1. Find the derivative of each function.
f (t )  5e
a)
x
d) f ( x )  e
2. f ( x ) 
t
2
4
 4 t  10
2x
27 x
3
3. f ( x ) 
4
f ( t )  sec ( 2 t )
b)

 4
f (t )   t  1

e)
f (t ) 
c)
2  2 t 
1
tan t  1
1 3

At what value of x is the tangent line to f(x) parallel to y = -9x+7?
x
x
2
 x2
find the domain of f and the domain of f '.
 x 2  2 x
x 1

1 x
4. a) f ( x )  
3x
Is f continuous at x=1? Is f differentiable at x=1? What is
seen in the graphs of f and of f ‘ at x = 1?
 a
b) f ( x )  
x
0 x  4
 bx  12
4 x
Find the values of a and b for which f is differentiable at
x = 4.
5.
a)
The tangent line to f ( x ) at x  2 is y  4 x  5 . Find the tangent line to f
1
( x ) at x  f ( 2 ).
b) f ( t )  t 2  t is one to one for t > -1/2. Let g(t) be the inverse function to f on this
domain. Find g'(2).
6. Find the derivative of h(x) at x=1 in each case. Some values for f(x) and f '(x) are given
x f(x) f '(x)
1 3
-4
2 2
5
3 4
-2
h ( x )  f ( f ( x ))
a)
b)
f (x
2 3
 1)
c)
x
2
f (x
7.
Find f " ( x ), the 2nd derivative
b)
e
f ( x )  sec( x  1)
of f ( x ), for
c)
f ( x )  xe
a)
4x
1
2
 1)
1
f(x) 
x
2
1
8. Find
d
58
dx
58
(cos 3 x )
9. a) For the curve x 2 y 2  3 x  1, find the slope of the tangent at the point (1,2).
b) Find y '(x) for sec( xy  1)  y 2  2 x 3
2
Evaluate
each limit.
10.
c)
lim
x 
a)
sin ( 5 x )
lim
x0
2
tan ( x )
3x
e
2x
e
5e
2x
 4e
d)
3x
b)
lim 3
lim
x  
3 x
e
2x
e
5e
2x
 4e
3x
 2 x 1 


 x2 
x 
11. a) Find the linear approximation to 4 15 .
 3 

 10 
b) Find the quadratic approximation to tan 
at a=π/4.
12. Two curves are given by the relations
3
2x y
2
 2 x  y  1
( x  1) y
and
2
1
Show that the curves intersect at the point ( 0 ,  1 ) and are orthogonal at that point.
13. The position of a traveling object is given by the vector equation

r (t )  e
4t

cos( 3 t ) i  e
4t

sin( 3 t ) j
a) Find the velocity vector at time t.
b) Find the speed at time t.
c) Find a unit tangent vector at time t.
14. Consider the curve given parametrically by
2
3
2
x ( t )  t  5t  8
y (t )  t  6t  t .
a) Show that the curve crosses itself at t  1 and t   6 .
b) Find the equations of the two tangent lines at the point where t  1 and t   6 .
15. Consider the curve given parametrically by x ( t )  t 2 e t
a) Find x '(t)
2 t
y (t )  t e
.
b) Find y '(t).
c) At what point(s) is the tangent to the curve horizontal? At what points is the tangent to
the curve vertical?
d) Evaluate lim
t 0
y ' (t )
x ' (t )
. Is the tangent line horizontal, vertical or neither at t=0?
16. A ball is dropped from the top of a 100 foot tall building. The height of the ball at
t seconds is h(t)= 100  16 t 2 . A boy who is lying on the ground 36 feet from the building
has his eye trained on the ball. How fast is the angle between the horizontal and the line
from his eye to the ball changing when t=2 seconds?