Determining Intervals on Increasing, Decreasing Functions - AP Calculus AB

1. What does it mean if $f'(x) = 0$ at some point?

   The function may have a local maximum, minimum, or saddle point. Critical points are candidates for local extrema or points of inflection.

2. What does it mean if $f'(x)$ changes sign at a point?

   There is a local extremum at that point. Sign changes in the derivative indicate local maxima or minima.

3. Explain the significance of the second derivative's sign.

   Indicates concavity and potential inflection points. Second derivative determines concavity: positive means concave up, negative means concave down.

4. What is the derivative of $f(x) = x^4$?

   $f'(x) = 4x^3$. Apply power rule: $(x^4)' = 4x^3$.

5. What is the first derivative test for determining intervals of decrease?

   If $f'(x) < 0$, the function is decreasing on that interval. When the slope is negative, the function is falling.

6. Find $f'(x)$ given $f(x) = 5x^3 - 2x + 7$.

   $f'(x) = 15x^2 - 2$. Apply power rule: $(5x^3)' = 15x^2$ and $(-2x)' = -2$.

7. Compute the derivative of $f(x) = x^2 + 4x - 5$.

   $f'(x) = 2x + 4$. Use power rule: $(x^2)' = 2x$ and $(4x)' = 4$.

8. Find $f'(x)$ for $f(x) = 4x^3 - 9x^2 + 6x - 1$.

   $f'(x) = 12x^2 - 18x + 6$. Apply power rule to each term of the polynomial.

9. What does the sign of $f'(x)$ indicate?

   The sign indicates whether the function is increasing or decreasing. Positive derivative means increasing, negative means decreasing.

10. What is the significance of a critical point?

    It is where $f'(x) = 0$ or $f'(x)$ is undefined. These points are where the derivative equals zero or doesn't exist.

11. What is the purpose of finding the critical points?

    To determine potential intervals of increase or decrease. Critical points divide the domain into intervals for monotonicity testing.

12. What is the significance of $f'(x)$ being zero at a point?

    Possible local maximum, minimum, or inflection point. Zero derivative indicates horizontal tangent line and potential extremum.

13. Find the critical points of $f(x) = -x^2 + 6x - 9$.

    $x = 3$. Set $f'(x) = -2x + 6 = 0$ to find where slope equals zero.

14. What is the first derivative test for determining intervals of increase?

    If $f'(x) > 0$, the function is increasing on that interval. This is the fundamental relationship between derivative sign and function behavior.

15. State the derivative of $f(x) = 2x^3 - 3x^2 + 5x - 4$.

    $f'(x) = 6x^2 - 6x + 5$. Apply power rule: derivative of $ax^n$ is $nax^{n-1}$.

16. How is the first derivative used to find intervals of monotonicity?

    By evaluating $f'(x)$ to determine where it is positive or negative. Sign of $f'(x)$ determines whether function increases or decreases.

17. What is the derivative of $f(x) = \frac{1}{2}x^2 - 3x$?

    $f'(x) = x - 3$. Apply power rule: $(\frac{1}{2}x^2)' = x$ and $(-3x)' = -3$.

18. Evaluate $f'(x)$ for $f(x) = \frac{1}{3}x^3 - 2x^2 + x$.

    $f'(x) = x^2 - 4x + 1$. Apply power rule to each term: $\frac{d}{dx}[\frac{1}{3}x^3] = x^2$, etc.

19. What is the derivative of a constant function?

    The derivative is 0. Constant functions have zero slope everywhere.

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