AP Calculus BC Flashcards: Connecting Differentiability And Continuity

What this deck covers

This deck focuses on Connecting Differentiability And Continuity, 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: Determine if $f(x) = \text{ln}(x)$ is differentiable at $x=0$.

Answer: No, it is not differentiable at $x=0$. Natural logarithm is undefined at zero, so no derivative exists.

Flashcard 2: What is the derivative of $f(x) = \text{ln}(x)$ for $x > 0$?

Answer: $f'(x) = \frac{1}{x}$. This is the standard derivative formula for natural logarithm.

Flashcard 3: Find the derivative of $f(x) = |x|$ at $x = 0$.

Answer: The derivative does not exist at $x = 0$. The absolute value function has a sharp corner at the origin.

Flashcard 4: What is the relationship between left-hand and right-hand derivatives for existence?

Answer: They must be equal for the derivative to exist. Left and right limits of the difference quotient must match.

Flashcard 5: Identify if differentiability implies continuity.

Answer: Differentiability implies continuity. If a function has a derivative, it must be continuous.

Flashcard 6: What is the definition of differentiability at a point?

Answer: A function is differentiable at $x=a$ if $f'(a)$ exists. The derivative exists when the limit of the difference quotient exists.

Flashcard 7: Find $f'(x)$ for $f(x) = 3x^3 - 4x + 1$ at $x=1$.

Answer: $f'(1) = 5$. Power rule gives $f'(x) = 9x^2 - 4$, so $f'(1) = 5$.

Flashcard 8: What is the derivative of $f(x) = e^x$ at any point $x$?

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

Flashcard 9: State the rule for the existence of derivatives related to sharp corners.

Answer: Derivatives do not exist at sharp corners. Sharp corners create different left and right-hand derivatives.

Flashcard 10: Is $f(x) = x^2\text{sin}(\frac{1}{x})$ continuous at $x=0$?

Answer: Yes, it is continuous at $x=0$. The squeeze theorem shows continuity despite oscillation.

Flashcard 11: Determine if $f(x) = x^4$ is differentiable at $x=0$.

Answer: Yes, $f(x) = x^4$ is differentiable at $x=0$. Even powers create smooth curves differentiable at all points.

Flashcard 12: Find if $f(x) = x^3$ is differentiable at $x=0$.

Answer: Yes, $f(x) = x^3$ is differentiable at $x=0$. Polynomial functions are differentiable everywhere in their domain.

Flashcard 13: Find if $f(x) = \text{sgn}(x)$ is continuous at $x=0$.

Answer: $f(x) = \text{sgn}(x)$ is not continuous at $x=0$. The sign function has a jump discontinuity at the origin.

Flashcard 14: Does $f(x) = \text{abs}(x)$ have a derivative at $x=0$?

Answer: No, it does not have a derivative at $x=0$. Absolute value creates a sharp corner with undefined derivative.

Flashcard 15: State if $f(x) = [x]$ (floor function) is differentiable at $x=1$.

Answer: No, it is not differentiable at $x=1$. Floor functions have jump discontinuities at integer values.

Flashcard 16: Identify if continuity implies differentiability.

Answer: Continuity does not imply differentiability. Continuous functions can have corners where derivatives don't exist.

Flashcard 17: Identify if $f(x) = \text{exp}(x)$ is differentiable at $x=1$.

Answer: Yes, $f(x) = \text{exp}(x)$ is differentiable at $x=1$. Exponential functions are differentiable everywhere in their domain.

Flashcard 18: Find if $f(x) = \frac{1}{x-1}$ is continuous at $x=1$.

Answer: $f(x)$ is not continuous at $x=1$. Division by zero creates a vertical asymptote and discontinuity.

Flashcard 19: State the rule for the existence of derivatives related to discontinuities.

Answer: Derivatives do not exist at discontinuities. Differentiability requires continuity as a prerequisite.

Flashcard 20: Is $f(x) = \text{step}(x)$ differentiable at $x=2$?

Answer: No, $f(x) = \text{step}(x)$ is not differentiable at $x=2$. Step functions have jump discontinuities at transition points.

Flashcard 21: State the rule for the existence of derivatives related to vertical tangents.

Answer: Derivatives do not exist at vertical tangents. Vertical tangents have infinite slope, making derivatives undefined.

Flashcard 22: Determine if $f(x) = \text{sin}(x)$ is differentiable at $x=0$.

Answer: Yes, $f(x) = \text{sin}(x)$ is differentiable at $x=0$. Sine function is smooth and differentiable everywhere.

Flashcard 23: Is $f(x) = x \times \text{sgn}(x)$ differentiable at $x=0$?

Answer: No, it is not differentiable at $x=0$. This equals $|x|$, which has a corner at the origin.

Flashcard 24: For $f(x) = \tan(x)$, find $f'(x)$ at $x=\frac{\text{π}}{4}$.

Answer: $f'(\frac{\text{π}}{4}) = 2$. Derivative of tangent is $\sec^2(x)$, which equals 2 at $\frac{\pi}{4}$.

Flashcard 25: Identify the differentiability of $f(x) = \frac{1}{2x}$ at $x=0$.

Answer: $f(x)$ is not differentiable at $x=0$. Division by zero makes the function undefined at the origin.

Flashcard 26: Identify if continuity implies differentiability.

Answer: Continuity does not imply differentiability. Continuous functions can have corners where derivatives don't exist.

Flashcard 27: Does $f(x) = \text{abs}(x)$ have a derivative at $x=0$?

Answer: No, it does not have a derivative at $x=0$. Absolute value creates a sharp corner with undefined derivative.

Flashcard 28: What is the derivative of $f(x) = e^x$ at any point $x$?

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

Flashcard 29: Identify if differentiability implies continuity.

Answer: Differentiability implies continuity. If a function has a derivative, it must be continuous.

Flashcard 30: Find the derivative of $f(x) = |x|$ at $x = 0$.

Answer: The derivative does not exist at $x = 0$. The absolute value function has a sharp corner at the origin.