Sketching Graphs of Functions and Derivatives - AP Calculus BC

1. What can be concluded if $f'(x) > 0$?

   $f(x)$ is increasing on that interval. Positive first derivative means function values are getting larger.

2. What occurs at a local minimum of $f(x)$?

   $f'(x) = 0$ and $f''(x) > 0$. Both conditions ensure a true local minimum exists.

3. What is the relationship between $f''(x)$ and concavity?

   $f''(x) > 0$: concave up; $f''(x) < 0$: concave down. Second derivative test determines the direction of curvature.

4. What does $f''(x) > 0$ imply about $f(x)$?

   Function is concave up. Positive second derivative indicates the function curves upward.

5. Determine the local maximum of $f(x) = -x^2 + 4x - 3$.

   Local maximum at $x = 2$. Find critical point: $f'(x) = -2x + 4 = 0$ gives $x = 2$.

6. What does $f'(x) < 0$ imply about $f(x)$?

   $f(x)$ is decreasing on that interval. Negative first derivative means function values are getting smaller.

7. What is the critical point of a function $f(x)$?

   Where $f'(x) = 0$ or $f'(x)$ is undefined. Critical points occur where the derivative equals zero or doesn't exist.

8. What is the significance of $f'(x) = 0$?

   Potential local max, min, or saddle point. Zero derivative indicates a horizontal tangent line at that point.

9. What does $f'(x) = 0$ imply about $f(x)$?

   Possible local max, min, or point of inflection. Zero derivative indicates a horizontal tangent line.

10. What does a zero $f'(x)$ indicate?

    Potential local max, min, or saddle point. Zero first derivative creates a horizontal tangent line.

11. What does the second derivative of a function indicate?

    The concavity of the function. $f''(x)$ describes whether the graph curves upward or downward.

12. What does $f''(x) = 0$ suggest about $f(x)$?

    Possible inflection point. Zero second derivative may indicate where concavity changes.

13. What is the effect of $f'(x) = 0$ on $f(x)$?

    Potential local extremum. Zero derivative creates a horizontal tangent, possibly an extremum.

14. When does $f(x)$ have a point of inflection?

    When $f''(x)$ changes sign. Sign change in second derivative indicates concavity reversal.

15. What does a positive $f''(x)$ indicate about $f(x)$?

    The graph of $f(x)$ is concave up. Positive second derivative means the graph curves upward like a bowl.

16. Calculate $f'(x)$ for $f(x) = x^3 - 3x + 1$.

    $f'(x) = 3x^2 - 3$. Apply power rule: derivative of $x^3$ is $3x^2$, derivative of $-3x$ is $-3$.

17. What does it mean for $f'(x)$ to be undefined?

    Possible critical point or cusp. Undefined derivative often occurs at corners, cusps, or vertical tangents.

18. Determine where $f(x) = -x^3 + 3x^2$ is decreasing.

    Decreasing on $(0, 2)$. Find where $f'(x) = -3x^2 + 6x = -3x(x-2) < 0$.

19. Compute $f''(x)$ for $f(x) = x^5 - 5x^3 + 10x$.

    $f''(x) = 20x^3 - 30x$. Differentiate twice: $f'(x) = 5x^4 - 15x^2$, then $f''(x) = 20x^3 - 30x$.

20. What is the relationship between $f(x)$ and $f'(x)$?

    $f'(x)$ gives the slope of $f(x)$. The derivative measures how steeply the function rises or falls.

21. Calculate $f'(x)$ for $f(x) = 2x^3 - 5x^2 + 4x - 1$.

    $f'(x) = 6x^2 - 10x + 4$. Apply power rule to each term: $6x^2 - 10x + 4$.

22. Find the critical points for $f(x) = x^2 - 4x + 3$.

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

23. State the definition of an inflection point.

    Where $f''(x)$ changes sign. Inflection points occur where concavity changes from up to down or vice versa.

24. Identify the local minimum of $f(x) = x^2 - 4x + 4$.

    Local minimum at $x = 2$. Complete the square: $f(x) = (x-2)^2$, minimum at vertex.

25. Find $f''(x)$ for $f(x) = x^4 - 4x^2 + 6$.

    $f''(x) = 12x^2 - 8$. Differentiate twice using power rule: $(x^4)'' = 12x^2$, $(-4x^2)'' = -8$.

26. Identify the intervals of increase for $f(x) = x^2 - 4x + 3$.

    Increasing on $(2, \text{infinity})$. Find where $f'(x) = 2x - 4 > 0$, so $x > 2$.

27. What does a negative $f''(x)$ indicate?

    The graph is concave down. Negative second derivative means the graph curves downward.

28. What is the first derivative of $f(x)$ used to determine?

    The slope of the tangent line to $f(x)$. $f'(x)$ represents the instantaneous rate of change at any point.

29. How do you determine a local maximum using derivatives?

    If $f'(x) = 0$ and $f''(x) < 0$. Second derivative test: negative second derivative confirms a maximum.

30. Find inflection points for $f(x) = x^3 - 3x + 1$.

    Inflection point at $x = 0$. Set $f''(x) = 6x = 0$ to find where concavity changes.

31. Calculate $f''(x)$ for $f(x) = x^3 - 3x^2 + 1$.

    $f''(x) = 6x - 6$. Differentiate twice: $f'(x) = 3x^2 - 6x$, then $f''(x) = 6x - 6$.

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