Homework 3 – Additional Exercises
Problems 1 – 5 regard general logarithmic functions (see section 6.4*), problems 6 – 12 regard
inverse trigonometric functions (see section 6.6), and problems 13 – 18 are miscellaneous review
problems on chapter 6 topics.
1.
If a is a positive number and a ≠ 1 , how is log a x defined?
What is the domain of the function f ( x ) = log a x (where a > 0 and a ≠ 1 ).
What is the range of this function?
If a > 1, sketch the general shapes of the graphs of
y = log a x and y = a x with a common set of axes.
(e) If 0 < a < 1, sketch the general shapes of the graphs of
y = log a x and y = a x with a common set of axes.
(a)
(b)
(c)
(d)
⎛1⎞
⎝a⎠
3. Evaluate the integral:
∫
(b) 10(log10 4 + log10 7 ) .
(a) log a ⎜ ⎟
2. Evaluate the expression:
log10 x
dx .
x
⎛
⎝
1⎞
x⎠
4.
Find the inverse function of f ( x ) = log10 ⎜1 + ⎟ .
5.
Calculate the limit:
6.
Find f ′ . Also, indicate the domain of f and that of f ′ given that
f ( x ) = arcsin( e x ) .
7.
Find y ′ if tan −1 ( x y ) = 1 + x 2 y .
lim x − ln x .
x→ + ∞
8. A lighthouse is located on a small island, 3 km away from the nearest point P
on a straight shoreline, and its light makes four revolutions per minute. How fast
is the beam of light moving along the shoreline when it is 1 km from P ?
1+ x
9. Evaluate the integral:
∫ 1+ x
10. Evaluate the integral:
∫
11. Evaluate the integral:
∫
2
dx .
1
x x2 − 4
e2 x
1 − e4 x
dx .
dx .
π
⎛ x −1 ⎞
.
⎟ = 2 arctan x −
1
2
x
+
⎝
⎠
12. Prove the identity: arcsin ⎜
1
13. Evaluate the integral:
∫
ex
dx .
1 + e2 x
∫
tan x ⋅ ln (cos x) dx .
0
14. Evaluate the integral:
15. Use properties of integrals to prove the inequality:
1
∫
1 + e 2 x dx > e – 1 .
0
2x
16. Find f ′ ( x ) if f ( x ) =
∫e
( −t 2 )
dt .
ln x
17. Show that cos{ arctan [ sin ( arccot x ) ] } =
18. If f is a continuous function such that
x
∫
0
x
f ( t ) dt = xe 2 x +
∫e
−t
f ( t ) dt for all x,
0
find an explicit formula for f ( x ) .
x2 + 1
x2 + 2
.