AP Calculus AB Flashcards: Rates Of Change In Applied Concepts

What this deck covers

This deck focuses on Rates Of Change In Applied Concepts, 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: What is the derivative of $h(x) = \frac{1}{x^5}$?

Answer: $-\frac{5}{x^6}$. Power rule: $x^{-5}$ becomes $-5x^{-6}$.

Flashcard 2: Determine the derivative of the function $f(x) = 5x^4 - 3x^2 + 7$.

Answer: $20x^3 - 6x$. Power rule applied: $20x^3$ from $5x^4$ and $-6x$ from $-3x^2$.

Flashcard 3: What is the derivative of $f(x) = x^3 - 2x + 4$ with respect to $x$?

Answer: $3x^2 - 2$. Power rule: $3x^2$ from $x^3$ and $-2$ from $-2x$.

Flashcard 4: What is the formula for the rate of change of the volume of a sphere with respect to its radius?

Answer: $4\pi r^2$. Derivative of sphere volume $\frac{4}{3}\pi r^3$.

Flashcard 5: What is the derivative of $g(x) = x^3 + \frac{1}{x}$?

Answer: $3x^2 - \frac{1}{x^2}$. Power rule: $3x^2$ from $x^3$ and $-x^{-2}$ from $x^{-1}$.

Flashcard 6: What is the rate of change of the surface area of a sphere with respect to its radius?

Answer: $8\pi r$. Derivative of sphere surface area $4\pi r^2$.

Flashcard 7: What is the derivative of the function $f(x) = x^5 - 4x^3 + 2$?

Answer: $5x^4 - 12x^2$. Power rule: $5x^4$ from $x^5$ and $-12x^2$ from $-4x^3$.

Flashcard 8: What is the rate of change of the perimeter of a square with respect to its side length?

Answer:

1. Each side contributes 1 to perimeter, so total rate is 4.

Flashcard 9: Find the rate of change of the circumference of a circle with respect to its radius.

Answer: $2\pi$. Derivative of circumference $2\pi r$ is constant $2\pi$.

Flashcard 10: What is the derivative of $y = x^3 + x^2$ with respect to $x$?

Answer: $3x^2 + 2x$. Apply power rule: $3x^2$ from $x^3$ and $2x$ from $x^2$.

Flashcard 11: Identify the derivative of $y = 3x^3 + 5x^2 - x$.

Answer: $9x^2 + 10x - 1$. Power rule: $9x^2$ from $3x^3$, $10x$ from $5x^2$, $-1$ from $-x$.

Flashcard 12: What is the derivative of $h(x) = \frac{1}{x^2}$?

Answer: $-\frac{2}{x^3}$. Power rule: $x^{-2}$ becomes $-2x^{-3}$.

Flashcard 13: What is the derivative of the function $f(x) = \frac{2}{x^3}$?

Answer: $-\frac{6}{x^4}$. Power rule: $2x^{-3}$ becomes $-6x^{-4}$.

Flashcard 14: Find the rate of change of the volume of a cylinder with respect to its height.

Answer: $\pi r^2$. For cylinder volume $\pi r^2 h$, derivative with respect to height.

Flashcard 15: Identify the rate of change of the volume of a cube with respect to its side length.

Answer: $3s^2$. Derivative of volume $s^3$ with respect to side length.

Flashcard 16: What is the derivative of the function $g(x) = x^4 - \frac{1}{x^2}$?

Answer: $4x^3 + \frac{2}{x^3}$. Power rule: $4x^3$ from $x^4$ and $2x^{-3}$ from $-x^{-2}$.

Flashcard 17: Find the derivative of $h(x) = 7x^5 - 3x^2 + 1$.

Answer: $35x^4 - 6x$. Power rule: $35x^4$ from $7x^5$ and $-6x$ from $-3x^2$.

Flashcard 18: Determine the derivative of the function $y = \frac{2}{x^2}$.

Answer: $-\frac{4}{x^3}$. Power rule: $2x^{-2}$ becomes $-4x^{-3}$.

Flashcard 19: Identify the rate of change of the volume of a rectangular prism with respect to its height.

Answer: Area of base. Volume formula is base area times height.

Flashcard 20: Find the derivative of $y = 2x^3 + 3x^2 - 5x + 1$.

Answer: $6x^2 + 6x - 5$. Power rule applied: $6x^2$, $6x$, and $-5$ from each term.

Flashcard 21: What is the derivative of $f(x) = 2x^3 - x^2 + 3$?

Answer: $6x^2 - 2x$. Power rule: $6x^2$ from $2x^3$ and $-2x$ from $-x^2$.

Flashcard 22: Identify the rate of change of the area of a rectangle with respect to its width.

Answer: Length. For rectangle area $lw$, derivative with respect to width is length.

Flashcard 23: What is the derivative of the function $g(x) = \frac{1}{x^4}$?

Answer: $-\frac{4}{x^5}$. Power rule: $x^{-4}$ becomes $-4x^{-5}$.

Flashcard 24: Identify the derivative of the function $g(x) = \frac{1}{x}$.

Answer: $-\frac{1}{x^2}$. Power rule: $x^{-1}$ becomes $-x^{-2}$.

Flashcard 25: Identify the derivative of $y = 6x^2 + 4x - 5$.

Answer: $12x + 4$. Power rule: $12x$ from $6x^2$ and $4$ from $4x$.

Flashcard 26: What is the rate of change of the area of a rectangle with respect to its length?

Answer: Width. For rectangle area $lw$, derivative with respect to length is width.

Flashcard 27: Determine the derivative of $f(x) = 4x^2 - 3x + 7$.

Answer: $8x - 3$. Power rule: $8x$ from $4x^2$ and $-3$ from $-3x$.

Flashcard 28: What is the derivative of $h(x) = 5x^3 - 4x + 8$?

Answer: $15x^2 - 4$. Power rule: $15x^2$ from $5x^3$ and $-4$ from $-4x$.

Flashcard 29: Find the rate of change of the area of a square with side length $s$.

Answer: $2s$. Derivative of area $s^2$ gives rate of change.

Flashcard 30: What is the derivative of $f(x) = x^4 + 2x^2 + 1$?

Answer: $4x^3 + 4x$. Power rule: $4x^3$ from $x^4$ and $4x$ from $2x^2$.