Key to Practice Midterm I

1.   A  true it is continuous since the three match up at 2:  2² + 2 = 6 = 6(2) - 6

B.   False.  The left side approaches -1 and the right side approaches 1

C.  False.  f(x) = x/x has a limit of 1 at x = 0, but the function is undefined at x = 1 so is not continuous.

2.  A.  [(x + h)²/2 - 3(x + h) - (x²/2 - 3x)]/h = [x²/2 + xh + h²/2 - 3x - 3h - x²/2 + 3x]/h

= [xh + h²/2 - 3h]/h = x + h/2 - 3.

B.  3(x²/2 - 3x)+ 6 = 3/2 x² - 9x + 6

C.  [x/3 - 2]²/2 - 3(x/3 - 2) = x²/18 - 2/3 x + 2 - x + 6 = x²/18 - 5/3 x + 8

3.  (There is no key for "your own words")

4.  A.  i.  -.8, ii.  DNE or infinity, iii.  1.8,  iv.  0

B.  x = -3,  -1, 3

C. -3, -1, 0, 3

5.  A.  -1.5

B.  DNE

C.  x² - 2

6.  A.  [0,100)

B. Yes it is continuous on the domain since 100 is not in the domain.

C.  30(50)/(100 - 50) = 30 billion dollars.

D.  No, since there are no fractional people.

7.  9x² + 5x^-2 + x^-1/2

8.  lim h -> 0 [sqrt(x + h - 3) - sqrt(x - 3)]/h

= lim h -> 0 [[sqrt(x + h - 3) - sqrt(x - 3)]/h][sqrt(x + h - 3)  + sqrt(x - 3)]/[sqrt(x + h - 3) + sqrt(x - 3)]

= lim h -> 0 [x + h - 3 - (x - 3)]/h[sqrt(x + h - 3) + sqrt(x - 3)]

= lim h -> 0 [h/h[sqrt(x + h - 3) + sqrt(x - 3)]

= lim h -> 0 [1/[sqrt(x + h - 3) + sqrt(x - 3)]

= 1/(sqrt(x - 3) + sqrt(x - 3)

= 1/[2sqrt(x - 3)]