AP Calculus BC Flashcards: Connecting A Function And Its Derivatives

What this deck covers

This deck focuses on Connecting A Function And Its Derivatives, 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 second derivative of $f(x) = \ln(x)$?

Answer: $f''(x) = -\frac{1}{x^2}$. Derivative of $\frac{1}{x}$ applied to $\ln(x)$.

Flashcard 2: Determine $f'(x)$ for $f(x) = e^x$.

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

Flashcard 3: What does the concavity test involve?

Answer: Analyzing $f''(x)$ to determine concavity. Examining where $f''(x)$ changes sign.

Flashcard 4: Determine $f''(x)$ for $f(x) = \text{cos}(2x)$.

Answer: $f''(x) = -4\text{cos}(2x)$. Apply chain rule: derivative of $\cos(2x)$ is $-2\sin(2x)$, then again.

Flashcard 5: Find $f''(x)$ for $f(x) = x^4$.

Answer: $f''(x) = 12x^2$. Apply power rule: $f'(x) = 4x^3$, then $f''(x) = 12x^2$.

Flashcard 6: What is the second derivative of $f(x) = \text{e}^x$?

Answer: $f''(x) = \text{e}^x$. Exponential function equals its own second derivative.

Flashcard 7: What is the second derivative of $f(x) = \text{cos}(x)$?

Answer: $f''(x) = -\text{cos}(x)$. Derivative of cosine is $-\sin(x)$; derivative again gives $-\cos(x)$.

Flashcard 8: Find $f'(x)$ for $f(x) = \frac{1}{x^2}$.

Answer: $f'(x) = -\frac{2}{x^3}$. Rewrite as $x^{-2}$ and apply power rule.

Flashcard 9: Calculate $f''(x)$ for $f(x) = \text{sin}(x)$.

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

Flashcard 10: What is the second derivative of $f(x) = \text{ln}(x^2)$?

Answer: $f''(x) = -\frac{2}{x^2}$. Use chain rule: $f'(x) = \frac{2}{x}$, then apply quotient rule.

Flashcard 11: What is the derivative of $f(x) = \text{cos}(x)$?

Answer: $f'(x) = -\text{sin}(x)$. Standard derivative of cosine function.

Flashcard 12: Calculate $f'(x)$ for $f(x) = \text{tan}(x)$.

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

Flashcard 13: Determine the first derivative of $f(x) = \text{e}^{2x}$.

Answer: $f'(x) = 2\text{e}^{2x}$. Apply chain rule: $\frac{d}{dx}[e^{u}] = e^{u} \cdot u'$.

Flashcard 14: What is the second derivative test used for?

Answer: To determine if a critical point is a local max or min. Uses sign of $f''(c)$ to classify critical points.

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

Answer: $f'(x) = 2\text{cos}(2x)$. Apply chain rule to $\sin(2x)$.

Flashcard 16: What is the first derivative of $f(x) = \text{sin}^2(x)$?

Answer: $f'(x) = 2\text{sin}(x)\text{cos}(x)$. Apply chain rule to $\sin^2(x) = (\sin(x))^2$.

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

Answer: $f'(x) = \text{e}^x + x \text{e}^x$. Apply product rule: $(uv)' = u'v + uv'$.

Flashcard 18: What is the second derivative of $f(x) = x^3$?

Answer: $f''(x) = 6x$. Apply power rule twice: $f'(x) = 3x^2$, then $f''(x) = 6x$.

Flashcard 19: Find the critical points of $f(x) = x^3 - 3x^2$.

Answer: At $x = 0$ and $x = 2$. Set $f'(x) = 3x^2 - 6x = 3x(x-2) = 0$.

Flashcard 20: What can be concluded if $f''(x) < 0$ at a point?

Answer: $f(x)$ is concave down. Negative second derivative means curve bends downward.

Flashcard 21: If $f''(x) > 0$, what can be said about $f(x)$?

Answer: $f(x)$ is concave up. Positive second derivative means curve bends upward.

Flashcard 22: Find $f'(x)$ for $f(x) = x^2 + 2x + 1$.

Answer: $f'(x) = 2x + 2$. Apply power rule to polynomial.

Flashcard 23: What is the derivative of $f(x) = x^2$?

Answer: $f'(x) = 2x$. Apply power rule: $\frac{d}{dx}[x^n] = nx^{n-1}$.

Flashcard 24: Identify the critical points of $f(x) = x^2 - 4x + 4$.

Answer: At $x = 2$. Set $f'(x) = 2x - 4 = 0$ to find critical points.

Flashcard 25: Identify the derivative of $f(x) = \frac{1}{x^3}$.

Answer: $f'(x) = -\frac{3}{x^4}$. Rewrite as $x^{-3}$ and apply power rule.

Flashcard 26: What is the first derivative of $f(x) = \text{ln}(x)$?

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

Flashcard 27: Find $f''(x)$ for $f(x) = \frac{1}{x}$.

Answer: $f''(x) = \frac{2}{x^3}$. Rewrite as $x^{-1}$, apply power rule twice.

Flashcard 28: State the power rule for differentiation.

Answer: $\frac{d}{dx}[x^n] = nx^{n-1}$. Fundamental rule for differentiating polynomial terms.

Flashcard 29: What does the second derivative of a function represent?

Answer: The concavity of the function. Second derivative determines curve shape and bending.

Flashcard 30: Find the derivative of $f(x) = \text{e}^{-x}$.

Answer: $f'(x) = -\text{e}^{-x}$. Apply chain rule with $u = -x$.