MATH 142 Business Math II, Week In Review
Spring, 2015, Problem Set 4 (4.1, 4.2, 4.3)
JoungDong Kim
1. Differentiate the following functions.
√
(a) f (x) = 2
(b) f (x) = 4π 2
(c) f (x) = x7
(d) f (x) = x3.1
(e) f (x) = x0.03
2
(f) f (x) = x 5
1
(g) f (x) = ex
3
1
√
(h) f (x) = 0.24 3 x
(i) f (x) = 4 − 3 ln x
(j) f (x) = 3ex − π 2 x
(k) f (x) = 3x2 + x − 1
√
1
(l) f (x) = π 2 x2 + √ x + 3 + 2 ln x
2
(m) f (x) = 0.02ex + 2x3 − 2x2 + 3x − 10
(n) f (x) = 1 − x4 − x8
(o) f (x) = ln 3 + ex − 0.001x2
2
(p) f (x) = 3ex − 0.003x4 + 0.01x3
x2 − 3x − 6
(q) f (x) =
3x
(r) f (x) =
1
+x
x
(s) f (x) = 3x3 +
3
x3
(t) f (x) = x−2 + x−3 + 3 ln
x
4
(u) f (t) = (2t)5 + et+2
3
(v) f (t) =
√
1
t− √
t
3
3
(w) f (u) = 2u 2 + 4u 4
(x) f (u) =
(y) y =
√
(u + 1)2
u
1
4x − √
9x
(z) y = x1.5 − x−1.5 + ln x−2
4
2. Find the equation of tangent line to f (x) = x−1 + x−2 + ln x at x = 1.
3. Find the value(s) of x where the tangent line is horizontal for f (x) = 7 + 12x − x3 .
4. The revenue function of x product is given by R(x) =
next item of 16 products.
5
√
x(x − 24). Estimate the revenue of the
5. Suppose the cost function is given by C(x) = 0.01x3 − x2 + 50x + 100, where x is the number of
items produced and C(x) is the cost in dollars to produce x items. Find the marginal cost for any
x.
6. Find the derivative of the following functions.
(a) f (x) =
√
xex
(b) f (x) = x4 ln x
(c) f (x) = (2x3 + 3) ln x
(d) y = (x3 − 3 ln x)(2ex + 3x)
1
1
x
(e) f (x) = e +
1+ 2
x
x
6
(f) f (x) =
ln x
+3
x2
(g) f (x) =
3
x+3
(h) f (x) =
3 − 2ex
1 − 2x
(i) y =
(j) y =
√
u
u2 + eu + 1
√
4
u2
u
+1
7
7. Find the derivative of following functions.
(a) y = (3x − 2)11
3
(b) y = (2ex − 3) 2
√
(c) f (x) = −15 3 2x3 − 3
(d) f (x) =
x2
4
+1
(e) f (x) = (2x − 3)5 (4x + 7)
8
(f) y = (1 − x3 )4 ln x
(g) y = (7x + 3)3 (x2 − 4)6
√
(h) y = e2x ln x
(i) y =
−3
(3ex + 1)3
(j) f (x) =
p
√
3
x+1
9
dy
if y =
8. Find
dx
qp
√
x+1+1
9. Suppose y = f (x) is a differentiable function and f ′ (4) = 7. Let h(x) = f (x2 ). Find h′ (2).
10