Test 1 Key, Fall 2001

Andy Long

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  Problem #1:
  • A removable discontinuity can be fixed by altering the graph at a single point to make it continuous.
  • When the function is increasing, the slope is positive; when the function is decreasing, the slope is negative.
  • When I ask for intervals, make sure that you give me intervals!
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  Problem #2:
  • When you multiply both sides of an inequality by a negative number, the inequality changes direction: if it was "greater than", it becomes "less than".
  • We were given

    

    When you substitute into the formula

    

    the symbol f disappears!

    

    Note also that it's h that tends to 0, and not x!

    Some of you had the right idea, of multiplying by the conjugate, but you must multiply the given expression by the conjugate divided by the conjugate (you're allowed to multiply any expression by 1:

    

  • Note:

    

    in general. Some of you are replacing the by =!

  
  Problem #3:

  • I noticed that when folks drew a graph of the function, they tended to connect the dots with a smooth curve. Here that is obviously not useful! The function is not defined anywhere but for these 7 values!

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  Problem #4:
  • 1. False: a rational function is defined as the ratio or quotient of two polynomials (not any two functions).
    2. False: for example, the square root function is defined on [0 ), but only differentiable on (0,), since at zero there is a vertical tangent.
    3. True.
    4. False: This is the squeeze theorem, rather than the intermediate value theorem.
    5. False: switch domain and range!

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  Problem #5:
  • For the average velocity, you need to use the formula

    

    where s is the position function, and T is the duration of the experiment (assuming that the experiment starts at 0).

    Don't simply compute the slopes of a bunch of secant lines and then average those!

  • Notice that for part 3, I didn't specify the secant line to use. Did you make a sensible choice?
  • Part 4 requires that you use the tangent line that you drew in part 1. Again, answers will vary.

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  Problem #6:
  • The major problem here was a failure to invoke any properties about compositions of continuous functions. The word ``composition'' was sorely missed!

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  Problem #7:

  • Use your calculators!
  • Does the tangent line you found look like a tangent line? Does it ``kiss'' the curve?

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  Problem #8:

  • There were really only two things that I was worried about in this problem: would you get the slope right at the origin, and would you taper off (slow gradually) at t=3.

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  Problem #9:

  • You need to justify your choice of the functions and . Show me (somehow!) that lies between them, in particular as .