MTH 251 – REVIEW PROBLEMS 1 – KEY
(1) B
(8) E
(15) B
(22) B
(2) B
(9) C
(16) D
(23) D
(3) A
(10) C
(17) B
(24) E
(4) B
(11) B
(18) B
(25) C
(5) E
(12) C
(19) B
(6) A
(13) C
(20) C
(7) E
(14) A
(21) A
(26) Let f : R → R be defined by:


if x < −2,
3x + 9,
f (x) = ax2 + bx + 1, if − 2 ≤ x ≤ 1,

 2
−x + 4x + 3, if x > 1.
Find a and b so that f is continuous on R.
a = 2, b = 3.
(27) Evaluate the following limits analytically, or determine that they are −∞, +∞, or do not exist:
(a)
(b)
(c)
(d)
(e)
√
2x − 24 + 12 x + 4
√
(g) lim
= −4
x→0 x + 18 − 9 x + 4
√
x − 10 + 4 x + 2
√
= −8
(h) lim
x→2 x + 8 − 5 x + 2
sin 3x 3
lim
=
x→0 2x
2
2
x − 9x + 14
lim
=2
x→1 x2 − 6x + 8
x2 + 2x − 24
1
lim 2
=
x→4 x + 12x − 64
2
2
x − 8x + 7
lim
=2
x→1 x2 − 5x + 4
x2 − 4x − 12
= −∞
lim+ 2
x +x−6
x→2
(f) lim
x→2−
3x4 − 12x3 + 9x + 2 1
=
x→∞ 12x4 − 8x3 − 9x + 1
4
4
3
3x − 12x + 9x + 2 1
(j) lim
=
x→−∞ 12x4 − 8x3 − 9x + 1
4
√
2
7x + 9x + 4
(k) lim
=5
x→+∞
2x + 1
√
7x + 9x2 + 4
(l) lim
=2
x→−∞
2x + 1
(i) lim
x2 − 4x − 12
= +∞
x2 + x − 6
(28) Compute the derivatives of the following functions using the DEFINITION of derivative:
(Note: You will be given no credit at all for using any rules of differentiation when you are asked
to use the definition of derivative.)
Date: Winter 2017.
1
MTH 251 – REVIEW PROBLEMS 1 – KEY
(a) f (x) = 2x2 + 3x + 1
f (x + h) − f (x)
h→0
h
2
2 (x + h) + 3 (x + h) + 1 − 2x2 + 3x + 1
= lim
h→0
h
f 0 (x) = lim
..
.
(Insert algebra here)
..
.
= lim (4x + 2h + 3)
h→0
= 4x + 3.
(b) g (x) =
1
2x + 1
g (x + h) − g (x)
h→0
h
1
1
−
2 (x + h) + 1 2x + 1
= lim
h→0
h
g 0 (x) = lim
..
.
(Insert algebra here)
..
.
−2
h→0 (2 (x + h) + 1) (2x + 1)
−2
=
2.
(2x + 1)
= lim
(c) h (x) =
√
x−1
h (x + k) − h (x)
k
√
√
x+k−1− x−1
= lim
k→0
k
h0 (x) = lim
k→0
..
.
(Insert algebra here)
..
.
= lim √
k→0
1
√
x+k−1+ x−1
1
= √
.
2 x−1
2
MTH 251 – REVIEW PROBLEMS 1 – KEY
(d) k (x) =
√
3
3
x
k (y) − k (h)
y−x
√
√
3 y − 3 x
= lim
y→x
y−x
z−w
= lim 3
z→w z − w 3
k 0 (x) = lim
y→x
z=
√
3
y, w =
√
3
x
..
.
(Insert algebra here)
..
.
1
3w2
1
= √
.
3
3 x2
=
(29) Consider the graph of a function f below:
y
6
5
4
3
2
1
1
2
3
4
5
6
7
8
9
10
x
(a) Determine the following values. Round to the nearest multiple of 0.25. If a limit does not
exist, write DNE:
c
f (c)
lim f (x)
2
5
2.5
3.5
DNE
4.5
2.5
5
5
5
6.5
4
4
4
4
8
3.5
3
3.5
DNE
x→c−
(b) Is f continuous at 2? No. 4.5? No. 6.5? Yes. 8? No.
lim f (x)
x→c+
lim f (x)
x→c
MTH 251 – REVIEW PROBLEMS 1 – KEY
4
(30) Let g be a function defined by:


3x + 7,
if x < −2,




1,
if x = −2,




1

 x2 + x + 1, if − 2 < x < 0,
g (x) = 2

2x + 1
if 0 < x < 2





4,
if x = 2,



3
7

 + x,
if x > 2.
2 4
(a) Evaluate the following limits, determine that they are +∞ or −∞, or that they do not exist.
Show all work. Your work must show that you are computing the limits correctly.
(i)
(ii)
lim g (x) = 1
x→−2−
(iv) lim− g (x) = 1
(vii) lim− g (x) = 5
(v) lim g (x) = 1
(viii) lim g (x) = 5
x→0
lim g (x) = 1
x→0+
x→−2+
(vi) lim g (x) = 1
(iii) lim g (x) = 1
x→0
x→−2
x→2
x→2+
(ix) lim g (x) = 5
x→2
(b) Is g continuous at −2? Yes. At 0? No. At 2? No.
(31) A rock is launched from a platform 256 feet above the. The height of the rock above the ground is
modelled by a function h, defined by:
h (t) = −16t2 + 96t + 256,
0 ≤ t ≤ 8,
where t is given in seconds, and h (t) is given in feet.
(a) What is the initial velocity of the rock when it is launched? h0+ (0) = 96 ft/sec.
(b) How fast is the rock going when it hits the ground? h0− (8) = −160 ft/sec.
(c) What is the average velocity of the rock over the time interval from 4 seconds to 7 seconds?
h (7) − h (4)
= −80 ft/sec.
7−4
(32) Consider the graph of a function f below and to the left:
y
y
6
5
4
3
2
1
−8 −7 −6 −5 −4 −3 −2 −−11
−2
−3
−4
−5
−6
1 2 3 4 5 6 7 8
x
6
5
4
3
2
1
−8 −7 −6 −5 −4 −3 −2 −−11
−2
−3
−4
−5
−6
1 2 3 4 5 6 7 8
x
Plot the graph of f 0 on the coordinate axes to the right. Make proper use of open circles
and closed circles.