AP Calculus BC Flashcards: Solving Optimization Problems

What this deck covers

This deck focuses on Solving Optimization Problems, 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: Calculate the derivative: $f(x) = 5x^4$.

Answer: $f'(x) = 20x^3$. Power rule: $4 \times 5x^{4-1} = 20x^3$.

Flashcard 2: State the method to solve optimization problems with constraints.

Answer: Use Lagrange multipliers. Technique for handling equality constraints.

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

Answer: $f'(x) = 2\text{e}^{2x}$. Chain rule applied to exponential with coefficient.

Flashcard 4: Identify the constraint in maximizing the volume of a box.

Answer: Surface area or material limits. Material availability restricts the design space.

Flashcard 5: Calculate the derivative: $f(x) = \text{tan}(x)$.

Answer: $f'(x) = \text{sec}^2(x)$. Standard derivative of tangent function.

Flashcard 6: Find the derivative of $g(x) = \frac{1}{x}$.

Answer: $g'(x) = -\frac{1}{x^2}$. Rewrite as $x^{-1}$ and apply power rule.

Flashcard 7: What step follows finding the critical points in optimization?

Answer: Evaluate the objective function at critical points. Determines which critical point gives optimal value.

Flashcard 8: Identify the formula for the derivative of $f(x) = x^2 + 3x$.

Answer: $f'(x) = 2x + 3$. Power rule applied: bring down exponent, reduce by 1.

Flashcard 9: State the condition for a relative maximum using $f''(x)$.

Answer: $f''(x) < 0$ at critical point. Negative second derivative indicates downward concavity.

Flashcard 10: Determine the derivative: $f(x) = x^3 - 3x^2 + 2$.

Answer: $f'(x) = 3x^2 - 6x$. Power rule applied to each term separately.

Flashcard 11: Determine the derivative: $f(x) = 3x^{-2}$.

Answer: $f'(x) = -6x^{-3}$. Power rule: $3 \times (-2)x^{-3} = -6x^{-3}$.

Flashcard 12: State the condition for a relative minimum using $f''(x)$.

Answer: $f''(x) > 0$ at critical point. Positive second derivative indicates upward concavity.

Flashcard 13: Identify the derivative of $f(x) = x^{\frac{1}{2}}$.

Answer: $f'(x) = \frac{1}{2}x^{-\frac{1}{2}}$. Power rule with fractional exponent: $\frac{1}{2}x^{-1/2}$.

Flashcard 14: State the critical points condition for $f'(x) = 0$.

Answer: Critical points occur where $f'(x) = 0$ or is undefined. These are potential locations for extrema.

Flashcard 15: What does $f'(x) = 0$ imply about $f(x)$?

Answer: Possible extremum or inflection point. Zero slope indicates potential maximum or minimum.

Flashcard 16: Calculate the derivative: $f(x) = e^x$.

Answer: $f'(x) = e^x$. Exponential function is its own derivative.

Flashcard 17: What is the derivative of $f(x) = \text{ln}(x)$?

Answer: $f'(x) = \frac{1}{x}$. Standard derivative of natural logarithm function.

Flashcard 18: Find the derivative of $f(x) = \text{arcsin}(x)$.

Answer: $f'(x) = \frac{1}{\text{√}(1-x^2)}$. Standard derivative of inverse sine function.

Flashcard 19: What is the derivative of $f(x) = \text{log}_a(x)$?

Answer: $f'(x) = \frac{1}{x \text{ln}(a)}$. General formula for logarithm base $a$ derivative.

Flashcard 20: State the chain rule for derivatives.

Answer: If $y = g(f(x))$, then $dy/dx = g'(f(x))f'(x)$. Rule for differentiating composite functions.

Flashcard 21: What is the condition for an inflection point?

Answer: $f''(x) = 0$ and $f''(x)$ changes sign. Point where concavity changes direction.

Flashcard 22: Calculate the derivative: $f(x) = \text{cos}(x)$.

Answer: $f'(x) = -\text{sin}(x)$. Derivative of cosine is negative sine.

Flashcard 23: Find the derivative of $f(x) = \frac{x^2}{2}$.

Answer: $f'(x) = x$. Coefficient $\frac{1}{2}$ remains, power rule on $x^2$.

Flashcard 24: What is the purpose of Lagrange multipliers?

Answer: To find extrema of functions subject to constraints. Method for optimization with equality constraints.

Flashcard 25: Find the critical points of $h(x) = x^2 - 4x + 4$.

Answer: $x = 2$. Set $h'(x) = 2x - 4 = 0$, solve for $x$.

Flashcard 26: What is the derivative of a constant function $f(x) = c$?

Answer: $f'(x) = 0$. Constants have zero rate of change.

Flashcard 27: What is the derivative of $f(x) = \text{sin}(x)$?

Answer: $f'(x) = \text{cos}(x)$. Standard trigonometric derivative rule.

Flashcard 28: Find the critical points of $h(x) = x^2 - 4x + 4$.

Answer: $x = 2$. Set $h'(x) = 2x - 4 = 0$, solve for $x$.

Flashcard 29: Calculate the derivative: $f(x) = 5x^4$.

Answer: $f'(x) = 20x^3$. Power rule: $4 \times 5x^{4-1} = 20x^3$.

Flashcard 30: What is the derivative of $f(x) = \text{sin}(x)$?

Answer: $f'(x) = \text{cos}(x)$. Standard trigonometric derivative rule.