Answers to HW4 problems
Problem 1.
(a) log4 (64) = 3
(b) 3log10 (10000) = 81
(c) 343log1000 (10) = 7
(d) log2048 (236 ) =
36
11
Problem 2.
(a) x = 52 log4 (49)
(b) x = 34
log3 (4)−1/2 −1
ln
Problem 3. x =
ln(7)
ln( ln(3) )
ln(5)
ln(2)
Problem 4. a ≥ 1 and a ≤ −1
1
Problem 5. (g −1 )0 (3) = p
√
1 + 1 + a2 +
a2
√
√
√
2
2 1+a
1+ 1+a2
Problem 6. 90 degrees or π/2 radians
2
2
(3a + 2a2 )ea − (3a + 2a2 − a3 )e2a
h00 (a)
Problem 7. (h ) (2) = − 0 3 = −
3
h (a)
(1 + (1 + a2 )ea2 )
−1 00
Problem 8. g 00 (x) = 30e−5x sin(3x) + 16e−5x cos(3x)
Problem 9. f 0 (x) =
1
x log(x) log log(x)
8
d 2x4x8 4x8 −1
4x8
7
7
2x4x
2x
+ 2x log(x) 4x + 32x log(x)
=x
Problem 10.
x
dx
√
Problem 11. tan(arccos(x)) =
1 − x2
,
x
cos(arctan(x)) = √
Problem 12. We see
√
d
1
1
arcsin
+ arctan
x2 − 1
=− q
dx
x
x2 1 −
+
1
x2
1
1 + x2
x
√
(1 + (x2 − 1)) x2 − 1
1
1
+ √
=0
= −√
4
2
x −x
x x2 − 1
which shows that the function is a constant C. Plugging in x = 1 gives C = π2 .
2