Test 1: Sample problem solutions

  1. Sketch the derivative of the function given by the following graph:

      figure55

    This is one where you have to sit down and do the graphing. As I said to several students in class, the important things to get are the points at which the slope is 0 (horizontal tangents), the points at which the slope is undefined, and the signs of the slopes between these points. Some relative magnitude of slopes would be nice, too, but that's in the bonus....

  2. For each of the following functions:
    1. find the domain.
    2. find the regions for which it is continuous, and differentiable.
    3. indicate what type of function it is.
    4. does it possess any symmetry?

  3. Sketch the following function, and draw its derivative from the graph:

    displaymath348

    Using the definition of the derivative, calculate its slope at x=1. Find the tangent line at that point, and draw it on your graph of f.

    I could simply compute tex2html_wrap_inline409 , but here's tex2html_wrap_inline411 :

    displaymath349

    displaymath350

    displaymath351

    Hence

    displaymath352

    The tangent line at tex2html_wrap_inline413 is thus

    displaymath353

      figure105

  4. Using the definition of the derivative function, find the derivative of the function

    displaymath354

    displaymath355

    displaymath356

    displaymath357

    displaymath358

  5. Evaluate the limits, and justify each step by indicating the appropriate limit laws:
  6. GIven the following table of a growing population:

    displaymath369

    Graph the data, and estimate the rate of change in size at day 2 in two different ways:

    How might you estimate the slopes at days 1 or 3 using secant lines?

    To begin with, we use the two points (1,3) and (3,9) to get a slope of the secant line of tex2html_wrap_inline417 for the point at x=2. We then draw in the tangent line:

      figure216

    I could draw a function which fits the data by hand also, and then estimate the slope using the standard ``rise over run'' formula.

    As for estimating the slopes at the endpoints, we simply use the points themselves and their nearest neighbors: for example, at x=1, we'd estimate the slope using the two points (1,3) and (2,4), for a slope of 1.





LONG ANDREW E
Wed Jan 31 23:39:34 EST 2001