Connecting Differentiability and Continuity - AP Calculus AB

1. What is the derivative of $f(x) = \text{cos}(x)$?

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

2. What is the derivative rule for a sum $f(x) + g(x)$?

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

3. For $f(x) = \frac{1}{x}$, is $f(x)$ differentiable at $x = 0$?

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

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

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

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

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

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

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

7. What is the derivative of $f(x) = \text{e}^x$?

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

8. What is the derivative of $f(x) = x^n$?

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

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

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

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

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

11. What is the derivative of $f(x) = \text{ln}(x)$ at $x = 1$?

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

12. Can a function be continuous but not differentiable at $x = a$?

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

13. Identify the differentiability of $f(x) = |x|$ at $x = 1$.

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

14. What is the derivative rule for a product $f(x)g(x)$?

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

15. What is the derivative of $f(x) = \ln(x)$ at $x = 1$?

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

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

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

17. What is the derivative of $f(x) = \text{sin}(x)$?

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

18. Can a function with a vertical tangent be differentiable there?

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

19. Identify whether $f(x) = |x|$ is differentiable at $x = 0$.

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

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

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

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

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

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

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

23. Does differentiability at $x = a$ imply continuity at $x = a$?

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

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

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

25. What is the derivative rule for a quotient $\frac{f(x)}{g(x)}$?

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

26. What is the chain rule for differentiation?

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

27. State one reason why a function might not be differentiable at a point.

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

28. Can a function have a corner at $x = a$ and still be differentiable there?

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

29. State the relationship between continuity and differentiability.

    Differentiability implies continuity. Differentiable functions are always continuous.

30. State the condition for differentiability at a point $x = a$.

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

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