Math Insight

Quiz 6

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Total points: 8
  1. Let $f(x) =\left(x^{2} + 6 x\right) e^{\frac{x}{4}} $.

    1. Calculate $f'(x)$.

      $f'(x) = $

    2. Find the critical points of $f$.

      Critical points =
      (If there are multiple answers, separate by commas; do not round.)

    3. Find the local extrema of $f$. (Extrema just means either a maximum or a minimum.) For each extremum calculate three things: the location of the extremum (i.e., value of $x$), the value of the extremum (i.e., value of $f(x)$), and whether it is a local maximum or a local minimum.

      Locations of the local extrema:
      (If multiple answers, enter in increasing order, separated by commas; if no answers, enter none.)

      Values of the local extrema:
      (If multiple answers, enter in the same order as above, separated by commas; if no answers, enter none. If rounding, keep at least four digits.)

      For each extremum, enter either maximum or minimum to indicate if the extremum is a local maximum or local minimum, respectively. (If multiple answers, enter in the same order as above, separated by commas; if no answers, enter none.)

      Characterization of extrema=

      Need help?

    4. Find the global maximum and global minimum of the function $f(x)$ on the interval $-13 \le x \le -1$. Also indicate the location (the value of $x$) of the global maximum and global minimum.

      Global maximum:
      (If rounding, keep at least four digits.)

      Location of global maximum: $x =$

      Global minimum:
      (If rounding, keep at least four digits.)

      Location of global minimum: $x= $

  2. After a nearby chemical spill, the pollution level of a bay is monitored. The pollution level $t$ weeks after the spill is given $p(t)=5 t e^{- 4 t}$.

    For how long after the spill does the pollution level in the bay continue to rise? What is the maximum pollution level?

    The pollution level continues to rise for
    weeks.

    The maximum pollution level is
    . (If rounding, keep at least four digits.)

  3. Let $f(x) =\left(x - 4\right) \left(x + 1\right) \left(3 x - 4\right) = 3 x^{3} - 13 x^{2} + 16$.

    1. Calculate $f'(x)$.

      $f'(x) = $

    2. Find the critical points of $f$.

      Critical points =
      (If there are multiple answers, separate by commas; do not round.)

    3. The critical points divide the number line into three intervals. What are these three intervals? Enter them from left to right. Enter oo for $\infty$.

      Interval 1 =

      Interval 2 =

      Interval 3 =

    4. On each of these intervals, $f'(x)$ does not change sign. Pick an auxiliary point in each interval, test the sign of $f'(x)$ at that point, and conclude whether $f$ is increasing or decreasing on that interval. Enter either increasing or decreasing in the answer blanks.

      On interval 1, $f$ is

      On interval 2, $f$ is

      On interval 3, $f$ is

    5. What are the roots of $f$ itself, i.e., at what points is $f(x)=0$?

      Roots =
      (Separate answers by commas, do not round.)

    6. Using this information, sketch the graph of $f(x)$.

      We can't grade this automatically, so you'll just have to check it yourself, either by checking the solution or by graphing the function using a computer or calculator.)

  4. Let $f(z) = -8z(z-k) +10$, where $k$ is a constant parameter.

    1. Find the critical points of $f$.

      Critical points: $z=$
      (If multiple answers, separate by commas; if no answers, enter none.)

    2. Find the local extrema of $f$. (Extrema just means either a maximum or a minimum.) For each extremum calculate three things: the location of the extremum (i.e., value of $z$), the value of the extremum (i.e., value of $f(z)$), and whether it is a local maximum or a local minimum.

      Locations of the local extrema: $z=$
      (If multiple answers, separate by commas; if no answers, enter none.)

      Values of the local extrema:
      (If multiple answers, enter in the same order as above, separated by commas; if no answers, enter none.)

      For each extremum, enter either maximum or minimum to indicate if the extremum is a local maximum or local minimum, respectively. (If multiple answers, enter in the same order as above, separated by commas; if no answers, enter none.)

      Characterization of extrema=

      Need help?

    3. Assume $k >0$. Find the global maximum and minimum of $f$ on the interval $0 \le z \le 3k$. Also indicate the location (the value of $z$) of the global maximum and global minimum.

      Global maximum:

      Location of global maximum: $z =$

      Global minimum:

      Location of global minimum: $z= $

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Math 1241, Fall 2020