Mathematical Scratchpad

Math 151 Practice

  1. The average annual rate $r$ (in percent form) for commercial bank credit cards from $1990$ through $1995$ can be modeled by MATHwhere $t=0$ represents $1990.$

    1. Find the derivative function of this interest rate function. MATH

    2. Explain the followings!

      1. How do we find the year which interest rate decreases the most? Estimate such year. [hint: draw $r^{\prime }(t)$ and find the minimum of $r^{\prime }(t),$it is about $t=3.151981544496$].

      2. How do we find the year which interest rate increases the most? Estimate such year. [hint: it will be increases the most near $t=5].$

  2. Suppose the rate of change for the number $N$ of bacteria in a culture after $t$ days is modeled by the following graph: [Note the following graph represents $y=N$ $^{\prime }(t)].$
    graphics/practice__16.png

    1. Does the function $N$ have max or min during $t=0$ and $t=10.$[hint: Since $N^{\prime }$(the derivative of $N)$ is positive over $[0,10],$ $N$ is increasing over $[0,10],$and the minimum of $N$ happens at $t=0,$and maximum happens at $t=10.]$

    2. When is the rate of change increasing the most? [It will be at the point where $N^{\prime }$reaches its maximum]

    3. Will the number of bacteria be stabilized eventually? [Yes, because the $N^{\prime }(t)$ tends to $0$ eventually].

  3. An open-top box is to be made from a piece of cardboard of $20$ inches long and $25$ inches wide by cutting out squares of equal size at each corner and bending up the flaps. What size of squares should be removed in order to produce the maximum volume?

    1. Describe how to set up the function.

    2. Describe the steps of how you would solve this problem if ClassPad is used.

  4. Find the dimension of the rectangle which achieves its maximum area under the curve $y=\sqrt{25-x^{2}}.$

  5. Suppose you are given the graph of the marginal cost (derivative for cost) for a company. [ The $x-axis$ represents the number of units in thousands, and the $y-axis$ represents the cost in millions]. And we assume the $x-intercepts$ are at $x=1,$ $3$ and $5.$
    graphics/practice__40.png

    1. When will the company achieve its largest and lowest cost respectively? $[hint:$ The cost reaches maximum at $x=1,5$ and minimum at $x=3].$

    2. When will the rate of cost increase the most during $x=2$ and $x=5?$ [It will be the local max of the graph $C^{\prime }$above]

    3. When will the rate of cost decrease the most during $x=0$ and $x=3?$ [It will be at the local min of the graph $C^{\prime }$above]

    4. Describe how you will find the inflection points for the cost function.[To find the max or min of $C^{\prime },$we need to find zeros for MATH

    5. Explain in your own words how inflection points are related to the cost function.

  6. Suppose $P(x)$ is a profit function (where $x$ denotes the number of units produced). Explain the followings:

    1. If MATH and $x_{1}<x_{2}$ then what should one do when they are producing $x_{2}$ units? Should one increase or decrease the number of units produced? [One may reduce the production level from $x_{2},$because it may just passed the maximum]

    2. If MATH and $x_{1}<x_{2}$ then what should one do when they are producing $x_{2}$ units? Should one increase or decrease the number of units produced? [From the given information, $P$ is decreasing at $x_{2},$ nne could increase the production level from $x_{2}$ and hope the profit will be increase again]

  7. If MATH represent the cost function.

    1. Find the marginal cost.

    2. Use the marginal cost to estimate the cost for the $34th$ unit.[Use $C^{\prime }(33)].$

    3. Find the number of units produced which will either maximize or minimize the cost.

  8. If MATH represent the profit function.

    1. Find the marginal profit.

    2. What is profit (or loss) initially?

    3. Use the marginal profit to estimate the profit for the $32nd$ unit. [Use $P^{\prime }(31)]$

    4. Find the number of units produced which will maximize the profit.[Find the vertex of this parabola].

    5. What is the maximum profit?

  9. Practice the homework problems on sections 3.4 and 3.5 (derivative for trig functions and chain rule).