AP Calculus BC Flashcards: Rate Of Change At A Point

What this deck covers

This deck focuses on Rate Of Change At A Point, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the derivative of $f(x) = x^3$ at $x = 2$?

Answer: $12$. Using power rule: $f'(x) = 3x^2$, so $f'(2) = 12$.

Flashcard 2: What is the instantaneous rate of change of $f(x) = \text{ln}(x)$ at $x = 1$?

Answer: $1$. The derivative of $\ln(x)$ is $\frac{1}{x}$.

Flashcard 3: Calculate the derivative of $f(x) = \text{tan}(x)$ at $x = 0$.

Answer: $1$. The derivative of $\tan(x)$ is $\sec^2(x) = 1$ at $x = 0$.

Flashcard 4: Find the derivative of $f(x) = e^x$ at $x = 0$.

Answer: $1$. The derivative of $e^x$ is $e^x$, and $e^0 = 1$.

Flashcard 5: Find the average rate of change of $f(x) = x^2$ from $x = 1$ to $x = 3$.

Answer: $4$. Using $\frac{f(3)-f(1)}{3-1} = \frac{9-1}{2} = 4$.

Flashcard 6: What is the derivative of $f(x) = \text{sin}(x)$ at $x = \frac{\text{π}}{2}$?

Answer: $0$. The derivative of $\sin(x)$ is $\cos(x)$, and $\cos(\frac{\pi}{2}) = 0$.

Flashcard 7: Identify $f'(x)$ for $f(x) = x^2 + 2x + 1$ using the power rule.

Answer: $2x + 2$. Applying power rule to each term separately.

Flashcard 8: Find $f'(x)$ for $f(x) = \text{e}^x + \text{e}^{-x}$ using differentiation.

Answer: $\text{e}^x - \text{e}^{-x}$. Differentiating each exponential term separately.

Flashcard 9: What is the slope of the tangent line to $f(x) = x^2$ at $x = 1$?

Answer: $2$. Using power rule: $f'(x) = 2x$, so $f'(1) = 2$.

Flashcard 10: What is the derivative of $f(x) = \text{e}^x \text{sin}(x)$ at $x = 0$?

Answer: $1$. Using product rule: $(e^x \sin x)' = e^x(\sin x + \cos x)$.

Flashcard 11: Calculate the average rate of change of $f(x) = \text{cos}(x)$ over $[0, \text{π}]$.

Answer: $-\frac{2}{\text{π}}$. Using $\frac{\cos(\pi) - \cos(0)}{\pi - 0} = \frac{-1-1}{\pi} = -\frac{2}{\pi}$.

Flashcard 12: What is the instantaneous rate of change of $f(x) = 2x^2 - x$ at $x = 2$?

Answer: $7$. Using $f'(x) = 4x - 1$, so $f'(2) = 7$.

Flashcard 13: Identify the instantaneous rate of change of $f(x) = 3x$ at $x = 2$.

Answer: $3$. The derivative of $3x$ is constant: $3$.

Flashcard 14: Calculate the derivative of $f(x) = \text{sin}(2x)$ at $x = 0$.

Answer: $2$. Using chain rule: $\frac{d}{dx}[\sin(2x)] = 2\cos(2x)$.

Flashcard 15: What is the derivative of $f(x) = \text{e}^{2x}$ at $x = 0$?

Answer: $2$. Using chain rule: $\frac{d}{dx}[e^{2x}] = 2e^{2x}$.

Flashcard 16: Find the derivative of $f(x) = \text{ln}(x^2)$ at $x = 1$.

Answer: $2$. Using chain rule: $\frac{d}{dx}[\ln(x^2)] = \frac{2x}{x^2} = \frac{2}{x}$.

Flashcard 17: What is the slope of the tangent line to $f(x) = x^3$ at $x = -1$?

Answer: $3$. Using $f'(x) = 3x^2$, so $f'(-1) = 3$.

Flashcard 18: What is the instantaneous rate of change of $f(x) = x^3 + x$ at $x = 1$?

Answer: $4$. Using $f'(x) = 3x^2 + 1$, so $f'(1) = 4$.

Flashcard 19: What is the instantaneous rate of change of $f(x) = x^2 - 4x$ at $x = 3$?

Answer: $2$. Using $f'(x) = 2x - 4$, so $f'(3) = 2$.

Flashcard 20: Determine the instantaneous rate of change of $f(x) = x^2 - 3x$ at $x = 1$.

Answer: $-1$. Using $f'(x) = 2x - 3$, so $f'(1) = -1$.

Flashcard 21: What is the derivative of $f(x) = \text{cos}(2x)$ at $x = \frac{\text{π}}{4}$?

Answer: $-2\text{sin}(2x)$. Using chain rule on $\cos(2x)$ gives $-2\sin(2x)$.

Flashcard 22: Determine $f'(x)$ for $f(x) = 7$ using basic derivative rules.

Answer: $0$. The derivative of any constant is zero.

Flashcard 23: Find the derivative of $f(x) = \text{ln}(x^2)$ at $x = 1$.

Answer: $2$. Using chain rule: $\frac{d}{dx}[\ln(x^2)] = \frac{2x}{x^2} = \frac{2}{x}$.

Flashcard 24: What is the derivative of $f(x) = \text{cos}(2x)$ at $x = \frac{\text{π}}{4}$?

Answer: $-2\text{sin}(2x)$. Using chain rule on $\cos(2x)$ gives $-2\sin(2x)$.

Flashcard 25: Find $f'(x)$ for $f(x) = 5x^2$ using differentiation.

Answer: $10x$. Using power rule: $\frac{d}{dx}[5x^2] = 10x$.

Flashcard 26: What is the average rate of change of $f(x) = x^2 + 4x$ over $[2, 5]$?

Answer: $11$. Using $\frac{(25+20)-(4+8)}{5-2} = \frac{45-12}{3} = 11$.

Flashcard 27: Determine the instantaneous rate of change of $f(x) = 3x^2 + 4$ at $x = 3$.

Answer: $18$. Using $f'(x) = 6x$, so $f'(3) = 18$.

Flashcard 28: What is the derivative of $f(x) = x^3$ at $x = 2$?

Answer: $12$. Using power rule: $f'(x) = 3x^2$, so $f'(2) = 12$.

Flashcard 29: Calculate the average rate of change of $f(x) = 2x + 3$ from $x = 1$ to $x = 4$.

Answer: $2$. Linear functions have constant rate of change equal to slope.

Flashcard 30: What is the rate of change of $f(x) = x^3$ from $x = 2$ to $x = 4$?

Answer: $28$. Using $\frac{f(4) - f(2)}{4 - 2} = \frac{64 - 8}{2} = 28$.

Flashcard 31: Determine the instantaneous rate of change of $f(x) = x^2 - 3x$ at $x = 1$.

Answer: $-1$. Using $f'(x) = 2x - 3$, so $f'(1) = -1$.

Flashcard 32: What is the derivative of $f(x) = \text{cos}(x)$ at $x = 0$?

Answer: $0$. The derivative of $\cos(x)$ is $-\sin(x)$, and $\sin(0) = 0$.

Flashcard 33: Identify the instantaneous rate of change of $f(x) = x^4$ at $x = 1$.

Answer: $4$. Using power rule: $f'(x) = 4x^3$, so $f'(1) = 4$.

Flashcard 34: Find the average rate of change of $f(x) = x^2$ from $x = 1$ to $x = 3$.

Answer: $4$. Using $\frac{f(3)-f(1)}{3-1} = \frac{9-1}{2} = 4$.

Flashcard 35: Find the average rate of change of $f(x) = \text{e}^x$ over $[0, 1]$.

Answer: $\text{e} - 1$. Using $\frac{e^1 - e^0}{1 - 0} = e - 1$.

Flashcard 36: Identify the instantaneous rate of change of $f(x) = 3x$ at $x = 2$.

Answer: $3$. The derivative of $3x$ is constant: $3$.

Flashcard 37: What is the rate of change of $f(x) = x^3$ from $x = 2$ to $x = 4$?

Answer: $28$. Using $\frac{f(4) - f(2)}{4 - 2} = \frac{64 - 8}{2} = 28$.

Flashcard 38: State the derivative definition for instantaneous rate of change at $x = a$.

Answer: $f'(a) = \frac{d}{dx}f(x)|_{x=a}$. The derivative evaluated at the specific point.

Flashcard 39: What is the slope of the tangent line to $f(x) = x^2$ at $x = 1$?

Answer: $2$. Using power rule: $f'(x) = 2x$, so $f'(1) = 2$.

Flashcard 40: Find the average rate of change of $f(x) = \text{e}^x$ over $[0, 1]$.

Answer: $\text{e} - 1$. Using $\frac{e^1 - e^0}{1 - 0} = e - 1$.

Flashcard 41: Determine $f'(x)$ for $f(x) = 7$ using basic derivative rules.

Answer: $0$. The derivative of any constant is zero.

Flashcard 42: Calculate the average rate of change of $f(x) = \cos(x)$ over $[0, \pi]$.

Answer: $-\frac{2}{\pi}$. Using $\frac{\cos(\pi) - \cos(0)}{\pi - 0} = \frac{-1-1}{\pi} = -\frac{2}{\pi}$.

Flashcard 43: Identify $f'(x)$ for $f(x) = x^2 + 2x + 1$ using the power rule.

Answer: $2x + 2$. Applying power rule to each term separately.

Flashcard 44: Calculate the average rate of change of $f(x) = 2x + 3$ from $x = 1$ to $x = 4$.

Answer: $2$. Linear functions have constant rate of change equal to slope.

Flashcard 45: What is the geometric interpretation of the average rate of change?

Answer: Slope of the secant line over $[a, b]$. Secant line connects two points on the curve.

Flashcard 46: Calculate the derivative of $f(x) = \text{sin}(2x)$ at $x = 0$.

Answer: $2$. Using chain rule: $\frac{d}{dx}[\sin(2x)] = 2\cos(2x)$.

Flashcard 47: What is the slope of the tangent line to $f(x) = x^3$ at $x = -1$?

Answer: $3$. Using $f'(x) = 3x^2$, so $f'(-1) = 3$.

Flashcard 48: What is the instantaneous rate of change of $f(x) = 2x^2 - x$ at $x = 2$?

Answer: $7$. Using $f'(x) = 4x - 1$, so $f'(2) = 7$.

Flashcard 49: What is the derivative of $f(x) = \text{e}^x \text{sin}(x)$ at $x = 0$?

Answer: $1$. Using product rule: $(e^x \sin x)' = e^x(\sin x + \cos x)$.

Flashcard 50: What is the geometric interpretation of the instantaneous rate of change?

Answer: Slope of the tangent line at $x = a$. Tangent line touches the curve at exactly one point.

Flashcard 51: State the derivative of $f(x) = \frac{1}{x}$ at $x = 1$.

Answer: $-1$. The derivative of $\frac{1}{x}$ is $-\frac{1}{x^2}$.

Flashcard 52: State the derivative of $f(x) = \frac{1}{x}$ at $x = 1$.

Answer: $-1$. The derivative of $\frac{1}{x}$ is $-\frac{1}{x^2}$.

Flashcard 53: What is the formula to find the derivative of $f(x)$ using limits?

Answer: $f'(x) = \text{lim}_{h \to 0} \frac{f(x+h) - f(x)}{h}$. The limit definition of derivative using difference quotient.

Flashcard 54: What is the geometric interpretation of the instantaneous rate of change?

Answer: Slope of the tangent line at $x = a$. Tangent line touches the curve at exactly one point.

Flashcard 55: What is the instantaneous rate of change of $f(x) = x^2 - 4x$ at $x = 3$?

Answer: $2$. Using $f'(x) = 2x - 4$, so $f'(3) = 2$.

Flashcard 56: What is the derivative of $f(x) = \text{cos}(x)$ at $x = 0$?

Answer: $0$. The derivative of $\cos(x)$ is $-\sin(x)$, and $\sin(0) = 0$.

Flashcard 57: What is the derivative of $f(x) = \text{sin}(x)$ at $x = \frac{\text{π}}{2}$?

Answer: $0$. The derivative of $\sin(x)$ is $\cos(x)$, and $\cos(\frac{\pi}{2}) = 0$.

Flashcard 58: Find $f'(x)$ for $f(x) = \text{e}^x + \text{e}^{-x}$ using differentiation.

Answer: $\text{e}^x - \text{e}^{-x}$. Differentiating each exponential term separately.

Flashcard 59: What is the formula for the average rate of change of $f(x)$ over $[a, b]$?

Answer: $\frac{f(b) - f(a)}{b - a}$. Standard formula: change in function divided by change in input.

Flashcard 60: What is the formula for the average rate of change of $f(x)$ over $[a, b]$?

Answer: $\frac{f(b) - f(a)}{b - a}$. Standard formula: change in function divided by change in input.

Flashcard 61: Identify the instantaneous rate of change of $f(x) = x^4$ at $x = 1$.

Answer: $4$. Using power rule: $f'(x) = 4x^3$, so $f'(1) = 4$.

Flashcard 62: State the derivative definition for instantaneous rate of change at $x = a$.

Answer: $f'(a) = \frac{d}{dx}f(x)|_{x=a}$. The derivative evaluated at the specific point.

Flashcard 63: Find the derivative of $f(x) = e^x$ at $x = 0$.

Answer: $1$. The derivative of $e^x$ is $e^x$, and $e^0 = 1$.

Flashcard 64: Find $f'(x)$ for $f(x) = 5x^2$ using differentiation.

Answer: $10x$. Using power rule: $\frac{d}{dx}[5x^2] = 10x$.

Flashcard 65: What is the derivative of $f(x) = \text{e}^{2x}$ at $x = 0$?

Answer: $2$. Using chain rule: $\frac{d}{dx}[e^{2x}] = 2e^{2x}$.

Flashcard 66: Determine the instantaneous rate of change of $f(x) = 3x^2 + 4$ at $x = 3$.

Answer: $18$. Using $f'(x) = 6x$, so $f'(3) = 18$.

Flashcard 67: What is the average rate of change of $f(x) = x^2 + 4x$ over $[2, 5]$?

Answer: $11$. Using $\frac{(25+20)-(4+8)}{5-2} = \frac{45-12}{3} = 11$.

Flashcard 68: What is the geometric interpretation of the average rate of change?

Answer: Slope of the secant line over $[a, b]$. Secant line connects two points on the curve.

Flashcard 69: What is the instantaneous rate of change of $f(x) = x^3 + x$ at $x = 1$?

Answer: $4$. Using $f'(x) = 3x^2 + 1$, so $f'(1) = 4$.

Flashcard 70: What is the instantaneous rate of change of $f(x) = \text{ln}(x)$ at $x = 1$?

Answer: $1$. The derivative of $\ln(x)$ is $\frac{1}{x}$.

Flashcard 71: Calculate the derivative of $f(x) = \text{tan}(x)$ at $x = 0$.

Answer: $1$. The derivative of $\tan(x)$ is $\sec^2(x) = 1$ at $x = 0$.