AP Calculus AB Flashcards: Introduction To Related Rates

What this deck covers

This deck focuses on Introduction To Related Rates, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus AB.

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: Find $\frac{dV}{dt}$ if $V = \frac{4}{3}\times \pi r^3$ and $\frac{dr}{dt} = 2$ at $r = 3$.

Answer: $\frac{dV}{dt} = 72\pi$. Volume formula for sphere, differentiated.

Flashcard 2: State the formula for the derivative of a product of two functions.

Answer: $(f \times g)' = f' \times g + f \times g'$. Product rule for derivatives.

Flashcard 3: What is the derivative of $\frac{d}{dx}(\text{sec}(x))$?

Answer: $\frac{d}{dx}(\text{sec}(x)) = \text{sec}(x)\text{tan}(x)$. Secant derivative involves secant and tangent.

Flashcard 4: Find $\frac{dV}{dt}$ for $V = \frac{4}{3}\text{π}r^3$ if $r = 6$ and $\frac{dr}{dt} = 0.5$.

Answer: $\frac{dV}{dt} = 72\text{π}$. Sphere volume: $\frac{dV}{dt} = 4\pi r^2 \cdot 0.5 = 72\pi$.

Flashcard 5: What is the derivative of $\frac{d}{dx}(a^x)$?

Answer: $\frac{d}{dx}(a^x) = a^x\text{ln}(a)$. General exponential function derivative.

Flashcard 6: Determine $\frac{dy}{dt}$ for $y = x^3$ if $x = 2$ and $\frac{dx}{dt} = 3$.

Answer: $\frac{dy}{dt} = 36$. Power rule: $\frac{dy}{dt} = 3x^2 \cdot 3 = 36$.

Flashcard 7: What is the formula for the derivative of $\frac{d}{dx}(\cos(x))$?

Answer: $\frac{d}{dx}(\cos(x)) = -\sin(x)$. Cosine derivative is negative sine.

Flashcard 8: Which rule is primarily used in related rates problems?

Answer: Chain rule. Links rates through composite functions.

Flashcard 9: Determine $\frac{dC}{dt}$ if $C = 2\text{π}r$ and $\frac{dr}{dt} = 5$ at $r = 4$.

Answer: $\frac{dC}{dt} = 10\text{π}$. Circumference formula differentiated.

Flashcard 10: Identify the formula for the derivative of $\text{sin}(x)$.

Answer: $\frac{d}{dx}(\text{sin}(x)) = \text{cos}(x)$. Sine derivative is cosine.

Flashcard 11: What is the relationship between rates of change in related rates problems?

Answer: They are connected by the chain rule. Chain rule connects changing variables.

Flashcard 12: What is the chain rule for finding derivatives?

Answer: $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Composes derivatives for nested functions.

Flashcard 13: What is the derivative of $\frac{d}{dx}(\text{arcsin}(x))$?

Answer: $\frac{d}{dx}(\text{arcsin}(x)) = \frac{1}{\text{sqrt}(1-x^2)}$. Inverse sine derivative formula.

Flashcard 14: What is the derivative of $\frac{d}{dx}(\text{arctan}(x))$?

Answer: $\frac{d}{dx}(\text{arctan}(x)) = \frac{1}{1+x^2}$. Inverse tangent derivative formula.

Flashcard 15: What is the derivative of $\frac{d}{dx}(\text{arccos}(x))$?

Answer: $\frac{d}{dx}(\text{arccos}(x)) = -\frac{1}{\text{sqrt}(1-x^2)}$. Inverse cosine derivative is negative.

Flashcard 16: Calculate $\frac{dA}{dt}$ for $A = \pi r^2$ if $r = 7$ and $\frac{dr}{dt} = 0.5$.

Answer: $\frac{dA}{dt} = 7\pi$. Circle area: $\frac{dA}{dt} = 2\pi r \cdot 0.5 = 7\pi$.

Flashcard 17: Determine $\frac{dC}{dt}$ for $C = 2\text{π}r$ if $r = 10$ and $\frac{dr}{dt} = 0.1$.

Answer: $\frac{dC}{dt} = 0.2\text{π}$. Circumference: $\frac{dC}{dt} = 2\pi \cdot 0.1 = 0.2\pi$.

Flashcard 18: Calculate $\frac{dA}{dt}$ for $A = \pi r^2$ if $\frac{dr}{dt} = 1$ and $r = 5$.

Answer: $\frac{dA}{dt} = 10\pi$. Circle area formula differentiated.

Flashcard 19: What is the basic strategy for solving related rates problems?

Answer: Identify variables, relate them, differentiate, solve. Standard approach for related rates.

Flashcard 20: Calculate $\frac{dh}{dt}$ for $h = \text{tan}(t)$ when $t = \frac{\text{π}}{4}$.

Answer: $\frac{dh}{dt} = 2$. At $t = \frac{\pi}{4}$, $\sec^2(\frac{\pi}{4}) = 2$.

Flashcard 21: What is the derivative of $\frac{d}{dx}(\text{log}_a{x})$?

Answer: $\frac{d}{dx}(\text{log}_a{x}) = \frac{1}{x\text{ln}(a)}$. Logarithm derivative with base $a$.

Flashcard 22: Determine $\frac{dA}{dt}$ for $A = s^2$ if $s = 10$ and $\frac{ds}{dt} = 2$.

Answer: $\frac{dA}{dt} = 40$. Square area formula: $\frac{dA}{dt} = 2s \cdot 2 = 40$.

Flashcard 23: What is the derivative of $\frac{d}{dx}(\text{ln}(kx))$?

Answer: $\frac{d}{dx}(\text{ln}(kx)) = \frac{1}{x}$. Chain rule cancels constant $k$.

Flashcard 24: What is the derivative of $\frac{d}{dx}(\text{cot}(x))$?

Answer: $\frac{d}{dx}(\text{cot}(x)) = -\text{csc}^2(x)$. Cotangent derivative uses cosecant squared.

Flashcard 25: What is the derivative of $\frac{d}{dx}(\text{csc}(x))$?

Answer: $\frac{d}{dx}(\text{csc}(x)) = -\text{csc}(x)\text{cot}(x)$. Cosecant derivative is negative.

Flashcard 26: What is the formula for the derivative of $\frac{f}{g}$?

Answer: $(f/g)' = \frac{f'g - fg'}{g^2}$. Quotient rule for derivatives.

Flashcard 27: What is the derivative of $\frac{d}{dx}(\text{e}^{kx})$?

Answer: $\frac{d}{dx}(\text{e}^{kx}) = k\text{e}^{kx}$. Chain rule with exponential function.

Flashcard 28: What is the derivative of $\frac{d}{dx}(\text{ln}(x))$?

Answer: $\frac{d}{dx}(\text{ln}(x)) = \frac{1}{x}$. Natural logarithm derivative formula.

Flashcard 29: What is the derivative of $\frac{d}{dx}(\frac{1}{x})$?

Answer: $\frac{d}{dx}(\frac{1}{x}) = -\frac{1}{x^2}$. Power rule with $n = -1$.

Flashcard 30: State the formula for the derivative of $\frac{d}{dx}(e^x)$.

Answer: $\frac{d}{dx}(e^x) = e^x$. Exponential function derivative equals itself.