Connecting Differentiability and Continuity - AP Calculus AB
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What is the derivative rule for a quotient $\frac{f(x)}{g(x)}$?
What is the derivative rule for a quotient $\frac{f(x)}{g(x)}$?
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$\frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$. Quotient rule for differentiation.
$\frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$. Quotient rule for differentiation.
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What is the derivative of $f(x) = x^n$?
What is the derivative of $f(x) = x^n$?
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$nx^{n-1}$. Power rule for differentiation.
$nx^{n-1}$. Power rule for differentiation.
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Determine the differentiability of $f(x) = x^2 + 3x + 2$ at $x = -1$.
Determine the differentiability of $f(x) = x^2 + 3x + 2$ at $x = -1$.
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Differentiable at $x = -1$. Polynomial functions are differentiable everywhere.
Differentiable at $x = -1$. Polynomial functions are differentiable everywhere.
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What is the derivative of $f(x) = \text{cos}(x)$?
What is the derivative of $f(x) = \text{cos}(x)$?
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$-\text{sin}(x)$. Derivative of cosine is negative sine.
$-\text{sin}(x)$. Derivative of cosine is negative sine.
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What is the derivative of $f(x) = \text{sin}(x)$?
What is the derivative of $f(x) = \text{sin}(x)$?
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$\text{cos}(x)$. Derivative of sine is cosine.
$\text{cos}(x)$. Derivative of sine is cosine.
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Determine the differentiability of $f(x) = \text{floor}(x)$ at $x = 2$.
Determine the differentiability of $f(x) = \text{floor}(x)$ at $x = 2$.
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Not differentiable at $x = 2$. Floor function has jump discontinuities at integers.
Not differentiable at $x = 2$. Floor function has jump discontinuities at integers.
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What is the chain rule for differentiation?
What is the chain rule for differentiation?
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If $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Chain rule for composite functions.
If $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Chain rule for composite functions.
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Identify the differentiability of $f(x) = |x|$ at $x = 1$.
Identify the differentiability of $f(x) = |x|$ at $x = 1$.
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Differentiable at $x = 1$. Absolute value is smooth away from zero.
Differentiable at $x = 1$. Absolute value is smooth away from zero.
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State the relationship between continuity and differentiability.
State the relationship between continuity and differentiability.
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Differentiability implies continuity. Differentiable functions are always continuous.
Differentiability implies continuity. Differentiable functions are always continuous.
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Determine if $f(x) = x^2 \text{cos}(1/x)$ is differentiable at $x = 0$.
Determine if $f(x) = x^2 \text{cos}(1/x)$ is differentiable at $x = 0$.
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Yes, differentiable at $x = 0$. Bounded oscillation makes it differentiable.
Yes, differentiable at $x = 0$. Bounded oscillation makes it differentiable.
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If $\text{lim}_{x \to a} f(x)$ does not exist, what about $f'(a)$?
If $\text{lim}_{x \to a} f(x)$ does not exist, what about $f'(a)$?
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$f'(a)$ does not exist. Discontinuity prevents differentiability.
$f'(a)$ does not exist. Discontinuity prevents differentiability.
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Determine the differentiability of $f(x) = \text{sgn}(x)$ at $x = 0$.
Determine the differentiability of $f(x) = \text{sgn}(x)$ at $x = 0$.
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Not differentiable at $x = 0$. Sign function has a jump discontinuity.
Not differentiable at $x = 0$. Sign function has a jump discontinuity.
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What is the derivative rule for a sum $f(x) + g(x)$?
What is the derivative rule for a sum $f(x) + g(x)$?
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$f'(x) + g'(x)$. Derivative of sum equals sum of derivatives.
$f'(x) + g'(x)$. Derivative of sum equals sum of derivatives.
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What is the derivative of $f(x) = x^{\frac{2}{3}}$ at $x = 0$?
What is the derivative of $f(x) = x^{\frac{2}{3}}$ at $x = 0$?
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The derivative does not exist at $x = 0$. Vertical tangent at the origin.
The derivative does not exist at $x = 0$. Vertical tangent at the origin.
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Can a function be continuous but not differentiable at $x = a$?
Can a function be continuous but not differentiable at $x = a$?
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Yes, a function can be continuous but not differentiable. Example: $f(x) = |x|$ at $x = 0$.
Yes, a function can be continuous but not differentiable. Example: $f(x) = |x|$ at $x = 0$.
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Does differentiability at $x = a$ imply continuity at $x = a$?
Does differentiability at $x = a$ imply continuity at $x = a$?
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Yes, differentiability implies continuity. A differentiable function must be continuous.
Yes, differentiability implies continuity. A differentiable function must be continuous.
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Identify the differentiability of $f(x) = x^{\frac{3}{2}}$ at $x = 0$.
Identify the differentiability of $f(x) = x^{\frac{3}{2}}$ at $x = 0$.
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Differentiable at $x = 0$. Fractional power greater than 1 is differentiable.
Differentiable at $x = 0$. Fractional power greater than 1 is differentiable.
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What is the derivative rule for a product $f(x)g(x)$?
What is the derivative rule for a product $f(x)g(x)$?
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$f'(x)g(x) + f(x)g'(x)$. Product rule for differentiation.
$f'(x)g(x) + f(x)g'(x)$. Product rule for differentiation.
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Can a function have a corner at $x = a$ and still be differentiable there?
Can a function have a corner at $x = a$ and still be differentiable there?
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No, corners are non-differentiable. Corners create non-differentiable points.
No, corners are non-differentiable. Corners create non-differentiable points.
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Identify whether $f(x) = x^2 \text{sin}(1/x)$ is differentiable at $x = 0$.
Identify whether $f(x) = x^2 \text{sin}(1/x)$ is differentiable at $x = 0$.
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Yes, it is differentiable at $x = 0$. The oscillating term is bounded by $x^2$.
Yes, it is differentiable at $x = 0$. The oscillating term is bounded by $x^2$.
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Identify whether $f(x) = |x|$ is differentiable at $x = 0$.
Identify whether $f(x) = |x|$ is differentiable at $x = 0$.
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No, $f(x) = |x|$ is not differentiable at $x = 0$. Has a sharp corner at the origin.
No, $f(x) = |x|$ is not differentiable at $x = 0$. Has a sharp corner at the origin.
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State the condition for differentiability at a point $x = a$.
State the condition for differentiability at a point $x = a$.
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$f(x)$ is differentiable at $x = a$ if $f'(a)$ exists. The derivative must exist for differentiability.
$f(x)$ is differentiable at $x = a$ if $f'(a)$ exists. The derivative must exist for differentiability.
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What is the derivative of $f(x) = \text{ln}(x)$ at $x = 1$?
What is the derivative of $f(x) = \text{ln}(x)$ at $x = 1$?
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- Derivative of $\ln(x)$ is $\frac{1}{x}$.
- Derivative of $\ln(x)$ is $\frac{1}{x}$.
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State one reason why a function might not be differentiable at a point.
State one reason why a function might not be differentiable at a point.
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A cusp or corner at the point. Sharp corners prevent differentiability.
A cusp or corner at the point. Sharp corners prevent differentiability.
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If $f'(a)$ does not exist, what can be said about $f(x)$ at $x = a$?
If $f'(a)$ does not exist, what can be said about $f(x)$ at $x = a$?
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$f(x)$ is not differentiable at $x = a$. No derivative means not differentiable.
$f(x)$ is not differentiable at $x = a$. No derivative means not differentiable.
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Can a function with a vertical tangent be differentiable there?
Can a function with a vertical tangent be differentiable there?
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No, vertical tangent implies non-differentiability. Vertical tangents are non-differentiable.
No, vertical tangent implies non-differentiability. Vertical tangents are non-differentiable.
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Identify the differentiability of $f(x) = \text{tan}(x)$ at $x = \frac{\text{pi}}{2}$.
Identify the differentiability of $f(x) = \text{tan}(x)$ at $x = \frac{\text{pi}}{2}$.
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Not differentiable at $x = \frac{\text{pi}}{2}$. Tangent is undefined at $\frac{\pi}{2}$.
Not differentiable at $x = \frac{\text{pi}}{2}$. Tangent is undefined at $\frac{\pi}{2}$.
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For $f(x) = \frac{1}{x}$, is $f(x)$ differentiable at $x = 0$?
For $f(x) = \frac{1}{x}$, is $f(x)$ differentiable at $x = 0$?
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No, $f(x)$ is undefined at $x = 0$. Function is undefined at $x = 0$.
No, $f(x)$ is undefined at $x = 0$. Function is undefined at $x = 0$.
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Identify the differentiability of $f(x) = \text{tan}(x)$ at $x = \frac{\text{pi}}{2}$.
Identify the differentiability of $f(x) = \text{tan}(x)$ at $x = \frac{\text{pi}}{2}$.
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Not differentiable at $x = \frac{\text{pi}}{2}$. Tangent is undefined at $\frac{\pi}{2}$.
Not differentiable at $x = \frac{\text{pi}}{2}$. Tangent is undefined at $\frac{\pi}{2}$.
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Identify whether $f(x) = |x|$ is differentiable at $x = 0$.
Identify whether $f(x) = |x|$ is differentiable at $x = 0$.
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No, $f(x) = |x|$ is not differentiable at $x = 0$. Has a sharp corner at the origin.
No, $f(x) = |x|$ is not differentiable at $x = 0$. Has a sharp corner at the origin.
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