Test 1

1. e
2.a) 1
   b) .6
   c) .8 = rate of increase of shopping centers in 1992
3. y = 6x - 5
4. 1/2
5. 1/4
6. You have to show that f(x) < x^3 sin(2/x) < h(x) and both the limit of both f(x) and g(x) is 0.
7. a) -9
    b) -6.3
    c) -6.03
    d) -6
8. a) parabola to the left of 2, line to the right of 2.
    b) limit as x approaches 2- = 0,  limit as x approaches 2+ = 0
    c) Yes, the limit exists because........
         No, the function is not continuous at x = 2 because.....
9. 3/4
10.a) x = 3
     b) y = 3
11. -4

Test 2

1. Graph of f '(x) will be the graph of the slope of f(x).  f '(x) = 0 when the tangent line of f(x) is horizontal.
2. x = -2, 1
3. dy/dx = (sin x sin y)/(cos x cos y)
4. f '(x) = -2/(2x - 3)^2
    f ''(x) = 8/(2x - 3)^3
    f '''(x) = -48/(2x - 3)^4
5. dy/dx = (-sin x ln x + (cos x)/x)*x^(cos x)
6. f ''(Pi/3) = -sqrt(3) - Pi/6   or   -2.255649583
7.  y = 5x + 4
8.a)  v = 6t^2 - 18t
        a = 12t - 18
   b) See instructor for graph
   c) v = 0 at t = 3
       a > 0 on (3/2, 6]
       a < 0 on [0, 3/2)
9.  (+/-2.99, +/-0.083)
10.a) dy/dx = 1/sqrt(1 - x^2)
     b) dy/dx = -(csc x)^2
     c) dy/dx = (cos x)(e^x) - (x)(sin x)(e^x) + (x)(cos x)(e^x)
     d) dy/dx = 2/x + 1
     e) dy/dx = (-sin(ln(2x)))/x


Test 3

1. dz/dt = 53.15
2.a) L(x) = 27x - 54
   b) approximately 27.27
3. x = e^(-1)
4. Abs.Min. = 3, Abs.Max. = 5
5. 0
6. 1
7.a) Yes
   b) Yes
   c) show this
   d) c = 1/3
   e) Rolle's Thm or Mean Value Thm.
8.a) Inc. (-infinity, -2) or (2, infinity)
       Dec. (-2, 2)
   b) local max at (-2, 21)
       local min at (2, -11)
    c) CD on (-infinity, 0)
        CU on (0, infinity)
    d) inf.pt. at (0, 5)
9.  x = 1/2
10.a) All reals
     b) y-int. = 2, x-int.'s = -2.4,-6.7,.12
     c) neither
     d) VA none, HA none
     e) Inc.(-5,-1)
         Dec. (-infinity, -5) or (-1, infinity)
     f) local min at (-5, -23)
        local max at (-1, 9)
     g) CU on (-infinity, -3)
         CD on (-3, infinity)
     h)  Sketch the graph.


Test 4

1.a) 3x^2/2 - 4x + C
   b) 4e^x + x^2/2 + C
2.  f(x) = x^3/3 + 5x^2/2 - 4x + 2
3.a) Downward parabola with vertex at (0,9)
   b) Area is approximately LHS = 25
4.a)  -1∫5 tan x dx
  
   b)        ∞

lim          ∑ (xi - 4e^xi )∆x  on [2, 4]

n-->∞    i = 1

5. -8
6.a) -8
   b) Yes, because the integral is equal to the limit of the Riemann Sum.
7.a) y' = x^3 - 4x
   b) y' = 16x^3 - 16x
8.  -26/3
9.a) & b) Sketch the graph and shade the region.
   c) 12
10.a) -3cos x + C
     b) ln |x| + C
     c) 5 ln |x| + C
     d) (2/3)3x^(2/3) - (2/5)x^(5/2) + C