(x) = -x^2 + 6x - 9$.
Find the critical points of $f(x) = -x^2 + 6x - 9$.
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$x = 3$. Set $f'(x) = -2x + 6 = 0$ to find where slope equals zero.
$x = 3$. Set $f'(x) = -2x + 6 = 0$ to find where slope equals zero.
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What is the significance of $f'(x)$ being zero at a point?
What is the significance of $f'(x)$ being zero at a point?
Tap to reveal answer
Possible local maximum, minimum, or inflection point. Zero derivative indicates horizontal tangent line and potential extremum.
Possible local maximum, minimum, or inflection point. Zero derivative indicates horizontal tangent line and potential extremum.
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Find $f'(x)$ given $f(x) = 5x^3 - 2x + 7$.
Find $f'(x)$ given $f(x) = 5x^3 - 2x + 7$.
Tap to reveal answer
$f'(x) = 15x^2 - 2$. Apply power rule: $(5x^3)' = 15x^2$ and $(-2x)' = -2$.
$f'(x) = 15x^2 - 2$. Apply power rule: $(5x^3)' = 15x^2$ and $(-2x)' = -2$.
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State the derivative of $f(x) = 2x^3 - 3x^2 + 5x - 4$.
State the derivative of $f(x) = 2x^3 - 3x^2 + 5x - 4$.
Tap to reveal answer
$f'(x) = 6x^2 - 6x + 5$. Apply power rule: derivative of $ax^n$ is $nax^{n-1}$.
$f'(x) = 6x^2 - 6x + 5$. Apply power rule: derivative of $ax^n$ is $nax^{n-1}$.
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What does it mean if $f'(x) = 0$ at some point?
What does it mean if $f'(x) = 0$ at some point?
Tap to reveal answer
The function may have a local maximum, minimum, or saddle point. Critical points are candidates for local extrema or points of inflection.
The function may have a local maximum, minimum, or saddle point. Critical points are candidates for local extrema or points of inflection.
← Didn't Know|Knew It →
Find $f'(x)$ for $f(x) = 4x^3 - 9x^2 + 6x - 1$.
Find $f'(x)$ for $f(x) = 4x^3 - 9x^2 + 6x - 1$.
Tap to reveal answer
$f'(x) = 12x^2 - 18x + 6$. Apply power rule to each term of the polynomial.
$f'(x) = 12x^2 - 18x + 6$. Apply power rule to each term of the polynomial.
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Explain the significance of the second derivative's sign.
Explain the significance of the second derivative's sign.
Tap to reveal answer
Indicates concavity and potential inflection points. Second derivative determines concavity: positive means concave up, negative means concave down.
Indicates concavity and potential inflection points. Second derivative determines concavity: positive means concave up, negative means concave down.
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What is the first derivative test for determining intervals of decrease?
What is the first derivative test for determining intervals of decrease?
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If $f'(x) < 0$, the function is decreasing on that interval. When the slope is negative, the function is falling.
If $f'(x) < 0$, the function is decreasing on that interval. When the slope is negative, the function is falling.
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What is the first derivative test for determining intervals of increase?
What is the first derivative test for determining intervals of increase?
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If $f'(x) > 0$, the function is increasing on that interval. This is the fundamental relationship between derivative sign and function behavior.
If $f'(x) > 0$, the function is increasing on that interval. This is the fundamental relationship between derivative sign and function behavior.
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What does the sign of $f'(x)$ indicate?
What does the sign of $f'(x)$ indicate?
Tap to reveal answer
The sign indicates whether the function is increasing or decreasing. Positive derivative means increasing, negative means decreasing.
The sign indicates whether the function is increasing or decreasing. Positive derivative means increasing, negative means decreasing.
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Find the critical points of $f(x) = -x^2 + 6x - 9$.
Find the critical points of $f(x) = -x^2 + 6x - 9$.
Tap to reveal answer
$x = 3$. Set $f'(x) = -2x + 6 = 0$ to find where slope equals zero.
$x = 3$. Set $f'(x) = -2x + 6 = 0$ to find where slope equals zero.
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Evaluate $f'(x)$ for $f(x) = \frac{1}{3}x^3 - 2x^2 + x$.
Evaluate $f'(x)$ for $f(x) = \frac{1}{3}x^3 - 2x^2 + x$.
Tap to reveal answer
$f'(x) = x^2 - 4x + 1$. Apply power rule to each term: $\frac{d}{dx}[\frac{1}{3}x^3] = x^2$, etc.
$f'(x) = x^2 - 4x + 1$. Apply power rule to each term: $\frac{d}{dx}[\frac{1}{3}x^3] = x^2$, etc.
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What is the derivative of a constant function?
What is the derivative of a constant function?
Tap to reveal answer
The derivative is 0. Constant functions have zero slope everywhere.
The derivative is 0. Constant functions have zero slope everywhere.
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Compute the derivative of $f(x) = x^2 + 4x - 5$.
Compute the derivative of $f(x) = x^2 + 4x - 5$.
Tap to reveal answer
$f'(x) = 2x + 4$. Use power rule: $(x^2)' = 2x$ and $(4x)' = 4$.
$f'(x) = 2x + 4$. Use power rule: $(x^2)' = 2x$ and $(4x)' = 4$.
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