c
Math 151 WIR, Spring 2010, Benjamin
Aurispa
Math 151 Exam 1 Night Before Drill
1. (a) Use the limit definition to find the derivative of the function f (x) =
√
3x + 7.
(b) Use part (a) to find the equation of the tangent line at x = 3.
2. Use the limit definition to find a vector tangent to the curve r(t) =< 3t2 + 2t, 2t + 1 > at the point
(16, 5).
3. The position function of an object in linear motion is given by f (t) =
in ft and time is measured in seconds.
t−1
where position is measured
t+2
(a) What is the average velocity of the object from t = 2 to t = 5?
(b) What is the velocity of the object at t = 3?
4. Sketch a graph of the function f (x) = |x2 − 4| and determine where f (x) is not differentiable by
examining the graph.
5. Find a unit vector which points in the same direction as the vector with initial point (−5, 2) and
terminal point (3, 7).
6. Given vectors a = 4i + 10j and b = −i + 5j, find a vector of length 2 that is orthogonal to a − 4b.
7. A plane heads in the direction N 60◦ E with an airspeed of 300 miles per hour. The wind is blowing
S 45◦ E at 20 miles per hour. Find the groundspeed of the plane.
8. Two forces are acting on an object placed at the origin. F1 =< −1, 3 > and F2 =< 5, 2 >. Find the
magnitude and direction of the resultant force F.
9. Find the scalar and vector projections of < 2, −9 > onto < 6, 3 >.
10. A port is located on a map at coordinates (2, 4). Two ships leave the port, one headed for an island
at (5, 9) and the other headed for an island at (4, −2). Find the angle between the two paths.
11. A force of 15 N is applied horizontally in moving an object up a ramp. The ramp has a horizontal
displacement of 8 meters and a vertical displacement of 3 meters. How much work is done by the
force?
12. Find a Cartesian equation for the curve r(t) =< −3 sin t, 3 cos t + 1 >. Sketch the curve and describe
the direction of motion as t increases.
13. Find parametric equations of the line passing through the points (8, 1) and (4, 5).
14. Find the value(s) of a such that the lines r(t) =< 2 − 5t, 6 + at > and r(s) =< 7 + 6s, −1 + 3as > are
perpendicular.
15. Compute the following limits or show why the limit does not exist.
(a) lim
x→−2
3
1
−
(x + 2)(x − 1) (x + 2)(x + 1)
2x − 3x2
(b) lim √
x→∞
x2 + 4x
x2 (x + 4)
(c) lim
x→−1− x2 − 4x − 5
1
c
Math 151 WIR, Spring 2010, Benjamin
Aurispa
7 − x2
x→∞ x3 − 3x
(d) lim
*
(e) lim r(t) where r(t) =
t→4
(f) lim x4 sin
x→0
1
x2
t−4
t2 − 16
√
, 2
t + 5 − 3 2t − 3t − 20
+
−4
(x − 3)(x + 6)
x→3
|x − 3|
√
(h) lim
x2 − 5x + x
(g) lim
x→−∞
16. Find all vertical and horizontal asymptotes of f (x) =
6x2 − 4x − 2
.
7x2 − 8x + 1
17. Show that the equation x4 − 3x2 + x − 3 = −2 has two real solutions.
18. Determine where the following function is discontinuous.
f (x) =

5 − x2





 3x + 7
5



x2 + 9



6x − 4
if
if
if
if
if
x ≤ −2
−2<x<1
x=1
1<x<3
x≥3
19. Find the value of m that makes the following function continuous everywhere, if possible.
f (x) =

2

 mx + 3
5

 2x + 2m
if x < −1
if x = −1
if x > −1
2