Linear and Quadratic Rates of Change - AP Precalculus
Card 1 of 30
Identify the average rate of change of $f(x) = 2x + 3$ over $[1, 4]$.
Identify the average rate of change of $f(x) = 2x + 3$ over $[1, 4]$.
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- Linear functions have constant slope, which equals the coefficient of $x$.
- Linear functions have constant slope, which equals the coefficient of $x$.
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What is the formula for the average rate of change of a function $f(x)$ over $[a, b]$?
What is the formula for the average rate of change of a function $f(x)$ over $[a, b]$?
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$\frac{f(b) - f(a)}{b - a}$. This is the slope formula for secant lines connecting two points.
$\frac{f(b) - f(a)}{b - a}$. This is the slope formula for secant lines connecting two points.
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What is the derivative of $f(x) = 3x^2 + 2x + 1$?
What is the derivative of $f(x) = 3x^2 + 2x + 1$?
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$6x + 2$. Apply power rule: $(x^n)' = nx^{n-1}$ and sum rule.
$6x + 2$. Apply power rule: $(x^n)' = nx^{n-1}$ and sum rule.
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State the formula for the derivative of $f(x) = ax^2 + bx + c$.
State the formula for the derivative of $f(x) = ax^2 + bx + c$.
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$2ax + b$. General form of derivative for quadratic functions.
$2ax + b$. General form of derivative for quadratic functions.
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Find the slope of the tangent line to $f(x) = x^2 - 5x + 6$ at $x = 3$.
Find the slope of the tangent line to $f(x) = x^2 - 5x + 6$ at $x = 3$.
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- Find $f'(x) = 2x - 5$, then substitute $x = 3$.
- Find $f'(x) = 2x - 5$, then substitute $x = 3$.
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What is the instantaneous rate of change of $f(x) = 4x^2$ at $x = 2$?
What is the instantaneous rate of change of $f(x) = 4x^2$ at $x = 2$?
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- Find $f'(x) = 8x$, then evaluate at $x = 2$.
- Find $f'(x) = 8x$, then evaluate at $x = 2$.
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What is the derivative of $f(x) = x^2 - 4x + 2$?
What is the derivative of $f(x) = x^2 - 4x + 2$?
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$2x - 4$. Apply power rule and linear term differentiation.
$2x - 4$. Apply power rule and linear term differentiation.
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Find the average rate of change of $f(x) = x^2 + 2x - 1$ over $[2, 4]$.
Find the average rate of change of $f(x) = x^2 + 2x - 1$ over $[2, 4]$.
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- Calculate $\frac{f(4) - f(2)}{4 - 2} = \frac{23 - 7}{2} = 8$.
- Calculate $\frac{f(4) - f(2)}{4 - 2} = \frac{23 - 7}{2} = 8$.
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What is the instantaneous rate of change of $f(x) = x^2 - x$ at $x = 0$?
What is the instantaneous rate of change of $f(x) = x^2 - x$ at $x = 0$?
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-1. Find $f'(x) = 2x - 1$, then substitute $x = 0$.
-1. Find $f'(x) = 2x - 1$, then substitute $x = 0$.
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State the formula for the derivative of $f(x) = 3x^2 + 4x + 5$.
State the formula for the derivative of $f(x) = 3x^2 + 4x + 5$.
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$6x + 4$. Differentiate each term using power rule and sum rule.
$6x + 4$. Differentiate each term using power rule and sum rule.
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Identify the slope of the tangent line to $f(x) = x^3 - 3x$ at $x = 1$.
Identify the slope of the tangent line to $f(x) = x^3 - 3x$ at $x = 1$.
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- Find $f'(x) = 3x^2 - 3$, then evaluate at $x = 1$.
- Find $f'(x) = 3x^2 - 3$, then evaluate at $x = 1$.
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What is the derivative of $f(x) = x^2 - 4x + 2$?
What is the derivative of $f(x) = x^2 - 4x + 2$?
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$2x - 4$. Apply power rule and linear term differentiation.
$2x - 4$. Apply power rule and linear term differentiation.
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Find the slope of the tangent line to $f(x) = 4x^2 - x$ at $x = 1$.
Find the slope of the tangent line to $f(x) = 4x^2 - x$ at $x = 1$.
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- Find $f'(x) = 8x - 1$, then evaluate at $x = 1$.
- Find $f'(x) = 8x - 1$, then evaluate at $x = 1$.
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Identify the instantaneous rate of change of $f(x) = x^3 - 3x^2$ at $x = 2$.
Identify the instantaneous rate of change of $f(x) = x^3 - 3x^2$ at $x = 2$.
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- Find $f'(x) = 3x^2 - 6x$, then substitute $x = 2$.
- Find $f'(x) = 3x^2 - 6x$, then substitute $x = 2$.
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What is the derivative of $f(x) = -x^2 + 3x$?
What is the derivative of $f(x) = -x^2 + 3x$?
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$-2x + 3$. Apply power rule and sum rule to find the derivative.
$-2x + 3$. Apply power rule and sum rule to find the derivative.
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State the derivative of $f(x) = x^2 + 5x - 3$.
State the derivative of $f(x) = x^2 + 5x - 3$.
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$2x + 5$. Differentiate each term using power rule and constant rule.
$2x + 5$. Differentiate each term using power rule and constant rule.
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What is the instantaneous rate of change of $f(x) = 2x^2$ at $x = -1$?
What is the instantaneous rate of change of $f(x) = 2x^2$ at $x = -1$?
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-4. Find $f'(x) = 4x$, then evaluate at $x = -1$.
-4. Find $f'(x) = 4x$, then evaluate at $x = -1$.
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Identify the average rate of change of $f(x) = -x^2 + 4$ over $[0, 2]$.
Identify the average rate of change of $f(x) = -x^2 + 4$ over $[0, 2]$.
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-2. Calculate $\frac{f(2) - f(0)}{2 - 0} = \frac{0 - 4}{2} = -2$.
-2. Calculate $\frac{f(2) - f(0)}{2 - 0} = \frac{0 - 4}{2} = -2$.
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What is the derivative of $f(x) = 7x^2$?
What is the derivative of $f(x) = 7x^2$?
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$14x$. Use power rule on $7x^2$ to get $7 \cdot 2x = 14x$.
$14x$. Use power rule on $7x^2$ to get $7 \cdot 2x = 14x$.
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Find the average rate of change of $f(x) = 5x^2 + 2x$ over $[1, 3]$.
Find the average rate of change of $f(x) = 5x^2 + 2x$ over $[1, 3]$.
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- Calculate $\frac{f(3) - f(1)}{3 - 1} = \frac{51 - 7}{2} = 22$.
- Calculate $\frac{f(3) - f(1)}{3 - 1} = \frac{51 - 7}{2} = 22$.
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What is the instantaneous rate of change of $f(x) = x^3$ at $x = 1$?
What is the instantaneous rate of change of $f(x) = x^3$ at $x = 1$?
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- Find $f'(x) = 3x^2$, then evaluate at $x = 1$.
- Find $f'(x) = 3x^2$, then evaluate at $x = 1$.
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Identify the instantaneous rate of change of $f(x) = x^2$ at $x = 3$.
Identify the instantaneous rate of change of $f(x) = x^2$ at $x = 3$.
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- Find $f'(x) = 2x$, then evaluate at $x = 3$.
- Find $f'(x) = 2x$, then evaluate at $x = 3$.
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State the relationship between secant and tangent lines in context of rates of change.
State the relationship between secant and tangent lines in context of rates of change.
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Secant: average rate, Tangent: instantaneous rate. Secant connects two points; tangent touches at one point.
Secant: average rate, Tangent: instantaneous rate. Secant connects two points; tangent touches at one point.
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What is the average rate of change of $f(x) = x^2$ over $[1, 3]$?
What is the average rate of change of $f(x) = x^2$ over $[1, 3]$?
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- Calculate $\frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4$.
- Calculate $\frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4$.
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Identify the slope of the tangent line to $f(x) = x^3 - 3x$ at $x = 1$.
Identify the slope of the tangent line to $f(x) = x^3 - 3x$ at $x = 1$.
Tap to reveal answer
- Find $f'(x) = 3x^2 - 3$, then evaluate at $x = 1$.
- Find $f'(x) = 3x^2 - 3$, then evaluate at $x = 1$.
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Find the slope of the tangent line to $f(x) = x^2 + 3x$ at $x = 2$.
Find the slope of the tangent line to $f(x) = x^2 + 3x$ at $x = 2$.
Tap to reveal answer
- Find $f'(x) = 2x + 3$, then substitute $x = 2$.
- Find $f'(x) = 2x + 3$, then substitute $x = 2$.
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State the formula for the instantaneous rate of change of $f(x)$ at $x = a$.
State the formula for the instantaneous rate of change of $f(x)$ at $x = a$.
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$f'(a)$. The derivative gives the instantaneous rate of change at any point.
$f'(a)$. The derivative gives the instantaneous rate of change at any point.
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State the formula for the derivative of $f(x) = 3x^2 + 4x + 5$.
State the formula for the derivative of $f(x) = 3x^2 + 4x + 5$.
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$6x + 4$. Differentiate each term using power rule and sum rule.
$6x + 4$. Differentiate each term using power rule and sum rule.
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What is the instantaneous rate of change of $f(x) = x^2 - x$ at $x = 0$?
What is the instantaneous rate of change of $f(x) = x^2 - x$ at $x = 0$?
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-1. Find $f'(x) = 2x - 1$, then substitute $x = 0$.
-1. Find $f'(x) = 2x - 1$, then substitute $x = 0$.
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Find the average rate of change of $f(x) = x^2 + 2x - 1$ over $[2, 4]$.
Find the average rate of change of $f(x) = x^2 + 2x - 1$ over $[2, 4]$.
Tap to reveal answer
- Calculate $\frac{f(4) - f(2)}{4 - 2} = \frac{23 - 7}{2} = 8$.
- Calculate $\frac{f(4) - f(2)}{4 - 2} = \frac{23 - 7}{2} = 8$.
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