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: Is the function $f(x) = \frac{1}{x}$ differentiable at $x=0$?

Answer: No, $f(x) = \frac{1}{x}$ is not differentiable at $x=0$. The function is undefined at $x=0$, so no derivative exists.

Flashcard 2: 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 3: Is the function $f(x) = \frac{1}{x}$ differentiable at $x=0$?

Answer: No, $f(x) = \frac{1}{x}$ is not differentiable at $x=0$. The function is undefined at $x=0$, so no derivative exists.

Flashcard 4: Identify if differentiability implies continuity.

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

Flashcard 5: 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 6: 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 7: 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 8: 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 9: 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 10: 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 11: 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 12: 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 13: 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 14: Identify the derivative of $f(x) = x^2$ at $x=2$.

Answer: $f'(2) = 4$. Power rule gives $f'(x) = 2x$, so $f'(2) = 4$.

Flashcard 15: 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 16: 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 17: 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 18: 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 19: What is the derivative of $f(x) = \text{cos}(x)$ at $x = 0$?

Answer: $f'(0) = 0$. Derivative of cosine is $-\sin(x)$, which equals 0 at $x=0$.

Flashcard 20: 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 21: What is the definition of continuity at a point?

Answer: $f$ is continuous at $x=a$ if $\lim_{x \to a} f(x) = f(a)$. The limit as x approaches a must equal the function value at a.

Flashcard 22: Identify the derivative of $f(x) = x^2$ at $x=2$.

Answer: $f'(2) = 4$. Power rule gives $f'(x) = 2x$, so $f'(2) = 4$.

Flashcard 23: 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 24: 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 25: 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 26: Identify if differentiability implies continuity.

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

Flashcard 27: 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 28: State the rule for the existence of derivatives related to cusps.

Answer: Derivatives do not exist at cusps. Cusps have vertical tangent lines with undefined slopes.

Flashcard 29: 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 30: 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 31: 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 32: 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 33: Determine the differentiability of $f(x) = \frac{1}{x^2}$ at $x=0$.

Answer: $f(x)$ is not differentiable at $x=0$. Function is undefined at zero due to division by zero.

Flashcard 34: Identify if continuity implies differentiability.

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

Flashcard 35: 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 36: 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 37: Which rule checks if a function $f(x)$ is differentiable at $x=a$?

Answer: Check if $\text{lim}_{h \to 0} \frac{f(a+h)-f(a)}{h}$ exists. This is the definition using the limit of the difference quotient.

Flashcard 38: 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 39: 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 40: Identify if continuity implies differentiability.

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

Flashcard 41: Determine if $f(x) = x^5$ is differentiable at $x=0$.

Answer: Yes, $f(x) = x^5$ is differentiable at $x=0$. Odd powers are differentiable everywhere in their domain.

Flashcard 42: Which rule checks if a function $f(x)$ is differentiable at $x=a$?

Answer: Check if $\text{lim}_{h \to 0} \frac{f(a+h)-f(a)}{h}$ exists. This is the definition using the limit of the difference quotient.

Flashcard 43: 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 44: State if $f(x) = x^{2/3}$ is differentiable at $x=0$.

Answer: No, $f(x) = x^{2/3}$ is not differentiable at $x=0$. Fractional powers less than 1 create vertical tangents at zero.

Flashcard 45: 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 46: 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 47: 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 48: Determine if $f(x) = x^5$ is differentiable at $x=0$.

Answer: Yes, $f(x) = x^5$ is differentiable at $x=0$. Odd powers are differentiable everywhere in their domain.

Flashcard 49: 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 50: 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 51: 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 52: 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 53: 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 54: 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 55: What is the definition of continuity at a point?

Answer: $f$ is continuous at $x=a$ if $\lim_{x \to a} f(x) = f(a)$. The limit as x approaches a must equal the function value at a.

Flashcard 56: State if $f(x) = x^{2/3}$ is differentiable at $x=0$.

Answer: No, $f(x) = x^{2/3}$ is not differentiable at $x=0$. Fractional powers less than 1 create vertical tangents at zero.

Flashcard 57: 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 58: 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 59: 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 60: 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 61: 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 62: 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 63: What is the derivative of $f(x) = \text{cos}(x)$ at $x = 0$?

Answer: $f'(0) = 0$. Derivative of cosine is $-\sin(x)$, which equals 0 at $x=0$.

Flashcard 64: 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 65: 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 66: Determine the differentiability of $f(x) = \frac{1}{x^2}$ at $x=0$.

Answer: $f(x)$ is not differentiable at $x=0$. Function is undefined at zero due to division by zero.

Flashcard 67: State the rule for the existence of derivatives related to cusps.

Answer: Derivatives do not exist at cusps. Cusps have vertical tangent lines with undefined slopes.

Flashcard 68: 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.