Preparation Sheet for Exam II, Math171, Fall 2015
Definitions and Theorems
Please learn all the definitions and theorems as stated on course webpage:
http://www.math.tamu.edu/∼thomas.schlumprecht/math171 15c.html
(1) Suppose that −1 ≤ x ≤ 1. What does it mean to say that sin−1 (x) =
θ?
(2) Simplify cot(sin−1 (x)) and cos(sin−1 (x)) , for 1 ≤ x ≤ 1.
(3) tan(cos−1 (.5) =
(4) Using the formula for the derivative of inverse functions derive the
formulae for
d
d
d
arcsin(x),
arccos(x), and
arctan(x).
dx
dx
dx
(5) Let C denote the parametric curve given by the vector equation
(r)(t) = tan−1 tet + 1 , t(t + 2) ,
−1 ≤ t ≤ 1.
a) Compute (r)0 (t), −1 ≤ t ≤ 1.
b) Determinethe value of the parameter t which corresponds to
π
the point
, 0 on C.
4
π c) Compute the slope of the tangent to C at the point
,0 .
4
sin 7t
(6) Evaluate lim
.
t→0 cos 8t
d √ x
(7) a)
2 e =
dx
d sec(x)
b) Determine the domain of f (x) = xsec(x) , and compute
x
.
dx
(8) Consider the function
f (x) = ln cos−1 (x − 3) .
Find the domain and its derivative.
(9) Find the domain of the function sin−1 (3x − 1)2 .
(10) Evaluate limx→0 x ln x.
(11) Let
π
,
h2 π i
Show that f is is one-to-one on the interval 0, . Let f −1 denote
2 h
πi
its inverse. Find the number x0 in the interval 0,
for which
2
f (x0 ) = 1. Use the Differentiation Theorem for Inverse Functions to
0
compute f −1 (1).
f (x) = esin x−cos x ,
1
0≤x≤
2
(12) Evaluate limt→1− e1/(t−1) .
(13) Find the inverse of f (x) =
x−2
.
x+2
4
(14) Compute the second derivative of ex .
(15) Compute
lim
x→0+
1 − cos2 x
1
ln x
.
Show all your steps clearly.
(16) Consider the function
x3
− 2 |2x − 1|,
−∞ < x < ∞.
3
Determine all the critical numbers of f .
Determine the interval(s) in which f is increasing.
Determine al local extrema.
f (x) =
(17) Show for all x, y that | cos(x) − cos(y)| ≤ |x − y|.
(18) Assume f is differentiable on (a, b) and there are three roots of the
equation f (x) = 0. Show that there are at least two points x for
which f 0 (x) = 0.
(19) Assume f is continuous on [a, b] and twice differentiable on (a, b)
and there are three roots of the equation f (x) = 0. Show that there
is at least one point x for which f 00 (x) = 0.
(20) Assume that f (x) is continuous on [a, b] and twice differentiable on
(a, b). Show that if f (a) = f (b) and if f (x) is concave up, then f (x)
has exactly one local minimum which is also the absolute minimum
of f (x) on [a, b].
(21) Suppose that f is twice continuously differentiable in an open interval I, and that f 00 (x) > 0 for every x in I. Prove that the function
G, defined by
G(x) := ef (x) ,
x ∈ I,
is concave upwards in I.
(22) Evaluate limx→∞ 3−x
(23) Find the domain for ln |x|.
(24) What is the range of the inverse f −1 , of f , with f (x) =
√
x − 1?
(25) If g is the inverse of f (x) = x5 + x3 . What is g 0 (2)?
(26) For the following functions find
(a) Intervals on which the function is increasing/ decreasing
(b) Intervals on which the function is concave up/down
(c) Local Maxima/Minima
(d) Inflection points,
3
(e) find horizontal and vertical asymptotes, if there are any
(f) and based on that information sketch the function
f (x) = x4 − 6x2
f (x) = x3 − 5x2 + 3x + 6
f (x) = e−(x−2)
2
f (x) = (x2 − 1))|2x − 1|
f (x) = x2 + 2x − 3

2

if x < 0
1 − x
f (x) = x + 1
if 0 ≤ x < 2

 2
x − 6x + 11 if 2 ≤ x ≤ 3
2
1 + 31/x
(28) Solve for x:
(27) lim
x→0−
(a) log2 (x + 2) = 3,
x
(b) 23 = 5,
(c) ln(x − 2e) + ln(x + 2e) = 2 + ln 2.
(d) ln(ln(x)) = 1,
(29) Compute the linear approximation to f (x) = esin x at x = π.
2
(30) Find the
√ linear approximation L(x) to the function f (x) = sin (x)
at a = π.
(31) Suppose x > 0 and let s denote distance between the points (x, 0)
dx
ds
and (0, 1). If x is changing with time and
= 2 compute
.
dt
dt
(32) Find the inverse of
10x
y= x
,
10 + 1
x
y = 210
ex + 1
y=
.
1 − ex
√
(33) Approximatively compute 36.1.
(34) Compute cos−1 cos(3π)
(35) Compute
ln3 x
x→∞ x
x
x
b) lim
,
x→∞ x + 1
a) lim
4
(36) Compute cos(2 tan−1 (3)) (Hint cos(2x) = cos2 (x) − sin2 (x)
(37) At which point in the closed interval [−1, 1] does the function f (x) =
x sin−1 (x) attain its absolute minimum value?
1
(38) Show that f (x) =
is one to one on the interval [0, ∞) and
2 + x2
compute its inverse.
2
e x − 1 − x2
(39) lim
.
x→0
x3
(40)
ln x
(41) Evaluate lim √ .
x→0
x
(42) Differentiate f (x) = ln(2x + 4).
(43) A can is to be made to contain 1 L of oil. Find the dimensions that
will minimize the cost of the material necessary to produce the can.
(44) Consider the function f (x) = 12x1/3 − 3x4/3 .
Find all local extreme points and classify them as local maxima
Find all inflection points
(45) An open box has do be made from a rectangular piece of material
(dimensions: 2 by 3 feet) by cutting equal squares from each corner
and turning up the sides.
Find the dimensions of the box with maximal volume.
(46) Find the (absolute) maximum value of the function f (x) = x5 − e−x
in the interval [−1, 1] .
(47) A poster is to have an area of 180 square inches with 1-inch margins
at the bottom and sides and a 2-inch margin at the top. What
dimensions will give the largest printed area?
(48) Find the point on the line y = 2x − 3 that is closest to the point
(−1, 3).
(49) Find the dimensions of the rectangle of largest area that has its base
on the x- axis and its other two vertices above the x-axis lying on
the parabola y = 8 − x2 .