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

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

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

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

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

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

Flashcard 5: What is the derivative of $f(x) = \text{sin}(x)$?

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

Flashcard 6: 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 7: 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 8: Identify the differentiability of $f(x) = |x|$ at $x = 1$.

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

Flashcard 9: State the relationship between continuity and differentiability.

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

Flashcard 10: 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 11: 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 12: 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 13: 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 14: 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 15: 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 16: Does differentiability at $x = a$ imply continuity at $x = a$?

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

Flashcard 17: 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 18: 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 19: 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 20: 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 21: 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 22: 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 23: What is the derivative of $f(x) = \text{ln}(x)$ at $x = 1$?

Answer:

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

Flashcard 24: 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 25: 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 26: Can a function with a vertical tangent be differentiable there?

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

Flashcard 27: 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 28: 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 29: 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 30: 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.