Homework 6
1
Calculus
Homework 6
Due Date: November 21 (Wednesday)
1. Reading assignments
(a) 3.3 Concavity and the Second-Derivative Test
(b) 3.4 Optimization Problems
(c) 3.5 Curve Sketching: A Summary
2. Find the second and the third derivative of the following functions.
4
(a) g(x) = − (x+2)
2
(b) g(x) = 5x(x + 3)3
Solution:
(a)
(b)
3. Find the slope of the graph of the function at the given point.
(a) Equation: x2 y + y 2 x = −2. Point: (2, -1).
(b) Equation: (x2 + y 2 )2 = 8x2 y. Point: (2, 2).
Solution:
2
Calculus H415611 Fall 2012
(a)
(b)
4. Find the critical numbers and the open intervals on which the function is increasing or
decreasing. Besides, sketch the graph of the function.
(a) f (x) = x2 + 8x + 10
(b) f (x) = x2/3 − 4
(c) f (x) =
x2
x+4
(d) f (x) =
−x3 + 1, x ≤ 0;
−x2 + 2x, x > 0.
Solution:
(a)
Homework 6
3
(b)
(c)
4
Calculus H415611 Fall 2012
(d)
5. Find all relative extrema of the function.
(a) f (x) = −4x2 + 4x + 1
(b) f (x) = 3x − 36x1/3
(c) f (x) = 2(x − 3)3
(d) f (x) = x +
1
x
(e) f (x) = x4 − 32x + 4
Solution:
(a)
(b)
Homework 6
5
(c)
(d)
(e)
6
Calculus H415611 Fall 2012
6. Find the absolute extrema of the function on the given closed interval.
(a) f (x) = x2 + 2x − 4, [-1, 1].
t
(b) f (x) = t−2
, [3,5]
(c) f (x) = 4 x + x1 + x12 , [-4, 5]
(d) f (x) = 8 − x24x+1 , [0, ∞)
Solution:
(a)
(b)
(c)
Homework 6
7
(d)
7. A retailer determines that the cost C of ordering and storing x units of a product can be
modeled by
20, 000
C = 3x +
, 0 < x ≤ 200.
x
The delivery truck can carry at most 200 units per order. Find the order size that will
minimize the cost. Use a graphing utility to verify your result.
Solution:
8. When soft drinks are sold for $1.00 per can at football games, approximately 6000 cans
are sold. When the price is raised to $1.20 per can, the quantity demanded drops to
5600. The initial cost is $5000 and the cost per unit is $0.50. Assuming that the demand
function is linear, what number of units and what price will yield a maximum profit?
Solution:
8
Calculus H415611 Fall 2012
9. A home gardener estimates that 16 apples trees will produce an average yield of 80 apples
per tree. But because of the size of the garden, for each additional planted, the yield will
decreasing by four apples per tree. How many trees should be planted to maximize the
total yield of apples? What is the maximum yield?
Solution: