Name: Math 241: Calculus & Analytic Geometry A
Exam 2
November 1 2002

There is always one moment in childhood when the door opens and lets the future in.
- Graham Greene, The Power and the Glory.

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

1. [15 pts] Find an equation of the tangent line to the curve
   $$y = \tan\left(\frac{\pi x^2}{4}\right)$$
   at the point $(1,1)$. Simplify your answer as much as possible.

2. [15 pts] Find the equation to the tangent line to the curve
   $$3 x^2 y^2 = (y+1)^2 (4-y^2)$$
   at $(2,1)$. Simplify your answer as much as possible.

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

3. [20 pts] A vertical post (located at the equator during Spring Equinox) stands 10 meters out of the ground so that the sun passes directly overhead at noon local time. How fast is the tip of the post's shadow moving at 3:00 pm? Simplify your answer as much as possible.

4. [10 pts] Find $f^{(50)}(x)$ where
   $$f(x) = (x+3)^{25} (x-2)^{25}.$$
   Justify your answer.

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

5. [10 pts] Find a general expression for $f^{(n)}(x)$ where
   $$f(x) = \frac{1}{(1-x)^2}.$$

6. [10 pts] Classify all the critical points of
   $$y = \sin (x) - \frac{1}{2} \sin^2 (x) + 3, \ \ \ \ -2 \pi \leq x \leq 2 \pi.$$

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

7. [5 pts] Find the inflection point(s) for the function and interval in problem 6.

8. [15 pts] Find the absolute minimum and maximum values of
   $$y = 2 x^3 - 3 x^2 -12 x +8$$
   on the interval $-2 \leq x \leq 2$.