Homework 4
1
Calculus
Homework 4
Due Date: November 7 (Wednesday)
1. Reading assignments
(a) 2.3 Rates of Changes: Velocity and Marginals
(b) 2.7 Implicit Differentiation
(c) 2.8 Related Rates
(d) 3.1 Increasing and Decreasing Functions
(e) 3.2 Extrema and the First-Derivative Test
2. The profit P (in dollars) from selling x units of a product is given by
√
1
P = 36, 000 + 2048 x − 2 ,
8x
150 ≤ x ≤ 275.
Find the marginal profit for each of the following sales.
(a) x = 150
(b) x = 200
(c) x = 300. (It does not make any sense when x = 300.)
Solution:
3. The profit P (in dollars) from selling x laptop computers is given by
P = −0.04x2 + 25x − 1500.
2
Calculus H415611 Fall 2012
(a) Find the additional profit when the sales increase form 150 to 151 units.
(b) Find the marginal profit when x = 150.
(c) Compare the results from parts (a) and (b).
Solution:
4. Finding the derivative of the function.
(a) f (x) = (x3 − 3x)(2x2 + 3x + 5)
(b) f (x) =
x3 +3x+2
x2 −1
(c) f (x) = (3x3 + 4x)(x − 5)(x + 1)
(d) f (x) =
(x+1)(2x−7)
2x+1
(e) f (x) = (2x3 − 1)2
(f) f (x) =
x2 −2x+5
√
x
(g) f (x) =
6−5x 2
2
x −1
(h) f (x) =
q
2x
x+1
5. The ordering and transportation cost C per unit (in thousands of dollars) of the components used in manufacturing a product is given by
C = 100
200
x
+
2
x
x + 30
where x is the order size (in hundreds). Find the rate of change of C with respect to x
for each order size. What do these rates of change imply about increasing the size of an
order? Of the given order sizes, which would you choose? Explain
(a) x = 10
(b) x = 20
(c) x = 30
Solution:
Homework 4
3
6. The value V of a machine t years after it is purchased is inversely proportional to the
square root of t + 1. The initial value of the machine is $10,000.
(a) Write V as a function of t.
(b) Find the rate of depreciation when t = 1.
Solution: