Homework 1
Math 15300 (section 51), Spring 2015
This homework is due in class on Wednesday, April 8th. You may cite results from class as appropriate.
Unless otherwise stated, you must provide a complete explanation for your solutions, not simply an
answer. You are encouraged to work together on these problems, but you must write up your solutions
independently.
You are encouraged to think about the problems marked with a (*), but they are not to be handed in.
0. (*) (a) Read sections 7.7-7.9 in the text.
(b) Look at the table in Section 8.1. You should already be familiar with integrals 1-15 (especially
1-8) from Math 152, and you should know 16-20 by the end of the week. If you have trouble
with any of these (or with any of the examples in that section), please review the material
from last term. You will need all of it for this term.
1. Determine the exact value:
√
(a) (Ex 7.7.1b) arcsin(− 3/2)
h
(c) (Ex 7.7.4b) sec arccos − 21
(e) (Ex 7.7.6b) arctan tan
(g) (Ex 7.7.8b) sec arctan
h
(b) (Ex 7.7.2a) arcsec(2)
i
11π
4
(d) (Ex 7.7.6a) arcsin sin
i
(f) (Ex 7.7.8a) cos arcsin
h i
4
3
h
11π
6
i
h i
3
5
(h) (Ex 7.7.9b) cos 2 arcsin
h i
4
5
2. Evaluate:
Z 1
(a) (Ex 7.7.39)
0
Z 8
(c) (Ex 7.7.44)
5
Z 5
(e)
4
Z 1
dx
1 + x2
dx
√
x x2 − 16
(b) (Ex 7.7.42)
0
Z 5
(d) (Ex 7.7.46)
2
dx
√
(x − 3) x2 − 6x + 8
dx
4 − x2
dx
9 + (x − 2)2
√
Z ln 3
(f) (Ex 7.7.50)
ln 2
√
e−x
1 − e−2x
3. Calculate:
x
dx
1 + x4
Z
sinh ax
(c) (Ex 7.8.37)
dx
cosh ax
Z
Z
(b) (Ex 7.7.58)
(a) (Ex 7.7.55)
cos x
dx
3 + sin2 x
4. Differentiate:
√
x
√
(c) (Ex 7.7.25) θ = arcsin( 1 − r2 )
(b) (Ex 7.7.13) y = arcsec(2x2 )
(a) (Ex 7.7.12) y = arctan
(e) (Ex 7.8.12) y = cosh
ln x3
(d) (Ex 7.7.29) y = sin [arcsec(ln x)]
(f) (Ex 7.8.18) y = xcosh x
1
5. Verify the identities:
(b) (Ex 7.8.20) sinh(t + s) = sinh t cosh s + cosh t sinh s
(b) (Ex 7.8.21) cosh(t + s) = cosh t cosh s + sinh t sinh s
(c) (Ex 7.8.22) sinh(2t) = 2 sinh t cosh t
tanh t + tanh s
(d) (Ex 7.9.14) tanh(t + s) =
(Hint: Use parts (a) and (b).)
1 + tanh t tanh s
(e) (*) Think of some more trig identities. Do they also correspond to identities for hyperbolic
trig functions?
6. (Ex 7.8.29) Show that for each positive integer n,
(cosh x + sinh x)n = cosh nx + sinh nx.
(Hint: What is cosh y + sinh y?)
7. (Ex 7.8.26) Find the absolute extreme values of y = 5 cosh x + 4 sinh x.
8. (Ex 7.9.19)
(a) Show that sinh x is one-to-one.
√
(b) Show that sinh−1 x = ln(x + x2 + 1)
d 1
(c) Show that
sinh−1 x = √
.
dx
1 + x2
9.
(Ex 7.7.34) Take x > 0, so that arctan x is a
positive (acute) angle. We can then construct
an angle of arctan x by drawing a right triangle
with legs of length x and 1, as shown. Use this
to compute the following:
x
arctan x
1
(a) tan(arctan x)
(b) cot(arctan x)
(c) sin(arctan x)
(d) cos(arctan x)
(e) sec(arctan x)
(f) csc(arctan x)
10. Sketch graphs of the following functions, on the interval [−2π, 2π]:
(a) arcsin(sin x)
(b) arctan(tan x)
(c) arcsec(sec x)
11. (*) (Ex 7.9.44) An object of mass m is falling through the atmosphere due to gravity. The force
exerted on it by air resistance is proportional to the square of the velocity.
dv
= mg − kv 2 where g is the gravitational
dt
constant, and k > 0 is the proportionality constant. (Hint: Remember Newton’s second law:
F = ma = m dv
dt . The only forces acting on the object are gravity and air resistance.)
(a) If v(t) is the velocity at time t, show that m
r
(b) Show that v(t) =
s
mg
tanh 
k

gk 
t is a solution to this equation with v(0) = 0.
m
(c) Compute lim v(t). This is called the terminal velocity of the object.
t→∞
2