Review for quiz 3

We’ll have our third quiz this Friday, October 31st. There will probably be four questions on the quiz that will be a lot like the problems you see here.

If you have questions about any of the problems on this sheet, you can use the Reply on Discourse button down below.

Problems

1. Let \(f(x) = 2 x^2 - 3 x - 1\) restricted to the interval \([0,3]\). Use the derivative of \(f\) to find the absolute maximum and minimum values of \(f\) and their locations.

2. Figure 1 shows the graph of \(f(x) = x e^{-3x}\).

   1. Use the first derivative of \(f\) to find the exact location of the maximum.
   2. Use the second derivative of \(f\) to find the exact location of the inflection point.

3. I need to fence off an enclosure divided into two portions, as shown in Figure 2. My requirements are:

   • The total area enclosed must be 100 square meters,
   • One side will be against an existing wall, and
   • The cost of the interior portion of the fence shown by the dashed line is only \(1/3\) the cost of the rest of the fence.

   What are the dimensions of the least expensive such enclosure?

4. Figure 3 shows the graph of \(f(x)=e^{-x}\) together with a rectangle inscribed under the graph of \(f\) with one vertex on the graph, one vertex at the origin, and two vertices on the coordinate axes. Find the maximum possible volume of that rectangle.

5. Figure 4 plots the data \[\{(-4,-1),(-1,-2),(1,2), (4,1)\},\] together with its regression line of the form \(f(x)=ax\).

   1. Write down the formula for \(E(a)\) representing the squared error of the regression as a function of the parameter \(a\).
   2. Find the value of \(a\) that minimizes that squared error.

6. Let \(f(x) = x^3-1\).

   1. Write down the corresponding Newton’s method iteration function \(N(x)\).
   2. Take two Newton steps from the initial seed \(x_0=2\).
   3. Suppose that we generate a sequence via iterated applications of \(N\). What is the limit of the resulting sequence?

Figures

Figure 1: The graph of \(f(x)=xe^{-3x}\)

Figure 2: A divided enclosure

Figure 3: A rectangle inscribed under the graph of \(y=e^{-x}\)

Figure 4: Some data and its regression line

Questions

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