Connecting a Function and Its Derivatives - AP Calculus AB

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

   $f''(x) = -\frac{1}{x^2}$. First derivative is $\frac{1}{x}$, then $-\frac{1}{x^2}$.

2. If $f''(x) = 0$, what can be inferred about $f(x)$?

   $f(x)$ may have a point of inflection. Second derivative test for inflection points.

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

   $f''(x) = -\text{sin}(x)$. First derivative is $\cos(x)$, then $-\sin(x)$.

4. What indicates a local maximum in terms of derivatives?

   $f'(x) = 0$ and $f''(x) < 0$. Critical point test: zero slope, negative concavity.

5. Find the derivative: $f(x) = 7x^5$.

   $f'(x) = 35x^4$. Factor out constant 7 and apply power rule.

6. State the relationship between concavity and the second derivative.

   Concave up if $f''(x) > 0$, concave down if $f''(x) < 0$. Second derivative test for concavity direction.

7. What is the derivative of $f(x) = \text{e}^{2x}$?

   $f'(x) = 2\text{e}^{2x}$. Use chain rule: derivative of $e^{2x}$ is $2e^{2x}$.

8. Evaluate $f'(x)$ for $f(x) = 5x^3 - 4x^2 + 2x - 7$.

   $f'(x) = 15x^2 - 8x + 2$. Apply power rule to each term separately.

9. What is the first derivative of $f(x) = \frac{1}{x^2}$?

   $f'(x) = -\frac{2}{x^3}$. Rewrite as $x^{-2}$, then apply power rule.

10. What is the derivative of $f(x) = \text{sin}(2x)$?

    $f'(x) = 2\text{cos}(2x)$. Use chain rule: derivative of $\sin(u)$ is $u'\cos(u)$.

11. If $f''(x) > 0$, what is the concavity of $f(x)$?

    $f(x)$ is concave up. Positive second derivative indicates upward curvature.

12. What does it mean if $f'(x) = 0$ for all $x$ in the interval?

    $f(x)$ is constant on the interval. Zero derivative means no rate of change.

13. Find $f'(x)$ for $f(x) = \frac{1}{2}x^3$.

    $f'(x) = \frac{3}{2}x^2$. Factor out $\frac{1}{2}$ and apply power rule.

14. What is $f''(x)$ for $f(x) = \text{e}^x + x^2$?

    $f''(x) = \text{e}^x + 2$. Sum rule: derivatives of $e^x$ and $x^2$ separately.

15. Identify the function whose first derivative is $f'(x) = 3x^2$.

    $f(x) = x^3 + C$, where $C$ is a constant. Antiderivative of $3x^2$ using power rule in reverse.

16. State the derivative of $f(x) = \text{ln}(x)$.

    $f'(x) = \frac{1}{x}$. Standard derivative formula for natural logarithm.

17. If $f'(x) < 0$ for all $x$, what is true about $f(x)$?

    $f(x)$ is decreasing. Negative derivative indicates downward slope.

18. Find $f''(x)$ for $f(x) = x^3 - 5x$.

    $f''(x) = 6x$. First find $f'(x) = 3x^2 - 5$, then differentiate.

19. Evaluate $f'(x)$ for $f(x) = \text{tan}(x)$.

    $f'(x) = \text{sec}^2(x)$. Standard trigonometric derivative formula.

20. Find the second derivative: $f(x) = \text{cos}(x)$.

    $f''(x) = -\text{cos}(x)$. First derivative is $-\sin(x)$, then $-\cos(x)$.

21. What is the derivative of a constant function $f(x) = c$?

    The derivative is $0$. Constants have zero rate of change.

22. Determine $f'(x)$ for $f(x) = \text{cos}(x)$.

    $f'(x) = -\text{sin}(x)$. Standard trigonometric derivative formula.

23. If $f'(x) > 0$ for all $x$, what can you say about $f(x)$?

    $f(x)$ is increasing. Positive derivative indicates upward slope.

24. Find the derivative of $f(x) = \frac{1}{x}$.

    The derivative is $f'(x) = -\frac{1}{x^2}$. Rewrite as $x^{-1}$ and apply power rule.

25. Determine the second derivative: $f(x) = 4x^2 + 3x$.

    $f''(x) = 8$. First derivative is $8x + 3$, then derivative is $8$.

26. Find the second derivative: $f(x) = 2x^4 - 3x^3 + x$.

    $f''(x) = 24x^2 - 18x$. First find $f'(x) = 8x^3 - 9x^2 + 1$, then differentiate.

27. Determine the derivative of $f(x) = \frac{1}{x^3}$.

    $f'(x) = -\frac{3}{x^4}$. Rewrite as $x^{-3}$ and apply power rule.

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

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

29. What is the derivative of $f(x) = e^{2x}$?

    $f'(x) = 2e^{2x}$. Use chain rule: derivative of $e^{2x}$ is $2e^{2x}$.

30. Determine the first derivative: $f(x) = \frac{x^3}{3}$.

    $f'(x) = x^2$. Derivative of $\frac{x^3}{3}$ using power rule.

31. State the Power Rule for differentiation.

    If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Bring down the exponent and reduce power by 1.

32. Evaluate the second derivative: $f(x) = 6x - 4x^2$.

    $f''(x) = -8$. First find $f'(x) = 6 - 8x$, then differentiate.

33. What is $f'(x)$ if $f(x) = \text{e}^{-x}$?

    $f'(x) = -\text{e}^{-x}$. Use chain rule with negative exponent.

34. What indicates a point of inflection in terms of the second derivative?

    A sign change in $f''(x)$. Concavity changes when second derivative changes sign.

35. Determine $f''(x)$ for $f(x) = 3x^4 + x^2$.

    $f''(x) = 36x^2 + 2$. First find $f'(x) = 12x^3 + 2x$, then differentiate.

36. What is the second derivative of $f(x) = x^3$?

    $f''(x) = 6x$. Apply power rule twice: $3x^2$ then $6x$.

37. What is $f''(x)$ if $f(x) = x^5 - 3x^2 + 1$?

    $f''(x) = 20x^3 - 6$. First find $f'(x) = 5x^4 - 6x$, then differentiate.

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