Groupwork

1) Take the derivatives of each of the following functions:

2) Use the fundamental Theorem of Calculus to evaluate the integrals given :

Some key words you may wish to remember are substitution(including trig substitutions), integration by parts, and partial fractions.

3) Evaluate each of the following integrals.
; n > 0

4) The following MAPLE program will differentiate f(x). Copy these commands into a new MAPLE worksheet.Note the restart command is used to erase any recently defined values. It is notnecessary to use the restart command after opening the MAPLE program, but it is a nice habit to get into when writing MAPLE programs. Also note that you must type in allmultiplication operators, the arrow is created with the minus symbol and the greater than symbol, and pressing shift and return simultaneously will allow you to group lines as with the last four lines.

> restart:
f:= x->5*x^3+1/sqrt(x);
Diff(f(x),x);
value(%);

5) Use MAPLE to check the rest of your derivatives from part 1. Note that the exponential function is abbreviated as exp(x). Note that all of the cut and paste editing that you have seen in word processors work in the MAPLE worksheets.

6) The following MAPLE program will evaluate integrals. Copy these commands into a MAPLE worksheet and see what results.

> p:= x->x*sin(x^2);
int(p(x),x);
int(p(x),x=1..4);

7) Use MAPLE to check your results from questions 2 and 3.  Note that you do not have to type ";n>0" for the first one (MAPLE will assume this).  You may need to ask MAPLE to simplify its output. This is done with the command simplify. Remember that % represents the last MAPLE output line, so if you want to simplify the last maple output type simplify (%).

8) In Calc I you learned several ways to estimate a definite integral using summations which estimated the area under a curve. Four specific types of summations were discussed: Ln, Rn, Mn, and Tn. The student package in MAPLE will produce each of these. The Type each of the following one at a time and see what happens. The student packagein MAPLEis also used when finding the Simpson approximation, Sn, for an integral (recall Simpson's rule from calc 2)Note thatevalfevaluates an answer using floating point decimal (thus the f on the end of eval), and %represents the last MAPLE output.

> with (student):
g:=x->x*cos(x);

> leftbox(g(x),x=0..2,4);

> rightbox(g(x),x=0..2,4);

> middlebox(g(x),x=0..2,4);

> value(leftsum(g(x),x=0..2,4));
L[4]:=evalf(%);
value(rightsum(g(x),x=0..2,4));
R[4]:=evalf(%);
value(middlesum(g(x),x=0..2,4));
M[4]:=evalf(%);

> T[4]:=evalf(trapezoid(g(x),x=0..2,4));

> S[4]:=evalf(simpson(g(x),x=0..2, 4));

9) Use the graph outputs in number 8 to determine which of the three approximation Ln, Rn, or Mn was closer to the actual definite integral. Explain your answer.

10) Use MAPLE to determine the error of each of the approximations found in number 8: Ln, Rn, Mn,Tn,Sn. You should not be retyping Maple outputs into new input lines. This could cause serious round-off error. Instead try either storing output values or stringing Maple commands together.