Math 151 In Class Exam 2 Review
1. Evaluate each limit.
a)
( x  2)
lim
2
tan( x  2 ) sin( x  2 )
x 2
b)
lim
(2  x )e
x 0
2 x
 2e
2
x
3x2
c)
lim 16 4 x  2
d)
x 
lim x sin
x 
1
x
2. Find a formula in x for the derivative of the inverse function to f(x)= tanx.
3. Find the tangent line at the point ( 1, 2) if x y  2 x  3 y  0 .
2
4.
f (x) 
2
3x  5
2x  4
a) Find the tangent line to f(x) at x=1.
b) Find f  1 ( x ).
c) Find the tangent line to f  1 ( x ) at x=f(1).
d) Is the inverse to the tangent line the tangent to the inverse?
5. A curve is parameterized by x  t 2 sin  t
line to the curve at the point where t = 1/2.
2
y  t cos  t
. Find an equation of the tangent
6. Find the points on the curve where the tangent line is horizontal and where the tangent
line is vertical.
a) x ( t )  ( t  2 ) 2 ( t  1)
b) x ( t )  t  t
3
2
y (t ) 
t
2
t 1
y (t )  t t  1
7. Find the 59th derivative with respect to x of
a) f(x)= sinx b) f ( x )  cos( 2 x ) c) f ( x )  e ax
8. For what values of a does
y e
ax
solve
y "  3 y ' 2 y
.
9. h ( x )  ( f ( x )) 2 and L(x)= 5x - 7 is the tangent line to f (x) at x=1. Find the linear
approximation to h(1.1).
10.
h ( x )  f ( g ( x )) The tangent
line to g at x  1 is T ( x )  4 x  2 . The tangent
L ( u )   3 u  4 . Find the tangent
line to h(x) at x  1.
line to f at u  6 is
11. Use linear approximation to approximate a)
3
b)
66
5
30 .
12. A snowball is melting at the rate of 4 cubic cm. per hour. Find the rate of change of
the surface area at the instant that the volume is 36  cubic cm.
13. A 4 ft tall child is walking away from a 10 ft tall lamp.
Let  be the angle between th e lamp post and a line from the light to
the top of his head. If  is increasing
by 9 /sec when  
o

4
fast is the child walking at that instant?
14. Find the derivative with respect to x for each function.
a) f ( x )  5 e
c) y ( x )
d) f ( x ) 
2
x  25
2
13
1 e
2
xy  cos( x  y )  1
given
x
b) f ( x )  sec( x 2 ) tan( x 2 )
2x
, how