MATH 1401 ANALYTIC GEOMETRY AND CALCULUS I

FALL 1998 UNIFORM FINAL

1. (6 points)

Let f(x) be defined by the graph above. Find the following limits. Use + and - where appropriate.



2. (3 points each) Evaluate the following limits:

3. (3 points each) Find the following derivatives. YOU DO NOT HAVE TO SIMPLIFY YOUR ANSWERS

FUNCTION: FIND:
y = x 3 - 2x
y = x 3 - 2x
y = e 5x
y = sinhx
y = tan(x 3)
y = ln(sinx)

4. (5 points each )Find the derivative of the following functions (You do not have to simplify your answers):

5. (5 points) Given y = x 3 - 3x 2 y + 2y 2 , find in terms of x and y only.

6. (5 points) Water escapes at the rate of 8 cubic feet/minute From an inverted conical container whose depth is 12 feet and whose base is a circle of radius 4 feet. At what rate is the level of the water sinking when there are 6 feet of water in the container?: ( )

7. (5 points) The strength of a rectangular beam is proportional to its width times the square of its depth. ( S = wd 2 ). Find the dimensions of the strongest beam that can be cut from a 12 inch diameter log.



8. (8 points) You are given the following information about y = f(x)

Sign graph for the derivative of y

Sign graph for the second derivative of y

Answer the following questions about the graph of y:

  1. Where is it increasing?__________________________________________
  2. Where is it concave down?______________________________________
  3. Where does it have a local maximum? _______________ minimum?_________
  4. Where does it have a point of inflection? ___________________________
  5. GRAPH y if
    Continuity: the function is continuous for all x except x = 2.

    Intercepts: (-2,0), (3,0) and (0,2).:

    6. 9. ( 5 points each) Let f(x) = - 3x 2 + 4x - 1.

    1. Find the equation of the tangent line to the graph of f(x) at the point (3,-16).
    2. Find the area bounded by f(x) and the x-axis. (Hint: you need to find the x-intercepts)

    10. (5 points each) Evaluate the following integrals:

    11. (3 points each) Find the following integrals. YOU DO NOT HAVE TO SIMPLIFY YOUR ANSWERS

    ( x 3 - 2x ) dx =
    ( e x + cosx ) dx =
    =
    tan(6x) dx =
    =
    =
    =

    MATH 1401 FALL 1998 UNIFORM FINAL: UNIVERSITY OF COLORADO AT DENVER: DECEMBER 1998