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: 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 2: 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 3: 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 4: 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 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 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 7: Determine $f'(x)$ for $f(x) = 7$ using basic derivative rules.

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

Flashcard 8: Find $f'(x)$ for $f(x) = 5x^2$ using differentiation.

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

Flashcard 9: 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 10: 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 11: 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 12: 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 13: 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 14: 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 15: 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 16: 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 17: 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 18: 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 19: 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 20: 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 21: 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 22: 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 23: 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 24: 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 25: 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 26: 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 27: 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 28: 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 29: 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 30: Calculate the derivative of $f(x) = \text{sin}(2x)$ at $x = 0$.

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