Math 151 Week in Review sections 3.2-3.8
1. Evaluate each limit.
a) lim
2x
x 0
sin
2
2
b) lim
x 0
x
sin
2
x
sin 2 x sin 3 x
c) lim
x 0
tan 4 x
tan 5 x
2. Find the equation of the tangent line to f at (a, f(a)).
a) f ( x )  sec x tan x
a 

3
b) f ( x )  sin x  cos x
a 

4
c) f ( x )  tan ( x )
4
a 

4
3. Find the derivative of each function.
2
a)
f ( x )  x sec ( x )
d)
g ( x )  sec
2
x
f ( x )  sin( 2 x ) cos( 3 x )
b)
e)
h ( x )  tan
2
c)
f ( x )  x csc
2
x
x
Why are the answers to d and e the same?
4. The tangent line to f(u) at u=2 is y  3 u  6 . The tangent line to g(x) at x=1 is
y   4 x  6 . Find the tangent line to f ( g ( x ))
@ x  1.
5. Find the derivative of each function.
a)
c)
f (x) 
f (x) 
tan
2
sec
2
x
first)
b)
x
1
2
(simplify
x 1
d)
f (x) 
2
x 1
f (x) 
sin x
1  cos
2
x
Find the domains of f and
f '.
The function, f(x), is differentiable with some values of f and f ' shown in the table.
Use this for problems 6 through 9..
x
1
2
f (x)
2
1
f '(x)
3
1
6. h ( x )  ( x 2  1) 3 f ( x ) Find h '(1).
2
7. h ( x ) 
x 1
f (x)
2
8 .Find the tangent line to h ( x )  f ( x  1) at ( 1, h(1)).
( x  y )  ax y . (from Stewart)
2
9. Find x' if
2
2
2
10. Find the tangent line at the point (3 , 1) to the curve described by xy
3
 xy  6 . (from Stewart)
11. Show that the curves x 2  y 2  5 x and x 2  y 2  10 y are orthogonal where they intersect.
12. r(t) = sec(t) i + tan(t) j
a) Find the tangent vector to the curve at t = π/4
b) Find the slope intercept equation of the tangent line at the point where t = π/4.

13. a) At what point do the curves traced by r1 ( t )  1  t , 3  t
2

and r2 ( s )  s  2 , s
intersect?
b) Find their angle of intersection. (Stewart, #20 pg 203)
14. r(t) = t cos t , t sin t .
a) Find the velocity vector and the speed. Compare the speed = |r ‘(t)| and |r(t)| ‘.
b) Find the acceleration vector.
15. a) Find
d
dx
83
83
sin x
b)
d
125
dx
125
cos x
2
16. Find the 4th derivative of f ( x )  x 5  7 x 4  g ( x ) where g(x) is a 3rd degree polynomial .
17.F the 3rd derivative of f ( x ) 
1
x
.