MATH 151, Fall 2015
Practice Exam
PART I: Multiple Choice
1. Find the work done by a force of 2N acting in the direction N 30o W (i.e. 30o west of the northerly direction) in
moving an object 6m due west.
√
(a) 3J
(b) 12J
√
(c) 6 3J
(d) 6J
√
(e) 12 3J
2. Find a unit vector perpendicular to the line described by the parametric equations x = −4t + 1, y = 6t + 5.
2
3
(a) √ , √
13
13
1
1
(b) − √ , √
5
5
(c) h3, 2i
2
3
√
√
(d)
,−
13
13
(e) h2, −3i
3. Which of the following is a cartesian equation for the curve described by the parametric equations x = 2 sin t,
y = 3 cos t, 0 ≤ t ≤ 2π?
(a) x2 + y 2 = 6
(b)
x2
y2
+
=1
4
9
(c) (x − 2)2 + (y − 3)2 = 1
(d) (x − 2)2 + y 2 = 1
(e) (x − 2)2 + y 2 = 9
1
4. If 8x + 7y + 12 = 0 is the equation of the tangent line to the graph of f (x) at x = 1, what is lim
h→0
(a)
8
7
(b) −
(c)
7
8
7
8
(d) −
8
7
(e) 8
√
16 − x − 4
5. lim
=
x→0
x
(a) −
1
8
(b) 0
(c) 1
(d)
1
8
(e) The limit does not exist.

4x + 1
if x ≤ −2


 x2 − 15 if − 2 < x ≤ 4
6. Where is f (x) =
1
2


 √ +
if x > 4
x x
not continuous?
(a) x = 4
(b) x = −2
(c) x = −2 and x = 4
(d) x = 0, x = −2 and x = 4
(e) x = 0 and x = −2
2
f (1 + h) − f (1)
?
h
7. Which of the following intervals contains a solution to the equation 3x3 − 2x2 = 2x + 5?
(a) (−1, 1)
(b) (1, 2)
(c) (2, 3)
(d) (−1, 0)
(e) (0, 1)
8. Find the distance from the point (1, 5) to the line y = 2x − 7.
10
(a) √
5
25
(b) √
5
10
(c) √
145
25
(d) √
145
14
(e) √
5
9. lim
x→3−
x2 − 4
=
x−3
(a) 1
(b) ∞
(c) −∞
(d) 3
(e) 0
3
10. By sketching the graph of f (x) = |x2 − 1|, where is f (x) not differentiable?
(a) x = 1
(b) x = −1
(c) x = −1 and x = 1
(d) x = 0
(e) f (x) is differentiable everywhere because it is a polynomial.
11. Find parametric equations for the line passing through the points (−1, 5) and (2, 7)
(a) x = −1 + 3t, y = 5 + 2t
(b) x = −1 + 2t, y = 5 + 7t
(c) x = −1 + t, y = 5 + 2t
(d) x = −1 + t, y = 5 + 13t
(e) x = −1 − 2t, y = 5 + 3t
12. Find the vertical asymptotes for the curve f (x) =
x2 + 2x − 15
x2 − x − 6
(a) x = 3 and x = −2
(b) x = −5 and x = 3
(c) x = 1
(d) x = 3
(e) x = −3 and x = 2
4
13. Find the value of x so the dot product of the vectors h3 + 2x, 5i and h−7, x + 1i is equal to 10.
16
9
26
19
8
3
26
−
9
13
−
19
(a) −
(b)
(c)
(d)
(e)
14. Find the average rate of change of f (x) = tan(2x) over the interval
(a)
(b)
(c)
(d)
(e)
24 √
( 3 − 1)
π
√
24 3
π
1
24
√ −1
π
3
1
24
1− √
π
3
√
24
(1 − 3)
π
5
hπ πi
,
8 6
PART II WORK OUT
Directions: Present your solutions in the space provided. Show all your work neatly and concisely and Box your
final answer. You will be graded not merely on the final answer, but also on the quality and correctness of the work
leading up to it.
√
15. (i) (9 pts) Using the definition of the derivative, find f ′ (x) for f (x) = 4 − 3x
(ii) (3 pts) Find the equation of the tangent line to the graph of f (x) at x = −2.
6
16. (6 pts) Two forces, G and F, act on an object located at the origin as shown below. The vector G has magnitude
40 N and measures an angle of 60◦ with the positive x-axis. The vector F has a magnitude 30 N and measures
an angle of 45◦ with the positive x-axis. Find the magnitude of the resultant force acting on the object. Do not
simplify.
y
G
x
F
17. (i) (2 pts) Determine whether the vectors h1, 2i and h−2, 3i are orthogonal, parallel, or neither.
(ii) (4 pts) Find the cosine of the angle between the vectors h1, 2i and h−2, 3i.
(iii) (4 pts) Find the vector projection of h1, 2i onto h−2, 3i.
7
18. Find the following limits:
√
10x2 − 5
(i) (6 pts) lim
x→−∞
2 − 3x
(ii) (6 pts)
lim −
x→−1
x2 + x
|x + 1|
1
1
−
(iii) (6 pts) lim x + 3 4
x→1
x−1
8
 2
x + 4cx + 3 if x < 3





−26
if x = 3
19. Consider f (x) =





cx − 6
if x > 3
(i) (3 pts) Find lim f (x) in terms of c.
x→3−
(ii) (3 pts) Find lim f (x) in terms of c.
x→3+
(iii) (3 pts) For what value of c does lim f (x) exist?
x→3
(iv) (3 pts) For the value of c found above, what is lim f (x)?
x→3
(v) (3 pts) For the value of c above, is f (x) continuous at x = 3? Support your answer.
9