Name: solutions
Discussion Section Number: all
Homework 4 — Due 10/4/2010 in class
This cover sheet must be attached as the top page of your homework.
1. Solve for x:
log3 x + log3 (2x + 1) = 1
2. Evaluate the following limits:
(a) =
lim e−x
3
x→0
lim x2 e−x
(b)
x→1
3. Show using the limit definition of the derivative that
d
(cos x) = − sin x
dx
4. Find the derivatives of the following functions:
f (x) = x3 sec x
√
s2 − s
g(s) =
3s
x2 + 3
h(x) =
x3 + 7
F (x) = ln x
G(t) = et tan t + cos t
5. For f (x) = x4 − 3x3 + 5x2 − x + 1, find f 0 (x), f 00 (x), f 000 (x), f (4) (x), and f (5) (x).
6. For g(α) = cos α, find g 0 (α), g 00 (α), g 000 (α), and g (4) (α).
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1. Solve for x:
log3 x + log3 (2x + 1) = 1
log3 x + log3 (2x + 1) = log3 3
log3 [x(2x + 1)] = log3 3
x(2x + 1) = 3
2x2 + x − 3 = 0
(2x + 3)(x − 1) = 0
3
x = − , 1
2
2. Evaluate the following limits:
(a) =
lim e−x
3
x→0
lim x2 e−x
(b)
x→1
(a)
1
1
=1
3 =
x
x→0 e
e0
3
lim e−x = lim
x→0
(b)
x2
1
1
= 1 =
x
x→1 e
e
e
lim x2 e−x = lim
x→1
3. Show using the limit definition of the derivative that
d
(cos x) = − sin x
dx
d
cos(x + ∆x) − cos x
(cos x) = lim
∆x→0
dx
∆x
cos(x) cos(∆x) − sin(x) sin(∆x) − cos x
= lim
∆x→0
∆x
cos(x) (cos(∆x) − 1)
sin(x) sin(∆x)
= lim
− lim
∆x→0
∆x→0
∆x
∆x
cos(∆x) − 1
sin(∆x)
= cos(x) lim
− sin(x) lim
∆x→0
∆x→0
∆x
∆x
= 0 − sin(x)
= − sin(x)
2
4. Find the derivatives of the following functions:
f (x) = x3 sec x
f 0 (x) = 3x2 sec x + x3 sec x tan x
g(s) =
g 0 (s) =
=
h(x) =
h0 (x) =
=
√
s2 − s
3s
(3s) 2s − 12 s−1/2 − 3 s2 − s1/2
s2
√
+ 12 s
3s2
9s2
x2 + 3
x3 + 7
x3 + 7 (2x) − x2 + 3 3x2
(x3 + 7)2
−x4 − 9x2 + 14x
(x3 + 7)2
F (x) = ln x
1
F 0 (x) =
x
G(t) = et tan t + cos t
G0 (t) = et sec2 t + et tan t − sin t
5. For f (x) = x4 − 3x3 + 5x2 − x + 1, find f 0 (x), f 00 (x), f 000 (x), f (4) (x), and f (5) (x).
f (x) = x4 − 3x3 + 5x2 − x + 1
f 0 (x) = 4x3 − 9x2 + 10x − 1
f 00 (x) = 12x2 − 18x + 10
f 000 (x) = 24x − 18
f (4) (x) = 24
f (5) (x) = 0
6. For g(α) = cos α, find g 0 (α), g 00 (α), g 000 (α), and g (4) (α).
g(α) = cos α
g 0 (α) = − sin α
g 00 (α) = − cos α
g 000 (α) = sin α
g (4) (α) = cos α = g(α)
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