MATH 1401 FALL 1999 UNIFORM FINAL

1. (9 points)

The graph of f(x) is given in figure 1. Using this graph, evaluate the following:

  1. lim x -2 f(x) = ____________
  2. lim x - 1 _ f(x) = ____________
  3. lim x 2 _ f(x) = ____________
  4. lim x 2+ f(x) = ____________
  5. lim x 2 f(x) = ____________
  6. f(-2 ) = ____________

    7. For parts g, h and i, state what part of the definition of continuity does not apply for f(x) at each of the following values.



    x = -2: x = 2 x = 3
    g) h) i)


2.(15 points) Evaluate the following limits. Justify your answer! (Use + infinity and - infinity where appropriate.)

3.(5 points) Let

  1. Find all values of x for which f is not continuous.
  2. State which of the discontinuities are removable.


4.(24 points) Evaluate the following derivatives. You do not have to algebraically simplify your answer.



5.(10 points) Evaluate the first derivative of the following. In each case, solve for

  1. xy + x 2 - y 2 = x + y


6.(4 points) A cookie sheet is being heated in an oven. At a particular instant in time, the length of the sheet is 20 cm and changing at a rate of 0.03 cm/sec and the width is 30 cm and changing at a rate of 0.01 cm/sec. At what rate is the area of the sheet changing at this instant.





7.(9 points) Let f(x) be defined by the graph in figure 3. List one letter (a through k) that corresponds to an x value that satisfy the following condition:

  1. When f ''(x) > 0 and f '(x) < 0, x = ____________
  2. When f ''(x) < 0 and f '(x) = 0, x = ____________
  3. When f ''(x) = 0 and f '(x) = 0, x = ____________
  4. When f ''(x) = 0 and f '(x) > 0, x = ____________
  5. When f ''(x) < 0 and f '(x) > 0, x = ____________
  6. The absolute maximum on [b,d] occurs at x = ____________
  7. The absolute minimum on [g,h] occurs at x = ____________
  8. Vertical asymptote(s) is (are) x = ____________
  9. Relative extrema occur at x = ____________


8.(5 points) A designer wants to build on open top box (4 sides and a bottom) where one side must be twice the length of the other. The amount of material to be used is 300 in 2. Find the length of the shortest side, x, if the dimensions of the box produce a maximum volume.



9.(9 points) Sketch a graph of a function with the following properties: (The function must satisfy ALL the properties.)

  1. f(0) = 2
  2. lim x -> f(x) = lim x -> - f(x) = 0
  3. There are vertical asymptotes at x = -3 and 3
  4. f '(0) = 0
  5. f "(x) 0 for any x
  6. f '(x) > 0 on (-3,0)
  7. f '(x) < 0 on (- ,-3), (0, 3) and (3,)
  8. f "(x) > 0 on (3,)
  9. f "(x) < 0 on (- ,-3) and (-3,3)

10.(12 points) Evaluate the following:

  1. )
  4. If y ' = 3x 2 - 4x + 6 and y(1) = 2, then y =


11.(8 points) Let y = 1 - x 2

  1. Find the equation of the tangent line at x = 2.


  2. Find the area bounded by the function and the x-axis.


12.(12 points) Evaluate the following integrals:

13.(3 points) Match the equations to their graphs:

  1. cos x _______
  2. cosh x _______
  3. e - x _______


A. B. C.
D. E. F.