Lab 4: Student Assignment

Week 4, February 1-7

MAT 229, Spring 2021

Exercises to submit

These must be done by hand on paper (or typed into Mathematica). Use Mathematica to check your work. Submit as pdf files in Canvas.

Exercise 1

Let f(x)=(x^(3x)). Its domain is (0,∞).

End behavior:

Is (lim_(x->(SuperPlus[0])))f(x) a determinate limit or indeterminate limit? If it is determinate, give its value. If it is indeterminate, write it in a form on which you can use L’Hospital’s rule and evaluate it.

Is (lim_(x->∞))f(x) a determinate limit or indeterminate limit? If it is determinate, give its value. If it is indeterminate, write it in a form on which you can use L’Hospital’s rule and evaluate it.

Find any critical numbers. Determine if each produces a maximum point, a minimum point, or neither.

Sketch the graph that shows the end behavior and the critical points.

Exercise 2

Consider the indefinite integral ∫((x+3)/(Sqrt[(x^2)+6x +7]))d x.

Find an appropriate substitution.

Using your substitution, evaluate the integral.

Exercise 3

Consider the indefinite integral ∫(((((x^2)-10))^2)/x)d x.

Rewrite it in a form that you can use the power/log rule.

Using your rewrite, evaluate the integral.

Exercise 4

Let g(x)=cos(x)(e^(2sin(x))). This is a periodic function, and we want to find the area of one of the bounded regions above the x-axis and y=g(x).

Find the two x-intercepts closes to the origin.

Set up the integral that gives the area and find an appropriate substitution.

Evaluate the integral to determine the area.

Plot this function and make sure your answer seems reasonable by estimating one of the bounded regions with a rectangle.

Exercise 5

Consider the region in the first quadrant bounded by y=(3/(Sqrt[4+(x^2)])) and y=1 that has finite area.

Provide a rough sketch from Mathematica.

Find any intersection points for these two curves.

The solid of revolution obtained by rotating the bounded region between these two curves around the x-axis has cross sections that are washers. Set up the integral that gives the volume.

Evaluate that integral to determine the solid’s volume.

From your sketch in part (a) make sure your answer seems reasonable by computing the volume of a cylinder with a radius that is roughly the average of the y-value for the outer radius minus the volume of the cylinder with inner radius 1.  (The volume of a circular cylinder is (base area)×height.)