MATH 172,Calculus II Name ___________________________________
Spring 2000
Final Exam,Take Home Part Date ____________________________________
At the end of ayear’ssequence of Calculus there are some skills,procedures and ideas that you should
be able to perform and/or understand.The questions below combine some material from Calculus Iand
Calculus II.
1.Some Derivative Rules from Calculus I
Find the derivative for each of the following functions.Show each step used in the process of finding
the derivative,stating the rule that you’re using,e.g., product rule,quotient rule,chain rule,etc.Then
use Scientific Notebook to find the derivative and compare your results.
a.fÝxÞ=e?3xcosÝ2xÞ.
b.gÝxÞ=3x?5
2x2+4x+1
c.hÝxÞ=x2+lnÝ1+x4Þ
2.Using Derivatives to find the Maximum (from Calculus I)
Consider the function fÝxÞ=xÝ1+xÞe?bx,for x³0,with bapositive constant.
a.Use the first derivative function to find the xvalue(s)that give(s)apossible maximum for fÝxÞ.
These xvalues will be in terms of the constant b.(Recall that amaximum or minimum might
occur where the slope of the function is zero.)
b.Then use the second derivative function to determine which xvalue will in fact give amaximum
(and not aminimum).
3.Some Antiderivative Techniques (from Calculus II)
Find the family of antiderivatives for each of the following functions using the specified technique.
Show each step used in the process of finding the antiderivative.Then use Scientific Notebook to
find the antiderivative and compare your results.
a.fÝxÞ=xcosÝ2xÞ,using integration by parts.
b.gÝxÞ=3x
2x2+1,using substitution.
c.hÝxÞ=2x?3
x2+3x?4,using atable of indefinite integrals.
4.Computing Definite Integrals (from Calculus II)
Compute the following definite integrals using the specified technique.Show the steps used in the
process of computing these definite integrals.Then use Scientific Notebook to compute the definite
integral and compare your results.
a.X0
24xe?x2dx,using the Fundamental Theorem of Calculus.
b.X0
24e?x3dx,using Simpson’srule.(Be sure to specify the number of subintervals used and why
you chose that number.)
c.X0
11?e?x2
xdx,using aTaylor series approximation for the integrand.(Be sure to specify the number
of nonzero terms used in the series and why you chose that number.)
5.Applications of integration (from Calculus II)
a.Computing aVolume
Let Rrepresent the region in the plane bounded by the function y=1?x
2
2and the x-axis.
i.Sketch the region R.
ii.Find the volume of the solid obtained by rotating the region Rabout the horizontal line y=3.
b.Computing Work
Suppose water is stored in atank that’sthe shape of apyramid with asquare base,sft.on each
side,and a height of hft.Compute the work done to pump afull tank of water out of the top.
