1. Find each limit, if it exists.
(a)
lim −3 x 4−2x35x2 4
x−1
(b)
2x2 −11x15
lim
x−3
x 3
(c)
6x 57x 3−3x2
lim
5
x −∞ 12x−3x
(d)
3x38x1
lim
x∞
3x 2−4
(e)
lim
sin 7x
x 0 sin  4x
2. A rocket is shot straight up so that its position after t seconds is given by
s t=−16t 220t25 (measured in feet).
(a) Find the velocity for any time t.
(b) Find the acceleration for any time t.
(c) When does the rocket reach its maximum height?
3. Find the equation of the tangent line to the curve
x=1 .
1
f  x =4x 2−5x2 at
4. Find the derivative of
y= x 6−3x54x 3−2x2 9x5
∫
4
3
2
5. Evaluate the integral
Y = 10x −4x 6x 3 dx
given that Y =5 when x=1 .
6. For
f  x =
2x−53x 2−2x−8
,
6x 2−7x−20
(a) For what x-values is
f  x  discontinuous?
(b) How should f  x  be defined to make it continuous at its discontinuities?
That is, redefine f  x  as a piecewise, continuous function.
7. Use the definition of the derivative, namely
the derivative for
f '  x=lim
h0
f  xh− f  x 
, to find
h
−5
f  x = 2 .
x
8. For line L given by 3x – 2y = 4,
(a) What is the slope of the line L?
(b) What is the slope of any parallel line to L?
(c) What is the slope of any line perpendicular to L?
(d) What is the equation of the line going through the point (5, -1) and
perpendicular to line L?
2
9. For this piecewise function.
{
3x1 −4≤x 0
g  x = x 2 −2 0 x≤2
−2x5 x 2
}
(a) Graph this function.
Now, evaluate the following, or state that the answer does not exist.
(b)
lim g  x  = _______________________________
x 2
(c)
lim g  x  = ________________________________
x5
(d)
(e)
lim g  x  = ________________________________
x 0
+
lim g  x  = ________________________________
x 0
-
(f)
g  2 = ___________________________________
(g)
g 0 = ___________________________________
3
10. Find the derivatives of the following functions.
Don't bother to simplify!
4
2x−34x2−7x
5x 5−3x 32x2 −1
(a)
f  x =
(b)
f  x = 4x37x2 −x 5x 48x5 
(c)
f  x =
sin  x tan  x 
x2