Rate of Change at a Point - AP Calculus BC
Card 1 of 30
State the derivative definition for instantaneous rate of change at $x = a$.
State the derivative definition for instantaneous rate of change at $x = a$.
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$f'(a) = \frac{d}{dx}f(x)|_{x=a}$. The derivative evaluated at the specific point.
$f'(a) = \frac{d}{dx}f(x)|_{x=a}$. The derivative evaluated at the specific point.
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What is the instantaneous rate of change of $f(x) = x^2 - 4x$ at $x = 3$?
What is the instantaneous rate of change of $f(x) = x^2 - 4x$ at $x = 3$?
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$2$. Using $f'(x) = 2x - 4$, so $f'(3) = 2$.
$2$. Using $f'(x) = 2x - 4$, so $f'(3) = 2$.
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What is the average rate of change of $f(x) = x^2 + 4x$ over $[2, 5]$?
What is the average rate of change of $f(x) = x^2 + 4x$ over $[2, 5]$?
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$11$. Using $\frac{(25+20)-(4+8)}{5-2} = \frac{45-12}{3} = 11$.
$11$. Using $\frac{(25+20)-(4+8)}{5-2} = \frac{45-12}{3} = 11$.
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What is the geometric interpretation of the average rate of change?
What is the geometric interpretation of the average rate of change?
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Slope of the secant line over $[a, b]$. Secant line connects two points on the curve.
Slope of the secant line over $[a, b]$. Secant line connects two points on the curve.
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Find the average rate of change of $f(x) = x^2$ from $x = 1$ to $x = 3$.
Find the average rate of change of $f(x) = x^2$ from $x = 1$ to $x = 3$.
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$4$. Using $\frac{f(3)-f(1)}{3-1} = \frac{9-1}{2} = 4$.
$4$. Using $\frac{f(3)-f(1)}{3-1} = \frac{9-1}{2} = 4$.
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What is the instantaneous rate of change of $f(x) = 2x^2 - x$ at $x = 2$?
What is the instantaneous rate of change of $f(x) = 2x^2 - x$ at $x = 2$?
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$7$. Using $f'(x) = 4x - 1$, so $f'(2) = 7$.
$7$. Using $f'(x) = 4x - 1$, so $f'(2) = 7$.
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Determine $f'(x)$ for $f(x) = 7$ using basic derivative rules.
Determine $f'(x)$ for $f(x) = 7$ using basic derivative rules.
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$0$. The derivative of any constant is zero.
$0$. The derivative of any constant is zero.
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Find $f'(x)$ for $f(x) = 5x^2$ using differentiation.
Find $f'(x)$ for $f(x) = 5x^2$ using differentiation.
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$10x$. Using power rule: $\frac{d}{dx}[5x^2] = 10x$.
$10x$. Using power rule: $\frac{d}{dx}[5x^2] = 10x$.
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Determine the instantaneous rate of change of $f(x) = 3x^2 + 4$ at $x = 3$.
Determine the instantaneous rate of change of $f(x) = 3x^2 + 4$ at $x = 3$.
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$18$. Using $f'(x) = 6x$, so $f'(3) = 18$.
$18$. Using $f'(x) = 6x$, so $f'(3) = 18$.
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Identify $f'(x)$ for $f(x) = x^2 + 2x + 1$ using the power rule.
Identify $f'(x)$ for $f(x) = x^2 + 2x + 1$ using the power rule.
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$2x + 2$. Applying power rule to each term separately.
$2x + 2$. Applying power rule to each term separately.
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Find the derivative of $f(x) = \text{ln}(x^2)$ at $x = 1$.
Find the derivative of $f(x) = \text{ln}(x^2)$ at $x = 1$.
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$2$. Using chain rule: $\frac{d}{dx}[\ln(x^2)] = \frac{2x}{x^2} = \frac{2}{x}$.
$2$. Using chain rule: $\frac{d}{dx}[\ln(x^2)] = \frac{2x}{x^2} = \frac{2}{x}$.
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What is the rate of change of $f(x) = x^3$ from $x = 2$ to $x = 4$?
What is the rate of change of $f(x) = x^3$ from $x = 2$ to $x = 4$?
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$28$. Using $\frac{f(4) - f(2)}{4 - 2} = \frac{64 - 8}{2} = 28$.
$28$. Using $\frac{f(4) - f(2)}{4 - 2} = \frac{64 - 8}{2} = 28$.
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Calculate the average rate of change of $f(x) = \cos(x)$ over $[0, \pi]$.
Calculate the average rate of change of $f(x) = \cos(x)$ over $[0, \pi]$.
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$-\frac{2}{\pi}$. Using $\frac{\cos(\pi) - \cos(0)}{\pi - 0} = \frac{-1-1}{\pi} = -\frac{2}{\pi}$.
$-\frac{2}{\pi}$. Using $\frac{\cos(\pi) - \cos(0)}{\pi - 0} = \frac{-1-1}{\pi} = -\frac{2}{\pi}$.
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What is the formula for the average rate of change of $f(x)$ over $[a, b]$?
What is the formula for the average rate of change of $f(x)$ over $[a, b]$?
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$\frac{f(b) - f(a)}{b - a}$. Standard formula: change in function divided by change in input.
$\frac{f(b) - f(a)}{b - a}$. Standard formula: change in function divided by change in input.
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Find $f'(x)$ for $f(x) = \text{e}^x + \text{e}^{-x}$ using differentiation.
Find $f'(x)$ for $f(x) = \text{e}^x + \text{e}^{-x}$ using differentiation.
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$\text{e}^x - \text{e}^{-x}$. Differentiating each exponential term separately.
$\text{e}^x - \text{e}^{-x}$. Differentiating each exponential term separately.
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What is the derivative of $f(x) = \text{e}^x \text{sin}(x)$ at $x = 0$?
What is the derivative of $f(x) = \text{e}^x \text{sin}(x)$ at $x = 0$?
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$1$. Using product rule: $(e^x \sin x)' = e^x(\sin x + \cos x)$.
$1$. Using product rule: $(e^x \sin x)' = e^x(\sin x + \cos x)$.
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Calculate the derivative of $f(x) = \text{tan}(x)$ at $x = 0$.
Calculate the derivative of $f(x) = \text{tan}(x)$ at $x = 0$.
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$1$. The derivative of $\tan(x)$ is $\sec^2(x) = 1$ at $x = 0$.
$1$. The derivative of $\tan(x)$ is $\sec^2(x) = 1$ at $x = 0$.
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What is the instantaneous rate of change of $f(x) = \text{ln}(x)$ at $x = 1$?
What is the instantaneous rate of change of $f(x) = \text{ln}(x)$ at $x = 1$?
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$1$. The derivative of $\ln(x)$ is $\frac{1}{x}$.
$1$. The derivative of $\ln(x)$ is $\frac{1}{x}$.
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What is the derivative of $f(x) = \text{e}^{2x}$ at $x = 0$?
What is the derivative of $f(x) = \text{e}^{2x}$ at $x = 0$?
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$2$. Using chain rule: $\frac{d}{dx}[e^{2x}] = 2e^{2x}$.
$2$. Using chain rule: $\frac{d}{dx}[e^{2x}] = 2e^{2x}$.
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What is the derivative of $f(x) = \text{cos}(2x)$ at $x = \frac{\text{π}}{4}$?
What is the derivative of $f(x) = \text{cos}(2x)$ at $x = \frac{\text{π}}{4}$?
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$-2\text{sin}(2x)$. Using chain rule on $\cos(2x)$ gives $-2\sin(2x)$.
$-2\text{sin}(2x)$. Using chain rule on $\cos(2x)$ gives $-2\sin(2x)$.
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What is the slope of the tangent line to $f(x) = x^3$ at $x = -1$?
What is the slope of the tangent line to $f(x) = x^3$ at $x = -1$?
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$3$. Using $f'(x) = 3x^2$, so $f'(-1) = 3$.
$3$. Using $f'(x) = 3x^2$, so $f'(-1) = 3$.
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What is the derivative of $f(x) = \text{sin}(x)$ at $x = \frac{\text{π}}{2}$?
What is the derivative of $f(x) = \text{sin}(x)$ at $x = \frac{\text{π}}{2}$?
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$0$. The derivative of $\sin(x)$ is $\cos(x)$, and $\cos(\frac{\pi}{2}) = 0$.
$0$. The derivative of $\sin(x)$ is $\cos(x)$, and $\cos(\frac{\pi}{2}) = 0$.
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Identify the instantaneous rate of change of $f(x) = x^4$ at $x = 1$.
Identify the instantaneous rate of change of $f(x) = x^4$ at $x = 1$.
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$4$. Using power rule: $f'(x) = 4x^3$, so $f'(1) = 4$.
$4$. Using power rule: $f'(x) = 4x^3$, so $f'(1) = 4$.
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What is the derivative of $f(x) = \text{cos}(x)$ at $x = 0$?
What is the derivative of $f(x) = \text{cos}(x)$ at $x = 0$?
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$0$. The derivative of $\cos(x)$ is $-\sin(x)$, and $\sin(0) = 0$.
$0$. The derivative of $\cos(x)$ is $-\sin(x)$, and $\sin(0) = 0$.
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Find the average rate of change of $f(x) = \text{e}^x$ over $[0, 1]$.
Find the average rate of change of $f(x) = \text{e}^x$ over $[0, 1]$.
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$\text{e} - 1$. Using $\frac{e^1 - e^0}{1 - 0} = e - 1$.
$\text{e} - 1$. Using $\frac{e^1 - e^0}{1 - 0} = e - 1$.
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Find the derivative of $f(x) = e^x$ at $x = 0$.
Find the derivative of $f(x) = e^x$ at $x = 0$.
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$1$. The derivative of $e^x$ is $e^x$, and $e^0 = 1$.
$1$. The derivative of $e^x$ is $e^x$, and $e^0 = 1$.
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Calculate the average rate of change of $f(x) = 2x + 3$ from $x = 1$ to $x = 4$.
Calculate the average rate of change of $f(x) = 2x + 3$ from $x = 1$ to $x = 4$.
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$2$. Linear functions have constant rate of change equal to slope.
$2$. Linear functions have constant rate of change equal to slope.
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What is the geometric interpretation of the instantaneous rate of change?
What is the geometric interpretation of the instantaneous rate of change?
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Slope of the tangent line at $x = a$. Tangent line touches the curve at exactly one point.
Slope of the tangent line at $x = a$. Tangent line touches the curve at exactly one point.
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What is the slope of the tangent line to $f(x) = x^2$ at $x = 1$?
What is the slope of the tangent line to $f(x) = x^2$ at $x = 1$?
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$2$. Using power rule: $f'(x) = 2x$, so $f'(1) = 2$.
$2$. Using power rule: $f'(x) = 2x$, so $f'(1) = 2$.
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Calculate the derivative of $f(x) = \text{sin}(2x)$ at $x = 0$.
Calculate the derivative of $f(x) = \text{sin}(2x)$ at $x = 0$.
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$2$. Using chain rule: $\frac{d}{dx}[\sin(2x)] = 2\cos(2x)$.
$2$. Using chain rule: $\frac{d}{dx}[\sin(2x)] = 2\cos(2x)$.
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