Test 1 Solution Key

First of all: if you're uncertain about what I want on a given problem, come up and ask! Many students took advantage of this opportunity on test 1 - but others who should have didn't!

• Problem 1:

  

  (the derivative does, in fact, exist at x=1, and is zero).

• Problem 2

  

  The trickiest part was the domain for g(f(x)), since the places where it is undefined are infinite (the places where the sine function is -1).

• Problem 3

  

• Problem 4

  The odd extension of f is a little funky in the following, but otherwise well done. Nice discussion of continuity and differentiability.

  
  The last line is "between (but not at) these points."

• Problem 5

  First of all, don't forget your units: some of you might have detected your mistakes, had you used them.

  

• Problem 6

  There are a variety of limit laws, and the order is not too important. Some folks used continuity, which was a nice idea, especially to handle the absolute value.

  

• Problem 7

  I cut off some of the top of this problem - the student worked it up the sideline, but didn't have much success (because s/he needed to consider the conjugate!). But the student figured out the slope via other means, and pushed ahead with the problem - good job! Don't let a little thing like a calculation stop you!

  I like the way that s/he has summarized the relevant information for the graph of the tangent line.

  

  Note: some students didn't even graph the function. For that, you could have used your calculator, and gotten a few easy points.