Lab10: Summary

Introduction

During this semester we have used Mathematica to investigate a variety of problems.  In this lab we will see how much you remember. ☺

Exercises

Use Mathematica to solve the following problems.

Graph f(x)=1/(x^(1/3)) on the interval [1, 15].  Find the area under the curve, volume of rotation about the x-axis on the interval [1, 15], and surface area of that volume.

Use Simpson’s Method to approximate the area found in part (1) accurate to five decimal digits.  This should include using the error formula to calculate the number of sub-intervals needed as well as the summation.

Find the first five nonzero terms in the Taylor series for the function given in part (1) centered at a=8.  Based on the Taylor inequality, how accurate is this polynomial in approximating the function on the interval [1, 15]?  Use this polynomial to approximate (7^(-1/3)) and check the approximation against the value that Mathematica or your calculator generates.

Graph the polar curve r=1/(θ^(1/3)) on the interval [0, 2π].  Find the arc-length of this curve.

Use limits to determine which function goes to zero faster as x→∞. f(x)=1/(x^(1/3)) or g(x)=ln(x/(x+1)).