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Math 151 WIR, Spring 2013, Benjamin
Aurispa
Math 151 Exam 1 Week in Review
1. Use the limit definition to find the derivative of f (x) = 3x2 + 2x − 1.
2. (a) Use the limit definition to find the derivative of the function f (x) =
√
3x + 7.
(b) Find an equation of the tangent line to the graph of f (x) at the point where x = 3.
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 seconds?
(b) What is the velocity of the object at t = 3 seconds?
4. Sketch a graph of the function f (x) = |x2 − 2x − 3| to determine where f (x) is not differentiable.
5. Compute the following limits or show why the limit does not exist.
3
1
lim
−
x→−2 (x + 2)(x − 1)
(x + 2)(x + 1)
2
x (x + 4)
lim
x→−1− x2 − 4x − 5
7 − x2
lim 3
x→∞ x − 3x
2x − 3x2
lim √
x→∞
x2 + 4x
*
+
t−4
t2 − 16
,
lim r(t) where r(t) = √
t→4
t + 5 − 3 2t2 − 3t − 20
(a)
(b)
(c)
(d)
(e)
(f) lim x4 sin
x→0
1
x2
(3 − x)(x + 6)
x→3
|x − 3|
√
(h) lim
x2 − 5x + x
(g) lim
x→−∞
6. Find all vertical and horizontal asymptotes of f (x) =
7. Consider the function f (x) =
6x2 − 4x − 2
.
7x2 − 8x + 1
x2 + 8x + a
.
x2 + x − 2
(a) For what value(s) of a would lim f (x) exist to a finite number?
x→−2
(b) For the value(s) of a found above, calculate lim f (x).
x→−2
8. Show that the equation x4 − 3x2 + x − 3 = −2 has a real solution.
9. Determine where the following function is not continuous.
f (x) =


5 − x2




3x + 7



5

x2 + 9




6x − 4



x−4
if
if
if
if
x ≤ −2
−2<x<1
x=1
1<x<3
if x ≥ 3
1
c
Math 151 WIR, Spring 2013, Benjamin
Aurispa
10. Find the values of m and c that make the following function continuous everywhere.
f (x) =

2

 mx + 3
c

 2x + 2m
if x < −1
if x = −1
if x > −1
11. Suppose a is a vector with initial point (−1, 2) and terminal point (3, 12) and that b = −i + 5j, find
a unit vector that is orthogonal to a − 4b.
12. 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 (true speed) of the plane.
13. Two forces are acting on an object placed at the origin. F1 =< −1, 3 > and F2 =< −5, −6 >. Find
the resultant force F and its direction from the positive x-axis.
14. Find the scalar and vector projections of the vector b = 2i − 9j onto the vector a = 6i + 3j.
15. 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). Use vectors to find the angle between the two
paths.
16. A force of 15 N is applied horizontally in moving an object 8 meters up a ramp. If the ramp is inclined
at a 30◦ angle, how much work is done by the force?
17. Find Cartesian equations for the following curves and sketch a graph.
√
(a) x = − t − 4, y = t + 7
(b) r(θ) =< sin2 θ + 1, 3 cos2 θ >
18. Find parametric equations of the line passing through the points (8, 1) and (4, 5).
19. Find the value(s) of a such that the lines r1 (t) =< 2 − 5t, 6 + at > and r2 (s) =< 7 + 6s, −1 + 3as >
are perpendicular.
20. Consider the vector function r(t) =
18
26
√
,
.
2
t +t−6 t+7
(a) What is the domain of the vector function?
(b) Does the curve represented by this vector function pass through the point (3, 2)?
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