Preparation Sheet for Exam I, Math171, Fall 2015
Definitions and Theorems
Please learn all the definitions and theorems as stated on the course webpage:
http://www.math.tamu.edu/∼thomas.schlumprecht/math171 13c.html
You will be asked to state them and you will be asked questions, which
will test your understanding of them.
Problems:
(1) Let v = 2i + 4j and w = −2i + 6j. Compute
1
v − w .
2
(2) Find the angle between (1, 2) and (3, 1).
(3) Let a = (−2, 3) and b = (1, 2). Find compa b and proja b.
(4) The vector a starts at the point (−3, 1) and ends at the point (6, 3).
Compute the unit vector in direction of a.
(5) For which value of a are the vectors ha, 3i and h−6, 10i orthogonal?
(6) Find the distance from the point (3, 2) to the line y = 2x − 1
(7) Find the domain of the following functions:
1 + x2
,
a) √
x2 + 5x − 14
p
b) 4 − |2 − x|,
p
c) sin(x).
(8) Let


6 − x
f (x) = 1


−8 + 2x
if x < 5
if x = 5
if x > 5.
Which of the following statements is true:
a) None of the other statements is true.
b) f is continuous only from the right at x = 5.
c) f is not continuous at x = 5, but limx→5 f (x) exists.
d) f is continuous only from the left at x = 5.
e) f is continuous at x = 5.
(9) Given that lim f (x) = 3 and lim g(x) = 0, find lim
x→a
x→a
1
x→a
2f (x)
.
g(x)2 + 1
2
(10) An equilateral triangle is expanding in such a way that it remains
equilateral at every instant. Given that the area of the triangle is
increasing at 9 square centimetres per second, determine the rate at
which the height of the triangle is increasing, when (each) side of
the triangle is 6 centimetres.
(11) Using the precise definition of limits, prove that
a) limt→2 4x − 5 = 3,
b) limx→1 4 − 3x = 1.
(12) Find the limits using the limit laws, if they do not exists say so. In
case the function diverges to ±∞ from the right and or from the left
say so. At each step mention which law you are using.
1
a) lim cos
,
x→0
x
1
,
b) lim x cos
x→0
x
3t2 − 5
.
t→+∞
4t4 − 6t3 + t2 + 3
√
√
d) limx→∞ 3x + 1 − 3x − 1,
c) lim √
3x
,
x→0 tan x
1 − cos x
f) lim
,
x→0
x2
e) lim
sin(x)
,
x2
#
"
x2 − 3x x2 − +2x
−
h) lim
x→∞
x+1
x+2
g) lim
x→0
1 − cos(x)
.
x
(13) Cyclist A starts at point P and rides west at 15 mph. At the same
time, cyclist B starts 10 miles north of point P and rides north at 15
mph. How fast is the distance between them changing after 2 hours?
i) lim
x→∞
(14) A television camera is positioned 4000 feet from the base of a rocket
launching pad. A rocket rises vertically and its speed is 600 feet per
second when it has risen 3000 feet. Assuming that the camera is
always kept focused on the rocket, how fast is the cameras angle of
elevation changing at that same moment? Include units with your
answer.
3
(15) Determine the values of the real numbers a and b so that
x2 + ax + b
1
= .
2
x→−2
x −4
3
lim
(16) Given f (3) = 4 and the slope of the line which passes through the
points (3, f (3)) and (3 + h, f (3 + h)) is 2h + h2 + 2, what is the
equation of the line tangent to f at x = 3?
(17) The temperature at 9:00 AM was 600 F,at 12:00 noon it was 900 F.
Was there any time the temperature was 800 F? Explain!
(18) Using the definition of derivatives find
a) f 0 (2) for f (x) = x3 + 2x2 ,
√
b) f 0 (3) for f (x) = 2x − 4,
d
1
√
c)
,
dx x + 1
dp 2
d)
x + 1.
dx
(19) A heap of rubbish in the shape of a cube is being compacted into
a smaller cube in such a way that the volume decreases at the rate
of 2 cubic meters per minute. (You may assume that the shape of
the heap remains cubical at every instant.) Find the rate of change
of the length of an edge of the cube when the volume is 27 cubic
meters.
How fast is the surface area of the cube changing at that instant
(i.e., when the volume is 27 cubic meters)?
(20) Using the fact that lim
t→0
sin(x)
= 1, compute,
x
1 − cos(x)
,
x→0
x
1 − cos(x)
b) lim
,
x→0
x2
1 − cos(x)
c) lim
,
x→0
x3
d
sin(x),
d)
dx
d
e)
tan(x),
dx
sin(x − 2)
f) lim
.
x→2
x−2
(21) Using the formulas for the derivatives of sin(x) and cos(x) compute
the derivatives of tan(x), cot(x), sec(x) and csc(x).
a) lim
4
(22) Find all horizontal and vertical asymptotes
a)
b)
c)
d)
(23)
3x2
,
+x−2
x3
,
x2 + x − 2
tan(3x),
sin x
.
x4 − 1
Show that every polynomial of degree 3 has at least one x-intercept.
x2
(24) Assume that the slope of the secant to a function through the point
(a, f (a)) and some other point (f (x), x) is 3ax+4. What is the slope
of the tangent to f (x) at the point (a, f (a)) ?
(25) The position of an object at time t is
a) r(t) = (sin(2t), cos(t2 )),
b) r(t) = (t2 , t3 )
Compute: the velocity at time t and the equation of tangent to the
curve at time t = 0 for (a) and t = 1 for (b).
(26) Decide whether the following functions are continuous, differentiable
at the given point a, and whether the derivative is continuous at the
given point: x2 + x − 2 at a = 1
a) f (x) = (
x3 if x ≤ 0,
at a = 0
x2 if x > 0,
(
sin(1/x) if x 6= 0,
at a = 0
f (x) =
0
if x = 0,
(
x sin(1/x) if x 6= 0,
f (x) =
at a = 0
0
if x = 0,
(
x2 sin(1/x) if x 6= 0,
f (x) =
at a = 0
0
if x = 0,
(
2 sin(1/x) if x 6= 0,
f (x) =
at a = 0
0
if x = 0,


|x| if x < 0,
f (x) = x2 if ∈ [0, 1] at a = 0 and a = 1


2x if x > 1

2

if x < 0,
x
f (x) = cos(x)
if 0 ≤ x < π/2, at a = 0 and a = π/2


(π/2) − x if π/2 ≤ x < ∞,
b) f (x) =
c)
d)
e)
f)
g)
h)
5
(27) Using the laws of Differentiation, compute
d
a)
sin(x2 ),
dx
d
b)
tan(x2 + 1),
dx
d
c)
cot(2x),
dx
d) sin(x) cos(1 + x2 ),
1
,
e)
1 + tan2 (x)
f) sin(sin(sin(x))),
x3 g) cos
,
1 + x2
sin2 (x)
h)
.
cos2 (x) + 1
(28) Find the equation of the tangent at the given point a.
1 − x2
, at a = 1,
1 + x2
p
p
b) cos(x2 ) at a = π/2 and a = π/3.
a)
dy
(29) Find dx
a) y 4 + x2 y 2 + yx4 = y + 1,
b) xy = cot(xy).
√
(30) Find the tangent to the point (−1, 4 2) to the curve
x2
y2
+
= 1.
9
36
(31)
a) State the exact definition of limx→a f (x) = L.
b) Using that definition prove that limx→1 3 − 2x = 1.