Connecting a Function and Its Derivatives - AP Calculus BC

1. Find $f''(x)$ for $f(x) = x^4$.

   $f''(x) = 12x^2$. Apply power rule: $f'(x) = 4x^3$, then $f''(x) = 12x^2$.

2. What can be concluded if $f''(x) < 0$ at a point?

   $f(x)$ is concave down. Negative second derivative means curve bends downward.

3. Find $f'(x)$ for $f(x) = x^2 + 2x + 1$.

   $f'(x) = 2x + 2$. Apply power rule to polynomial.

4. Determine the inflection point of $f(x) = x^3 - 3x$.

   At $x = 0$. Set $f''(x) = 6x = 0$ to find inflection points.

5. Find the critical points of $f(x) = x^3 - 3x^2$.

   At $x = 0$ and $x = 2$. Set $f'(x) = 3x^2 - 6x = 3x(x-2) = 0$.

6. Calculate $f'(x)$ for $f(x) = \tan(x)$.

   $f'(x) = \sec^2(x)$. Standard derivative of tangent function.

7. Find the derivative of $f(x) = \text{e}^{-x}$.

   $f'(x) = -\text{e}^{-x}$. Apply chain rule with $u = -x$.

8. If $f''(x) > 0$, what can be said about $f(x)$?

   $f(x)$ is concave up. Positive second derivative means curve bends upward.

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

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

10. Determine $f''(x)$ for $f(x) = \text{cos}(2x)$.

    $f''(x) = -4\text{cos}(2x)$. Apply chain rule: derivative of $\cos(2x)$ is $-2\sin(2x)$, then again.

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

    $f'(x) = -\text{sin}(x)$. Standard derivative of cosine function.

12. What does the concavity test involve?

    Analyzing $f''(x)$ to determine concavity. Examining where $f''(x)$ changes sign.

13. Identify the critical points of $f(x) = x^2 - 4x + 4$.

    At $x = 2$. Set $f'(x) = 2x - 4 = 0$ to find critical points.

14. Determine $f'(x)$ for $f(x) = \text{ln}(2x)$.

    $f'(x) = \frac{1}{x}$. Chain rule: derivative of $\ln(2x)$ is $\frac{2}{2x} = \frac{1}{x}$.

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

    $f''(x) = -\text{cos}(x)$. Derivative of cosine is $-\sin(x)$; derivative again gives $-\cos(x)$.

16. What is the second derivative of $f(x) = \text{ln}(x^2)$?

    $f''(x) = -\frac{2}{x^2}$. Use chain rule: $f'(x) = \frac{2}{x}$, then apply quotient rule.

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

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

18. What is the second derivative of $f(x) = x^5$?

    $f''(x) = 20x^3$. Apply power rule: $f'(x) = 5x^4$, then $f''(x) = 20x^3$.

19. What does the first derivative of a function represent?

    The slope of the tangent line to the curve at a point. Derivative measures instantaneous rate of change.

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

    $f''(x) = \text{e}^x$. Exponential function equals its own second derivative.

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

    $f'(x) = 2\text{sin}(x)\text{cos}(x)$. Apply chain rule to $\sin^2(x) = (\sin(x))^2$.

22. What is the second derivative test used for?

    To determine if a critical point is a local max or min. Uses sign of $f''(c)$ to classify critical points.

23. Find $f''(x)$ for $f(x) = \frac{1}{x}$.

    $f''(x) = \frac{2}{x^3}$. Rewrite as $x^{-1}$, apply power rule twice.

24. What does $f'(x) = 0$ indicate about $f(x)$ at $x$?

    Possible local maxima, minima, or inflection point. Zero derivative indicates horizontal tangent line.

25. Determine the first derivative of $f(x) = \text{e}^{2x}$.

    $f'(x) = 2\text{e}^{2x}$. Apply chain rule: $\frac{d}{dx}[e^{u}] = e^{u} \cdot u'$.

26. Determine $f'(x)$ for $f(x) = e^x$.

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

27. Identify 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 second derivative of $f(x) = \ln(x)$?

    $f''(x) = -\frac{1}{x^2}$. Derivative of $\frac{1}{x}$ applied to $\ln(x)$.

29. What is the first derivative of $f(x) = \ln(x)$?

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

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

    $f'(x) = \text{e}^x + x \text{e}^x$. Apply product rule: $(uv)' = u'v + uv'$.

31. Identify the derivative of $f(x) = \frac{1}{x}$.

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

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

    $f''(x) = -\frac{1}{x^2}$. Derivative of $\frac{1}{x}$ applied to $\ln(x)$.

33. What does the second derivative of a function represent?

    The concavity of the function. Second derivative determines curve shape and bending.

34. Calculate $f'(x)$ for $f(x) = \text{tan}(x)$.

    $f'(x) = \text{sec}^2(x)$. Standard derivative of tangent function.

35. State the power rule for differentiation.

    $\frac{d}{dx}[x^n] = nx^{n-1}$. Fundamental rule for differentiating polynomial terms.

36. Calculate $f''(x)$ for $f(x) = \text{sin}(x)$.

    $f''(x) = -\text{sin}(x)$. Derivative of sine is cosine; derivative of cosine is $-\sin(x)$.

37. What is the derivative of $f(x) = x^2$?

    $f'(x) = 2x$. Apply power rule: $\frac{d}{dx}[x^n] = nx^{n-1}$.

38. What is the first derivative test used for?

    To find intervals of increase or decrease. Uses sign of $f'(x)$ to determine monotonicity.

39. Determine the second derivative of $f(x) = \text{tan}(x)$.

    $f''(x) = 2\text{sec}^2(x)\text{tan}(x)$. Apply product rule to $\sec^2(x)$ derivative.

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

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

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

    $f'(x) = 2\text{cos}(2x)$. Apply chain rule to $\sin(2x)$.

Study these flashcards