AP Calculus AB 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 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: Can a function with a vertical tangent be differentiable there?

Answer: No, vertical tangent implies non-differentiability. Vertical tangents are non-differentiable.

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

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

Flashcard 3: Identify whether $f(x) = |x|$ is differentiable at $x = 0$.

Answer: No, $f(x) = |x|$ is not differentiable at $x = 0$. Has a sharp corner at the origin.

Flashcard 4: What is the derivative of $f(x) = \text{e}^x$?

Answer: $\text{e}^x$. Exponential function is its own derivative.

Flashcard 5: What is the derivative rule for a sum $f(x) + g(x)$?

Answer: $f'(x) + g'(x)$. Derivative of sum equals sum of derivatives.

Flashcard 6: What is the derivative rule for a product $f(x)g(x)$?

Answer: $f'(x)g(x) + f(x)g'(x)$. Product rule for differentiation.

Flashcard 7: Does differentiability at $x = a$ imply continuity at $x = a$?

Answer: Yes, differentiability implies continuity. A differentiable function must be continuous.

Flashcard 8: For $f(x) = \frac{1}{x}$, is $f(x)$ differentiable at $x = 0$?

Answer: No, $f(x)$ is undefined at $x = 0$. Function is undefined at $x = 0$.

Flashcard 9: What is the derivative rule for a quotient $\frac{f(x)}{g(x)}$?

Answer: $\frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$. Quotient rule for differentiation.

Flashcard 10: Determine the differentiability of $f(x) = x^2 + 3x + 2$ at $x = -1$.

Answer: Differentiable at $x = -1$. Polynomial functions are differentiable everywhere.

Flashcard 11: Determine the differentiability of $f(x) = \text{floor}(x)$ at $x = 2$.

Answer: Not differentiable at $x = 2$. Floor function has jump discontinuities at integers.

Flashcard 12: If $f'(a)$ does not exist, what can be said about $f(x)$ at $x = a$?

Answer: $f(x)$ is not differentiable at $x = a$. No derivative means not differentiable.

Flashcard 13: If $\text{lim}_{x \to a} f(x)$ does not exist, what about $f'(a)$?

Answer: $f'(a)$ does not exist. Discontinuity prevents differentiability.

Flashcard 14: Identify the differentiability of $f(x) = x^{\frac{3}{2}}$ at $x = 0$.

Answer: Differentiable at $x = 0$. Fractional power greater than 1 is differentiable.

Flashcard 15: Determine the differentiability of $f(x) = x^2 + 3x + 2$ at $x = -1$.

Answer: Differentiable at $x = -1$. Polynomial functions are differentiable everywhere.

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

Answer: $\text{cos}(x)$. Derivative of sine is cosine.

Flashcard 17: What is the derivative of $f(x) = x^{\frac{2}{3}}$ at $x = 0$?

Answer: The derivative does not exist at $x = 0$. Vertical tangent at the origin.

Flashcard 18: Identify whether $f(x) = x^2 \text{sin}(1/x)$ is differentiable at $x = 0$.

Answer: Yes, it is differentiable at $x = 0$. The oscillating term is bounded by $x^2$.

Flashcard 19: What is the derivative rule for a sum $f(x) + g(x)$?

Answer: $f'(x) + g'(x)$. Derivative of sum equals sum of derivatives.

Flashcard 20: What is the derivative of $f(x) = x^n$?

Answer: $nx^{n-1}$. Power rule for differentiation.

Flashcard 21: What is the derivative of $f(x) = \text{e}^x$?

Answer: $\text{e}^x$. Exponential function is its own derivative.

Flashcard 22: What is the derivative of $f(x) = x^n$?

Answer: $nx^{n-1}$. Power rule for differentiation.

Flashcard 23: Identify the differentiability of $f(x) = x^{\frac{3}{2}}$ at $x = 0$.

Answer: Differentiable at $x = 0$. Fractional power greater than 1 is differentiable.

Flashcard 24: Identify the differentiability of $f(x) = \text{tan}(x)$ at $x = \frac{\text{pi}}{2}$.

Answer: Not differentiable at $x = \frac{\text{pi}}{2}$. Tangent is undefined at $\frac{\pi}{2}$.

Flashcard 25: What is the chain rule for differentiation?

Answer: If $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Chain rule for composite functions.

Flashcard 26: What is the derivative of $f(x) = \text{ln}(x)$ at $x = 1$?

Answer:

1. Derivative of $\ln(x)$ is $\frac{1}{x}$.

Flashcard 27: Can a function be continuous but not differentiable at $x = a$?

Answer: Yes, a function can be continuous but not differentiable. Example: $f(x) = |x|$ at $x = 0$.

Flashcard 28: State one reason why a function might not be differentiable at a point.

Answer: A cusp or corner at the point. Sharp corners prevent differentiability.

Flashcard 29: Determine the differentiability of $f(x) = \text{sgn}(x)$ at $x = 0$.

Answer: Not differentiable at $x = 0$. Sign function has a jump discontinuity.

Flashcard 30: Identify the differentiability of $f(x) = |x|$ at $x = 1$.

Answer: Differentiable at $x = 1$. Absolute value is smooth away from zero.

Flashcard 31: State the relationship between continuity and differentiability.

Answer: Differentiability implies continuity. Differentiable functions are always continuous.

Flashcard 32: What is the derivative rule for a product $f(x)g(x)$?

Answer: $f'(x)g(x) + f(x)g'(x)$. Product rule for differentiation.

Flashcard 33: Determine if $f(x) = x^2 \text{cos}(1/x)$ is differentiable at $x = 0$.

Answer: Yes, differentiable at $x = 0$. Bounded oscillation makes it differentiable.

Flashcard 34: What is the derivative of $f(x) = \ln(x)$ at $x = 1$?

Answer:

1. Derivative of $\ln(x)$ is $\frac{1}{x}$.

Flashcard 35: Determine the differentiability of $f(x) = \text{sgn}(x)$ at $x = 0$.

Answer: Not differentiable at $x = 0$. Sign function has a jump discontinuity.

Flashcard 36: What is the derivative of $f(x) = \text{sin}(x)$?

Answer: $\text{cos}(x)$. Derivative of sine is cosine.

Flashcard 37: Can a function with a vertical tangent be differentiable there?

Answer: No, vertical tangent implies non-differentiability. Vertical tangents are non-differentiable.

Flashcard 38: Identify whether $f(x) = |x|$ is differentiable at $x = 0$.

Answer: No, $f(x) = |x|$ is not differentiable at $x = 0$. Has a sharp corner at the origin.

Flashcard 39: Can a function have a corner at $x = a$ and still be differentiable there?

Answer: No, corners are non-differentiable. Corners create non-differentiable points.

Flashcard 40: If $\text{lim}_{x \to a} f(x)$ does not exist, what about $f'(a)$?

Answer: $f'(a)$ does not exist. Discontinuity prevents differentiability.

Flashcard 41: Determine if $f(x) = x^2 \text{cos}(1/x)$ is differentiable at $x = 0$.

Answer: Yes, differentiable at $x = 0$. Bounded oscillation makes it differentiable.

Flashcard 42: Identify the differentiability of $f(x) = \text{tan}(x)$ at $x = \frac{\text{pi}}{2}$.

Answer: Not differentiable at $x = \frac{\text{pi}}{2}$. Tangent is undefined at $\frac{\pi}{2}$.

Flashcard 43: State the condition for differentiability at a point $x = a$.

Answer: $f(x)$ is differentiable at $x = a$ if $f'(a)$ exists. The derivative must exist for differentiability.

Flashcard 44: Identify whether $f(x) = x^2 \text{sin}(1/x)$ is differentiable at $x = 0$.

Answer: Yes, it is differentiable at $x = 0$. The oscillating term is bounded by $x^2$.

Flashcard 45: What is the derivative of $f(x) = \text{cos}(x)$?

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

Flashcard 46: Does differentiability at $x = a$ imply continuity at $x = a$?

Answer: Yes, differentiability implies continuity. A differentiable function must be continuous.

Flashcard 47: If $f'(a)$ does not exist, what can be said about $f(x)$ at $x = a$?

Answer: $f(x)$ is not differentiable at $x = a$. No derivative means not differentiable.

Flashcard 48: What is the derivative rule for a quotient $\frac{f(x)}{g(x)}$?

Answer: $\frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$. Quotient rule for differentiation.

Flashcard 49: What is the chain rule for differentiation?

Answer: If $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Chain rule for composite functions.

Flashcard 50: Determine the differentiability of $f(x) = \text{floor}(x)$ at $x = 2$.

Answer: Not differentiable at $x = 2$. Floor function has jump discontinuities at integers.

Flashcard 51: State one reason why a function might not be differentiable at a point.

Answer: A cusp or corner at the point. Sharp corners prevent differentiability.

Flashcard 52: Can a function have a corner at $x = a$ and still be differentiable there?

Answer: No, corners are non-differentiable. Corners create non-differentiable points.

Flashcard 53: For $f(x) = \frac{1}{x}$, is $f(x)$ differentiable at $x = 0$?

Answer: No, $f(x)$ is undefined at $x = 0$. Function is undefined at $x = 0$.

Flashcard 54: What is the derivative of $f(x) = x^{\frac{2}{3}}$ at $x = 0$?

Answer: The derivative does not exist at $x = 0$. Vertical tangent at the origin.

Flashcard 55: Can a function be continuous but not differentiable at $x = a$?

Answer: Yes, a function can be continuous but not differentiable. Example: $f(x) = |x|$ at $x = 0$.

Flashcard 56: State the relationship between continuity and differentiability.

Answer: Differentiability implies continuity. Differentiable functions are always continuous.

Flashcard 57: Identify the differentiability of $f(x) = |x|$ at $x = 1$.

Answer: Differentiable at $x = 1$. Absolute value is smooth away from zero.

Flashcard 58: State the condition for differentiability at a point $x = a$.

Answer: $f(x)$ is differentiable at $x = a$ if $f'(a)$ exists. The derivative must exist for differentiability.