MATH 121  FINAL EXAM, Part 1  NAME ______________________

                                        December 1996

You will have up to 30 minutes to complete this part of the exam.  No calculators are permitted on this part.  Point values are shown.

1.(5@)  State the definitions.

a.  derivative of a function f at x.

b.  definite integral of f from a to b.

2.(5)  State the Fundamental Theorem of Calculus.  Be sure to define necessary quantities.

3.(5@)  Find derivatives

a.  f(x) = [(sinx)/( e^x+1)]   f ¢(x) =

b.  g(z) = arctan(Öz)  g ¢(z) =

c.  y = ln(cosx)   [(dy)/( dx)] =

d.  x²y³+y = 5   [(dy)/( dx)] =

4.(5@)  Find the exact value of the definite integrals

a.  ò₁⁴x²+Öx dx =

b.  ò₀^p/3sinx dx =

c.  ò₀²e^-2x dx =

5.(5@)  Sketch graphs of the functions

a.  y = ln(¹/_x)    b.  y = e^-2x

 MATH 121  FINAL EXAM, Part 2  NAME ______________________

                                        December 1996

Calculators are permitted on this portion of the exam.  Write any formula that you use your machine to evaluate.  A correct solution with no work shown may receive zero points.

1.(5)  Let f be a continuous function with f(x) ³ 0.  Using the definition of definite integral, explain why  ò_a^bf(x) dx can be interpreted as the area under the graph of f.

2.(5@)  Here is a short table of values for a continuous function f.

b.  Estimate f^-1(8.5)

c.  Approximate   ò₈¹⁰f(t) dt.  

d.  What is your best estimate of the error in your approximation?  State the assumptions you are using in your error estimate in part (c).

e.  Evaluate  ò₈¹⁰f ¢(t) dt.  Tell why your answer is correct.

3.(5)  You do not know an antiderivative of the function y = |x|.  Find the exact value of  and explain why your answer is correct.  ( ``The box told me so'' is not an acceptable explanation.)

4.(5@)  An experimental jet car runs along a track for 6 seconds before exploding into a giant fireball that can be seen for miles.  From its starting time, t = 0 seconds, its speed in feet per second, is given by the formula V(t) = 0.08te^t.  Make sure your answers contain the correct units.

a.  How fast was the car going at t = 6 seconds?

b.  What was the average acceleration over the first 6 seconds?

c.  How far did the car travel during the first 6 seconds?

5.(10)  A landowner wants to enclose part of a field.  Since the field is along a busy road, it is decided that a brick wall there would be best.  The brick wall costs $6/foot.  The other three sides of the rectangular region can be enclosed with a cheaper fence costing $4/foot.  If the owner has $500 in the budget for this project, find the dimensions to enclose the largest area.  Tell how you know that the solution you have found is a maximum.  Show your work.

6.(5)  Find the intervals on which y = e^-x2 is concave down.  An exact answer must be given.

7.(5@)  Below is a graph of f ¢(x) on the interval 0 £ x £ 4. graph

 a.  State the interval(s) on which f is increasing.

b.  If there are local maxima or minima of f(x), specify the location.  Explain your reasoning.

c.  Where in the interval 0 £ x £ 4 does f achieve its global maximum?  Explain.

d.  Suppose that you are told that f(0) = 1, write down an exact expression (not a number) for f(2).  (No computations are required here.)

e.  Still assuming that f(0) = 1.  Estimate f(2).

f.  The graph shows that f ¢(x) has a local minimum at x = 1.  Explain why f(x) has a point of inflection at x = 1.

8.(5@)   Let f(x) = x⁵-x-1 .

a.  Find the equation of the line tangent to f(x) where x = 1.

b.  Approximate the value of y corresponding to x = 1.1 by using the linear approximation to the curve.

c.  Use your calculator to find the exact value of f(1.1).  Explain why is your estimate in (b) is so bad.

9.(5@)  The curve shown is the derivative f ¢(x) for some function.  Sketch the original function and the second derivative.  Assume that f(0) = -1. graph

10.(2@)  True/False Questions. If your answer is ``False,'' explain why the statement is false.

_____  a.  If f(x) is continuous at x = a, then f ¢(a) exists.

_____  b.  If f(x) is continuous on [a,b], then  exists.

_____  c.  If f ¢(x) > 0 on a £ x £ b, then a ``right-hand'' sum estimate of the definite integral of f(x) from a to b will be positive.

_____  d.  If [(d)/( dz)]B(z) = C(z) and [(d)/( dz)]A(z) = G(z), then [(d)/( dx)]B(A( x³)) =  C(A(x³))·G(x³)

_____  e.  The instantaneous acceleration at t = a may be interpreted as the slope of the tangent line to the graph of the velocity function at t = a.

_____  f.  If f(x) > 0 and if f ¢(3) = 0, then f(3) is a local maximum.

_____  g.  Every increasing function has an inverse.

_____  h.  The derivative of a product of two functions is the product of the derivatives.