Sample Final Exam MAT 131, Fall 1998
The final will be held on Thursday, December 17 at 7:00 pm. Be
sure to bring your Stony Brook ID card and your calculator. You may also
bring a single
" by 11" sheet of handwritten notes.
This sheet must not be a photocopy or computer printout. The locations of
the final exam are given in the table below.
1 
Barcus/Perez 
Physics P113 

2 
Barcus/Perez 
Physics P118 
3/4 
Sutherland/Kim 
Humanities 101 

5 
Mandell/HerreraGuzman 
SB Union 236 
6 
Mandell/HerreraGuzman 
SB Union 237 

7/8 
Bernhard 
Earth+Space 001 
9 
Bishop/Panafidin 
Lt Engineer 152 

10 
Bishop/Panafidin 
Hvy Engineer 201 
11/12 
Barcus/Weinberg 
SB Union Aud. 

13 
Schalm/Cheng 
SB Union Aud. 
14 
Schalm/McKenzie 
SB Union Aud. 

15 
Schalm/Cheng 
SB Union Aud. 
16 
Schalm/Moraru 
SB Union Aud. 

17/18 
Bernhard/Teo 
Earth+Space 001 
This sample is a collection of problems similar to the type of
questions which will be on the final. However, you are responsible
for all the material we have covered this semester just
because a topic isn't on this sample doesn't mean it won't be on the
final. Also, merely completing this sample exam is not
adequate preparation for the final. You must also do a large number
of additional problems, both those assigned in the homework and others
like them. You should ensure that you know how to do all the problems on
the midterms and the previous sample exams, as well. This sample contains
more problems than the actual final.
1. Write the equation of the linear function f with f(0)=1 and
f(3)=3. Also write the equation of the exponential function g with
g(0)=1 and g(3)=3.
2.
Compute the derivatives with respect to x for each of the following:
 (a)

x^{1/2} + x + x^{1/2}
 (b)

 (c)

 (c)

 (d)

 (e)

 (f)

 (g)

 (h)

3.
Compute each of the following antiderivatives (indefinite integrals):
 (a)

 (b)

 (c)

 (d)

 (e)

 (f)

4.
Evaluate each of the following definite integrals:
 (a)

 (b)

 (c)

 (d)

 (e)

 (f)

5. What is the average value of y=x^{3} over the interval [0,2]?
6. Find the point on the graph of the curve
that is
closest to the point (5,0). (Hint: if d is the distance from
(5,0) to a point on the curve, then it is permissible (and easier!)
to minimize d^{2}.)
7. Give the lefthand sum, righthand sum, and trapezoid
approximation for the integral
,
using n=4 rectangles. What should n be to ensure that the
righthand sum is accurate to within 0.001? (Hint: compare the
expressions for the lefthand and righthand sums for arbitrary n
what does this tell you about the exact value of the integral?)
8.
Write the equation of the line tangent to the curve
y=3x^{2} + 2x + 1at the point (1,6).
9.
Write the equation of the line tangent to the curve
y^{3}2xy+x^{3}=0at the point (1,1).
10. From physics, we know that the illumination at a point xwhich is provided by a light source at L is proportional to the
intensity of the light at L divided by the square of the distance
between x and L. Suppose that two lights L_{1} and L_{2} are
placed 20 meters apart, and that the intensity of L_{2} is 8 times
the intensity of L_{2}. Where is the point on the line between
L_{1} and L_{2} where the illumination is at a minimum?
11. Find the maximimum and minimum values of the function
f(x)=x^{3} + 3x^{2}  42x  22 on the interval
.
12. Calculate the area of the region bounded by the graphs of
y=x/2 and x=y^{2}3.
13.
Write the equation of the parabola which best approximates at
(that is, the second Taylor polynomial). Use your
polynomial to find approximations of the nonzero solutions to
. (Hint: graph the relevant functions for
to make sure your answers make sense. The
fact that
is helpful.)
14.
Use Newton's method to determine the nonzero solutions of
to within 0.000005. You may either use your answer to the
previous problem as x_{0}, or use x_{0}=2.
15. Compute the following limits. Distinguish between ,
, and ``does not exist''.
 (a)

 (b)

 (c)

 (d)

 (e)

 (f)

16.
The figure below is the graph of a function f(x). Use it to sketch
the graph of f'(x) and the graph of
.
17. For each differential equation on the left, indicate which
function on the right is a solution.
(a) y'=5y 

(1) y=x^{2} 
(b) y'=y(y1) 

(2) 
(c)
x^{2} y'' + 2x y' = 1 

(3)
y=(1+e^{x})^{1} 
(d)
yy'' = xy' 

(4) y=e^{5x} 
18.
A coffee filter has the shape of an inverted cone. Water drains from
it at a constant rate of
. When the depth is , the
water level drops at a rate of
. What is the ratio of the
height of the cone to its radius? You may find it useful to recall that
the volume of a cone of radius r and height h is
.
19.
Let be the parametric curve given by
What is the slope of at the point (4,2), when t=4?
20.
An angle is known to vary periodically with time, in such a
way that its rate of change is proportional to the product of itself
and the cosine of the time t. Write a differential equation which
expresses this relationship. Show that
is a
solution to the differential equation. If you know that
and
, what is the equation for ?
Scott Sutherland
19981207