MATH 221 Solutions to Final Exam Review Exercises

Part I

1. a)

b)

c)

d)

e)

f)

g)

2. a)

b)

c) Ignore problem 2c.

3. a)

b) =32/81

8/3

e)

Part II

1. a) b) x = -2.

2. a) Point = (-1,3), Slope = 3(-1)² – 4 = -1 . Equation (y-3) = (-1)(x+1)

b) x = 1, -1. Points (-1,3) and (1,-3)

c) Slope of perpendicular = 1. 3x² – 4 = 1. .

Points

d) (2,3) is not on the graph of y = x². Let (x,x²) be a point on the graph whose tangent line passes through (2,3). Then the slope of the tangent is given by 2x and by . Setting these equal yields x = 3 and x = 1. The respective slopes are 6 and 2. The equations are (y-3) = 6(x-2) and (y-3) = 2(x-2).

3. a) Equation of line y = 2-x. Curves cross at x = -2 and at x = 1.

Area =

b) Area =

4. 486 square inches.

5. a) . C = 3.

b) , ,

C = 1, D = 3.5.

6. a)

Critical points -3, 4, 1 Local Max at -3, Local Min at 1

b) First volume =

Second Volume =

7. Removable discontinuity at x = -2, Removable discontinuity at x = -1, Infinite or Essential discontinuity at x = 2.

8. a) Area = 1/2

b) i)

ii)

9. a) Point (1,1), Slope = -1/2. Equation

b) , x = 1, -1. Points (1,4), (-1,-4). Slope = 6

Equations (y-4) = 6(x-1) and (y+4) = 6(x+1)

c) , x = 1, -3

Points (1,3), (-3, 3), Slopes 4, -4

Equations (y-3) = 4(x-1) and (y-3) = -4(x+3)