Math 151 Week in Review
Review For Exam 1 (1.1-3.2)
Monday Sept. 27, 2010
Instructor: Jenn Whitfield
Thanks to Amy Austin for contributing some problems.
All prolbems in this set are copywrited
1. Using the limit definition for slope, find the
equation of the tangent line to the graph of
f (x) at the indicated value:
√
(a) f (x) = x + 1 at the point (3, 2)
x
at x = 0
(b) f (x) =
1−x
x2 − x − 2
x→−1 x2 + 13x + 12
√
√
x + 2 − 2x
9. lim
x→2
x2 − 2x
8. lim
10.
lim ~r(t) where
t→−∞
~r(t) = h
t2 − 4
3t
,√
i
2
3t − 6t + 3 t2 − 4t
2 − cos x
x+3
√
lim ( x2 + x + 1 + x)
11. lim
x→0
12.
− 2i:
x→−∞
(a) Find the tangent vector to the curve
r(t) at the point (1, 0).
13. Find lim
(b) Find parametric equations for the tangent line to the curve at the point
(1, 0).
14. Find the vertical and horizontal asymptotes
for
6x
f (x) =
x − x3
2. Given r(t) =
ht2 , 2t
(c) Eliminate the parameter to find a
cartesian equation of the tangent line.
3. Find the derivative of the following functions.
x2 + x
x3 − 1
√
√
t3 − 5 t + π 2 t
(b) f (t) =
t
√
16
(c) ~r(t) = 10 t, 30t − 2
t
(a) g(x) =
4. If f (3) = 4, g(3) = 2, f ′ (3) = −6, and
3x2
g′ (3) = 5, find h′ (3) given h =
+ fg
f
5. Draw a diagram to show there are two tangent lines to the parabola y = x2 that pass
thru the point (0, −4). Find the x coordinates where these tangent lines touch the
parabola.
6. For what values of a and b is the line 6x+y =
b tangent to the parabola y = ax2 when
x = −2?
7. A particle moves according to the equation
of motion s = 4t3 − 9t2 + 6t + 2, t ≥ 0, where
s is measured in feet and t in seconds.
a) Find the instantaneous velocity of the
particle at t = 3
b) When is the particle at rest?
x(4 − x)
x−3
x→3−
15. Find the angle between the vectors 2i + 4j
and 3i − 5j.
16. If ~a = h−1, 2i, ~b = h2, 3i, and ~c = h2, 5i, find
the vector and scalar projection of 3~a + ~b
onto ~c.
17. Show there exists a solution to the equation
x3 + 2x + 1 = 0
18. Let f (x) =
 2
x −2





 2
if x < 0
x3 − 4
if x ≥ 2
x −4


x−2




if 0 ≤ x < 2
(a) Find lim f (x)
x→2+
(b) Find lim f (x)
x→0
19. Let f (x) =

|2x − 6|





x−3











−x + 1
if x ≤ 3
if 3 < x < 5
1
if x ≥ 5
x−6
Find all values of x where f (x) is discontinuous and not differentiable.
20. f (x) =
ax + 3
if x ≤ 3
,
bx2 − 2x + 8 if x > 3
find the values of a and b that make f (x)
continuous and differentiable everywhere.
21. Obtain a vector equation of the straight line
passing through the point (1, 3) and perpendicular to the vector joining the points
(−2, 1) and (1, −1).
22. Find the work done by a force, F~ , of 2 N
acting in the direction N 300 W in moving
an object 6 m due west.