Name: Math 241: Calculus & Analytic Geometry A
Exam 3
November 27 2002

Were it not for Occam's Razor, which always demands simplicity, I'd be tempted to believe that human beings are more influenced by distant causes than immediate ones. This would be especially true of overeducated people, who are capable of thinking past the immediate and becoming obsessed by the remote.
- from ``Straight Man'' by Richard Russo

Instructions: Show all work to receive full or partial credit. All University rules and guidelines for student conduct are applicable.

1. [10 pts] Find
   $$\lim_{x \rightarrow \infty} \frac{(2x+1)^2}{3x^2-1}$$
   if it exists.

2. [10 pts] Sketch the curve
   $$y = \frac{(x+4)(x-3)^2}{x^4 (x-1)}$$
   using asymptotes and intercepts, but no derivatives. Label all asymptotes and intercepts.

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

3. [15 pts] Find the area under the curve
   $$f(x) = 4 x^3 + 1$$
   between $x=0$ to $x=1$.

4. [15 pts] Find the area under the curve
   $$f(x) = \csc^2 x$$
   between $x=\frac{\pi}{4}$ and $x=\frac{3 \pi}{4}$.

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

5. [15 pts] Find $\frac{df}{dx}$ if
   $$f(x) = \int_2^{x^2} (\cos s + 1) ds.$$

6. [15 pts] Approximate the root of
   $$f(x) = x^3 + x +1$$
   using two steps of Newton's Method with an initial guess of $x_0 = 0$.

7. [20 pts] A cardboard box is constructed from a sheet of cardboard by folding along the dotted lines and taping along the matching edges (1-7). Maximize the volume enclosed by the box if one has 120 inches of tape, and that the tape must cover each of the matching edges.
   
   .