School of Mathematics and Statistics
Carleton University
Math. 69.104
ASSIGNMENT 2
SOLUTIONS
Due: November 16, 1999 (or Nov. 15, 1999 Mr. Dubé),

• 1. [10] Let y=f(x) where f is given by

  

• [1] Find all intercepts,
• [3] Intervals where f is increasing or decreasing,
• [3] Intervals where f is concave up or down,
• [1] All points of inflection,
• [1] All critical points,
• [1] Sketch the graph.

Solution
a) The intercepts are found as follows: y =0 when x=-1. When x=0 we have y=1.

b) , so f is increasing when x < 1 and decreasing when x > 1.

c) . So, the graph of f is concave up when , that is, when or when . The graph is concave down in the interval

d) The points of inflection occur when there is a change in concavity. In this case they are found to be at and at .

e) There is only one critical point (a local maximum) at x=1 (since ). Use your plotter to sketch the graph. You'll find:

[5] 2a)

Sketch the graph of over the interval [-1,2].

[5] 2b)

Evaluate and interpret your result as an area.

The integral is the area under the curve above between the lines x=-1, x=2 and above the x-axis.

Solution Use the Box method to show that

Now the integral breaks up into 3 pieces as follows:

[5] 3a)

Evaluate using any method: . Solution Use the Substitution Rule: Let so that . The integral now looks like,

[5] 3b)

Evaluate

Solution Use the Substitution Rule: Let so that . The integral now looks like,

or

[5] 3c)

Evaluate

Solution Use the Substitution Rule first: Let so that . The integral now looks like,

Bring this to a more recognizable form where the Table Method may be used..., that is, let Then, and

[5] 3d)

Evaluate

Solution The denominator factors as where the quadratic is irreducible. Thus, since the degree of the numerator is less than the degree of the denominator, we get

where A = 1 by the cover-up method. To get B, C we simply multiply throughout by the denominator and find

since A=1. Comparing coefficients we see that B=1 and C=-1. It follows that

Integrating we get

Total: /40

• About this document ...