Connecting Differentiability and Continuity - AP Calculus BC

1. Determine if $f(x) = \text{sin}(x)$ is differentiable at $x=0$.

   Yes, $f(x) = \text{sin}(x)$ is differentiable at $x=0$. Sine function is smooth and differentiable everywhere.

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

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

3. Identify if differentiability implies continuity.

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

4. Find if $f(x) = \text{sgn}(x)$ is continuous at $x=0$.

   $f(x) = \text{sgn}(x)$ is not continuous at $x=0$. The sign function has a jump discontinuity at the origin.

5. What is the relationship between left-hand and right-hand derivatives for existence?

   They must be equal for the derivative to exist. Left and right limits of the difference quotient must match.

6. Is $f(x) = \text{step}(x)$ differentiable at $x=2$?

   No, $f(x) = \text{step}(x)$ is not differentiable at $x=2$. Step functions have jump discontinuities at transition points.

7. Is $f(x) = x^2\text{sin}(\frac{1}{x})$ continuous at $x=0$?

   Yes, it is continuous at $x=0$. The squeeze theorem shows continuity despite oscillation.

8. Identify the derivative of $f(x) = x^2$ at $x=2$.

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

9. Identify if $f(x) = \text{exp}(x)$ is differentiable at $x=1$.

   Yes, $f(x) = \text{exp}(x)$ is differentiable at $x=1$. Exponential functions are differentiable everywhere in their domain.

10. What is the derivative of $f(x) = \text{cos}(x)$ at $x = 0$?

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

11. State the rule for the existence of derivatives related to sharp corners.

    Derivatives do not exist at sharp corners. Sharp corners create different left and right-hand derivatives.

12. Find if $f(x) = \frac{1}{x-1}$ is continuous at $x=1$.

    $f(x)$ is not continuous at $x=1$. Division by zero creates a vertical asymptote and discontinuity.

13. Find the derivative of $f(x) = |x|$ at $x = 0$.

    The derivative does not exist at $x = 0$. The absolute value function has a sharp corner at the origin.

14. Identify the differentiability of $f(x) = \frac{1}{2x}$ at $x=0$.

    $f(x)$ is not differentiable at $x=0$. Division by zero makes the function undefined at the origin.

15. For $f(x) = \tan(x)$, find $f'(x)$ at $x=\frac{\text{π}}{4}$.

    $f'(\frac{\text{π}}{4}) = 2$. Derivative of tangent is $\sec^2(x)$, which equals 2 at $\frac{\pi}{4}$.

16. State the rule for the existence of derivatives related to discontinuities.

    Derivatives do not exist at discontinuities. Differentiability requires continuity as a prerequisite.

17. Identify if continuity implies differentiability.

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

18. Which rule checks if a function $f(x)$ is differentiable at $x=a$?

    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.

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

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

20. Determine if $f(x) = x^4$ is differentiable at $x=0$.

    Yes, $f(x) = x^4$ is differentiable at $x=0$. Even powers create smooth curves differentiable at all points.

21. Determine if $f(x) = \text{ln}(x)$ is differentiable at $x=0$.

    No, it is not differentiable at $x=0$. Natural logarithm is undefined at zero, so no derivative exists.

22. Determine if $f(x) = x^5$ is differentiable at $x=0$.

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

23. State if $f(x) = [x]$ (floor function) is differentiable at $x=1$.

    No, it is not differentiable at $x=1$. Floor functions have jump discontinuities at integer values.

24. Is $f(x) = x \times \text{sgn}(x)$ differentiable at $x=0$?

    No, it is not differentiable at $x=0$. This equals $|x|$, which has a corner at the origin.

25. State the rule for the existence of derivatives related to vertical tangents.

    Derivatives do not exist at vertical tangents. Vertical tangents have infinite slope, making derivatives undefined.

26. Find $f'(x)$ for $f(x) = 3x^3 - 4x + 1$ at $x=1$.

    $f'(1) = 5$. Power rule gives $f'(x) = 9x^2 - 4$, so $f'(1) = 5$.

27. What is the definition of continuity at a point?

    $f$ 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.

28. What is the derivative of $f(x) = e^x$ at any point $x$?

    $f'(x) = e^x$. The exponential function is its own derivative everywhere.

29. What is the derivative of $f(x) = \text{ln}(x)$ for $x > 0$?

    $f'(x) = \frac{1}{x}$. This is the standard derivative formula for natural logarithm.

30. Find if $f(x) = x^3$ is differentiable at $x=0$.

    Yes, $f(x) = x^3$ is differentiable at $x=0$. Polynomial functions are differentiable everywhere in their domain.

31. Does $f(x) = \text{abs}(x)$ have a derivative at $x=0$?

    No, it does not have a derivative at $x=0$. Absolute value creates a sharp corner with undefined derivative.

32. Determine the differentiability of $f(x) = \frac{1}{x^2}$ at $x=0$.

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

33. State the rule for the existence of derivatives related to cusps.

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

34. What is the definition of differentiability at a point?

    A function is differentiable at $x=a$ if $f'(a)$ exists. The derivative exists when the limit of the difference quotient exists.

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