Final Examination
Calculus I Science
1. (10 points) Evaluate each of the following limits.
1
1
(a) lim+
−√
x
x
x→0
√
x2 − 3
(b) lim
x→−∞ 4x + 1
1
2
−
(c) lim x + 2 x + 4
x→0
sin(x)
May 16, 2016
8. (4 points) Find the equation of the line tangent to the
curve defined by x2 + y 2 = (2x2 + 2y 2 − 1)2 at the point
(0, 21 ).
√
2− 7−x
(d) lim 2
x→3 x − 5x + 6
9. (4 points) (a) State (or explain) the Mean Value Theorem.
ln(e + h) − 1
, (by
h
interpreting it as a
derivative)
(b) Suppose f (x) is a twice-differentiable function, and
f (x) has at least two critical values. Show that f 00 (x)
has at least one root.
(e) lim
h→0
Find the intervals of increase/decrease of
2. (3 points) Find the horizontal and vertical asymptotes of 10. (4 points) 2/3
x
f
(x)
=
9x
(x − 20), and coordinates of all local ex3e + 1
f (x) = x
.
trema.
5e − 2
2
x + 6x + 12
if x < −2,
3. (3 points) Let f (x) =
11. (4 points) Given
ax + b
if x ≥ −2.
Find all pairs of values a and b so that f is differentiable
−2(1 + 2 sin(x))
p
f (x) = (1 + sin(x))2/3 ,
f 00 (x) =
everywhere.
9 3 1 + sin(x)
4. (3 points) Give the rule of a function of the form f (x) =
find the intervals of concavity and points of inflection of
(Ax − B)(Cx − D)
that has all of the following properf (x) over the interval [0, 2π].
(Ex − F )(Gx − H)
ties:
12. (5 points) Given the following information about a func• lim f (x) = 1.
tion f (x):
x→∞
• lim f (x) exists, but f (3) does not.
Domain: R \ {0}
x-intercepts: x = −1, x = 2, x = 9,
f (−5) = 7, f (−3) = 5, f (6) = −3, f (10) = 4
x→3
• lim− f (x) = ∞.
x→2
dy
5. (16 points) Find
for each of the following. Do not
dx
simplify your answers.
√
9
x9
− 9 + 9x + log9 (x) + 9 x + 99
(a) y =
9
x
q
ln(x) sec(2x − 1) (c) y = (ex − 3)2 +
(b) y =
7
(d) y = tan(x9 + xx )
(Hint: what is
d
dx
lim f (x) = −∞,
x→0−
0
lim f (x) = 6,
x→∞
lim f (x) = 1
x→−∞
f (x) > 0 : (−∞, −5) ∪ (6, ∞)
f 0 (x) < 0 : (−5, 0) ∪ (0, 6)
f 00 (x) > 0 :(−∞, −5) ∪ (−5, −3) ∪ (0, 10)
x
sin(x)
(xx )?)
f 00 (x) < 0 :(−3, 0) ∪ (10, ∞)
Sketch a graph of f which takes all of this information
into account. Label all intercepts, asymptotes, extrema
and inflection points.
6. (3 points) Let f (x) = xg(x). If g 0 (3) = 2, and the slope
of the line tangent to f (x) at x = 3 is 14, determine the
13. (6 points) A 600m by 800m rectangular park has a bivalue of g(3).
cycle path cutting though it diagonally (see below). A
rectangular play area is to be fenced off in the space be7. (5 points) Allan and Balla both start walking from the
low the bike path. What dimensions of the play area will
same point. Allan walks east at a rate of 1.5 km/h, while
allow it to have the largest possible area?
Balla jogs π3 radians north of east at a rate of 4 km/h.
How fast is the distance between them increasing after 2
hours?
[Hint: Recall the law of cosines: c2 = a2 + b2 −
2ab cos C for a triangle of with side lengths a, b and c,
and where C is the angle opposite the side c.]
B
π
3
C
A
14. (4 points) Use differentiation to verify that the following formula is correct.
Z
1
x
dx = √
+ C.
2
3/2
(x + 4)
4 x2 + 4
Page 1 of 2
Final Examination
Calculus I Science
Z
3
15. (3 points) Find
p
Z
x cos(2x) dx as the limit of a Rie0
mann Sum. Do not evaluate the limit.
n X
4
4
4
17. (3 points) Express
lim
5 + i sin 2 + i
n→∞
n
n n
i=1
as a definite integral starting at a = 3.
18. (11 points) Evaluate each of the following integrals
Z
√
(a)
( x − 3)2 dx
Z
tan(θ) − cos2 (θ) + sec(θ)
(b)
dθ
cos(θ)
Z 2
3
y
(c)
e − 2 dy
y
1
Z 5
(d)
cot(3x2 − 9) dx
−1
x
2 (2(ex
sin x )
8
x
− 3)ex +
7.
y
e−t dt = 4 +
km/h
1
2
9. (a) If f is a function continuous on [a, b] and differentiable on (a, b) then there exists a number c ∈ (a, b)
with f (b) − f (a) = f 0 (c)(b − a).
(b) Since f has two critical values, and f 00 exists everywhere, we know that f 0 has at least two roots, say
x1 < x2 . Then f 0 is continuous and differentiable
on [x1 , x2 ], and f 0 (x1 ) = f 0 (x2 ). So by Rolle’s theorem, there exists c ∈ (x1 , x2 ) with f 00 (c) = 0.
10. Increasing: (−∞, 0), (8, ∞), Decreasing: (0, 8)
Local Max: (0, 0), Local Min: (8, −432)
11π
11. CU: [ 7π
6 , 6 ],
POI: ( 7π
6 ,
Z
0
7
2
8. y =
19. (3 points) Given the relation
12.
x2
11π
CD: [0, 7π
6 ], [ 6 , 2π]
11π
1
1
√
3 ); ( 6 , √
3 )
4
4
(−5, 7)
2
sin (t) dt,
(−3, 5)
2
(10, 4)
dy
.
use implicit differentiation to find
dx
20. (4 points) Answer true or false, justifying your answer
with an explanation or counterexample:
f (x)
= 1.
x→a g(x)
−1
• If lim f (x) = lim g(x), then lim
x→a
x→a
2
9
• If f (x) is differentiable at x = a, lim f (x) must exx→a
ist.
(6, −3)
0
• If f is increasing on [a, b], then f (x) > 0 for every
x in (a, b).
• If f and g are both continuous on [a, b], then
Z b
Z b
Z b
f (x) · g(x) dx =
f (x) dx ·
g(x) dx.
a
a
a
13. The play area should be 300m × 400m.
15.
x→∞
Z
17.
1. (a) ∞
(e)
π
n
(2 + x) sin(−1 + x) dx
1 2
2x
− 4x3/2 + 9x + C
(c) e2 − e −
3
2
(d) 0
− 21 , 35 ,
VA: x = ln(2/5)
19.
20.
5. (a) x8 − 81x−10 + 9x ln 9 +
1
x ln 9
= 2x sin2 (x2 )ey
• False
• False
+ 19 x−8/9
sec(2x − 1) + sec(2x − 1) tan(2x − 1)(2) ln(x))
Total: 100 points
dy
dx
• True
4. Answers will vary.
1 1
7(x
cos
2iπ
n
(b) sec θ − sin θ + tan θ + C
3. a = 2, b = 8
(b)
n
i=1
7
18. (a)
− 18
1
4
1
e
2. HA: y =
n X
iπ
3
(b) − 14
(d)
9π
4
16. lim
Answers:
(c)
sin x−x cos x
))
sin2 x
(d) sec (x + xx )(9x + x (ln x + 1))
5
Z
3)2 +
6. g(3) = 8
π
16. (2 points) Express
1
x
2 ((e −
2 9
(c)
9 − x2 dx, by interpreting in terms
0
of area.
May 16, 2016
Page 2 of 2
• False