First Derivative Test - AP Calculus AB

1. Find the derivative of $f(x) = \sin x$.

   $f'(x) = \cos x$. Derivative of sine function is cosine.

2. Identify the type of extrema at a critical point if $f'$ changes from positive to negative.

   Relative maximum. Function rises then falls, creating a local peak.

3. What is the First Derivative Test conclusion for $f'(x) = 0$ and no sign change?

   No relative extrema. When $f'$ doesn't change sign, there's no local extremum.

4. What is the relative extrema at $x = 2$ for $f(x) = x^3 - 3x^2 + 2$?

   Relative minimum. $f'$ changes from negative to positive at $x = 2$.

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

   Decreasing on $(-\text{infinity}, 0)$ and $(0, 2)$.. $f'(x) < 0$ when $x < 0$ or $0 < x < 2$.

6. Determine the critical points of $f(x) = \sin x$ on $[0, 2\pi]$.

   Critical points: $x = \frac{\pi}{2}, \frac{3\pi}{2}$. $\cos x = 0$ when $x = \frac{\pi}{2}, \frac{3\pi}{2}$.

7. What is the relative extrema at $x = -2$ for $f(x) = x^2 + 4x + 4$?

   Relative minimum. $f'$ changes from negative to positive at $x = -2$.

8. Find the critical points of $f(x) = x^2 + 4x + 4$.

   Critical point: $x = -2$. $f'(x) = 2x + 4 = 0$ gives $x = -2$.

9. Determine the critical points of $f(x) = \frac{1}{3}x^3 - 2x + 1$.

   Critical points: $x = \pm \sqrt{2}$. Solve $x^2 - 2 = 0$ to get $x^2 = 2$.

10. Find $f'(x)$ for $f(x) = 2x^3 - 3x^2 - 12x + 1$.

    $f'(x) = 6x^2 - 6x - 12$. Use power rule: $6x^2$ from $2x^3$, $-6x$ from $-3x^2$.

11. State the condition for a critical point of a function.

    The derivative $f'(x) = 0$ or $f'(x)$ is undefined. Critical points occur where the slope is zero or undefined.

12. Find $f'(x)$ for $f(x) = \frac{1}{4}x^4 - x^2 + 2$.

    $f'(x) = x^3 - 2x$. Derivative of $\frac{1}{4}x^4$ is $x^3$, of $x^2$ is $2x$.

13. What does $f'(x) < 0$ indicate about a function on an interval?

    The function is decreasing on that interval. Negative derivative means the function has negative slope.

14. Determine the intervals where $f(x) = \cos x$ is decreasing on $[0, 2\pi]$.

    Decreasing on $(0, \pi)$. $-\sin x < 0$ when $0 < x < \pi$.

15. Identify the critical points of $f(x) = \frac{1}{4}x^4 - x^2 + 2$.

    Critical points: $x = 0, \pm 1$. Solve $x^3 - 2x = x(x^2 - 2) = 0$.

16. Identify the critical points of $f(x) = \cos x$ on $[0, 2\backslashpi]$.

    Critical points: $x = \pi$. $-\sin x = 0$ when $x = \pi$ in the given interval.

17. What does $f'(x) > 0$ indicate about a function on an interval?

    The function is increasing on that interval. Positive derivative means the function has positive slope.

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

    Increasing on $(2, \, \text{infinity})$. $f'(x) > 0$ when $x > 2$ (test a point like $x = 3$).

19. What is the relative extrema at $x = \frac{\pi}{2}$ for $f(x) = \sin x$?

    Relative maximum. $f'$ changes from positive to negative at $x = \frac{\pi}{2}$.

20. What is the First Derivative Test used for in calculus?

    To determine relative (local) extrema of a function. Identifies where functions reach local peaks and valleys.

21. Identify the critical points of $f(x) = 2x^3 - 3x^2 - 12x + 1$.

    Critical points: $x = -1, 2$. Solve $6x^2 - 6x - 12 = 6(x^2 - x - 2) = 0$.

22. What is the relative extrema at $x = -1$ for $f(x) = 2x^3 - 3x^2 - 12x + 1$?

    Relative maximum. $f'$ changes from positive to negative at $x = -1$.

23. Identify the type of extrema at a critical point if $f'$ changes from negative to positive.

    Relative minimum. Function falls then rises, creating a local valley.

24. What is the relative extrema at $x = -\sqrt{2}$ for $f(x) = \frac{1}{3}x^3 - 2x + 1$?

    Relative maximum. $f'$ changes from positive to negative at $x = -\sqrt{2}$.

25. Find the derivative of $f(x) = x^3 - 3x^2 + 2$.

    $f'(x) = 3x^2 - 6x$. Use power rule: derivative of $x^3$ is $3x^2$, $x^2$ is $2x$.

26. What is the relative extrema at $x = 2$ for $f(x) = 2x^3 - 3x^2 - 12x + 1$?

    Relative minimum. $f'$ changes from negative to positive at $x = 2$.

27. What is the relative extrema at $x = \frac{3\pi}{2}$ for $f(x) = \sin x$?

    Relative minimum. $f'$ changes from negative to positive at $x = \frac{3\pi}{2}$.

28. Determine the intervals where $f(x) = \cos x$ is increasing on $[0, 2\pi]$.

    Increasing on $(\pi, 2\pi)$. $-\sin x > 0$ when $\pi < x < 2\pi$.

29. What is the relative extrema at $x = \pi$ for $f(x) = \cos x$?

    Relative minimum. $f'$ changes from negative to positive at $x = \pi$.

30. Find the derivative of $f(x) = \frac{1}{3}x^3 - 2x + 1$.

    $f'(x) = x^2 - 2$. Use power rule on each term of the polynomial.

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

    Critical points: $x = 0, x = 2$. Set $f'(x) = 3x^2 - 6x = 3x(x-2) = 0$.

32. Identify the critical points of $f(x) = \cos x$ on $[0, 2\pi]$.

    Critical points: $x = \pi$. $-\sin x = 0$ when $x = \pi$ in the given interval.

33. What conclusion is drawn if $f'(x)$ does not change sign at a critical point?

    No relative extrema at that point. No sign change means no local maximum or minimum.

34. State the derivative of $f(x) = \cos x$.

    $f'(x) = -\sin x$. Derivative of cosine function is negative sine.

35. What is the relative extrema at $x = \sqrt{2}$ for $f(x) = \frac{1}{3}x^3 - 2x + 1$?

    Relative minimum. $f'$ changes from negative to positive at $x = \sqrt{2}$.

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