True/False Review Problems from Chapter 7
Calculus 2
Determine whether each of the statements are True or False. The answers are included
below.
3
1. The inverse function of f (x) = e1+ln(x
−1)
is g(x) = (1 + x/e)1/3 .
2. The domain of the function f in problem 1 is the interval (1, ∞).
3. The range of the function f in problem 1 is the interval (1, ∞).
R1
4. 2 t−1 dt = ln(1/2).
5.
d
dx {ln(f (x))}
6.
d
f (x)
}
dx {e
=
= ef
0
f 0 (x)
f (x) .
(x)
.
7. A number M exists such that ln x ≤ M for all x > 0.
8. The derivative of ln(2x) with respect to x is 1/x.
√
R x
9.
1 + x2 ) + C.
dx
=
ln(
2
1+x
10. Let f be a one-to-one function. If (a, b) is a point on the graph of y = f (x) then (b, a) is
a point on the graph of y = f −1 (x).
11. exp(−x) = −exp(x) for all x.
R ln 5
12. 0 e2x dx = 12.
13. 2π = e2 ln(π) .
14. 2π = eπ ln(2) .
15. log10 (e) = 1/ ln(10)
R π/4
16. 0 tan(x) dx = − ln 2
17. The derivative of 2x with respect to x is 2x .
18. arctan(−1) = 3π/4.
19. arctan(x) = arcsin(x)/ arccos(x)
20. cos(sin−1 (x)) is always greater or equal to 0.
√
21. If f (x) = arctan( x) then f 0 (1) = 1/2.
Z 2
1
π
22.
dx =
2
2
−2 4 + x
23. tan−1 (tan x) = x for all x in the domain of the tangent function.
2
24. If f (x) = sin−1 (cos x), then f 0 (4) = −1.
R
√1
25.
dx = sec−1 (x) + C
x x2 −1
26. The function f (x) = sin(x), 0 ≤ x ≤ π is one-to-one.
27. If a < b then ea < eb .
28. If a > 0 then ln(a) > 0.
29. You can always divide by ex .
30. If x > 0 then (ln x)8 = 8 ln(x).
31. If x < 0 then ln(x8 ) = 8 ln(−x).
32.
d
dx {ln(10)}
33.
d
x
dx {10 }
=
1
10 .
= x10x−1 .
x
34. The function f (x) = ln(ln(x)) is one-to-one and its inverse function is g(x) = ee .
35. log5 (x) = ln(x)/ ln(5)
36. The domain of the function f (x) = (x + 1)x is the interval (0, ∞).
2
37. The range of the function f (x) = ex is the interval [1, ∞).
R1
38. −e x1 dx = −1.
39. If a > 0 and b > 0 then ln(ab) = ln(a) + ln(b).
40. If a > 0 and b > 0 then ln(a + b) = ln(a) ln(b).
Z 8
1
41.
dx = 2 ln 2
2 x
Z ex
dt
42.
=x
t
1
ANSWERS:
1. True
2. True (ln(x3 − 1) si only defined if x > 1.)
3. False (The range of f equals the domain of g = f −1 which is (−∞, ∞))
4. True
5. True
6. False (The derivative of ef (x) would equal f 0 (x)ef (x) .)
7. False
8. True
9. True (Use the substitution u = 1 + x2 .)
10. True
11. False
3
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
True
False
True
True (In general, loga (b) = ln(b)/ ln(a).)
False
d
False (In general, dx
{ax } = ln(a)ax for a > 0.)
False (By definition arctan(x) is always between −π/2 and π/2, so arctan(−1) = −π/4
False (Try x = 1: the left-hand side is π/4 but the right-hand side is undefined.)
True (Because sin−1 (x) is always between −π/2 and π/2, and the cosine of an angle
between −π/2 and π/2 is always greater or equal to 0.)
False (f 0 (1) = 1/4.)
False (The integral evaluates to π/4.)
False (example: tan−1 (tan 3π/4) = −π/4.)
False (f 0 (4) = 1.)
True
False
True
False
True (ex is always strictly larger than 0.)
False
True
d
False (Since ln(10) is a constant dx
{ln(10)} = 0.)
False (See problem 17.)
True
True
False (The domain is the interval (−1, ∞).)
True (The function f has an absolute minimum value of 1 at x = 0.)
R −1
False (The integral is undefined. Note that −e x1 dx = −1 though.)
True (One of the elementary laws of logarithms!)
False
True
True