Math 103. Homework 5. Solutions

1. [Exercise 8, page 171]

(a) This occurs at t=0, the slope is zero there.

(b) This occurs at t3. As you move along the graph, the slope is more and more negative; it does not exist at t3.

(c) This occurs at t1, where the two graphs meet.

(d) This occurs at t2 because the slope of the tangent line to the graph of f is greatest at the point (t2, f(t2)).

2. [Exercise 26 (a)(b), page 171]

(a) Use the four step process

Step 1.


f(x+h) = 1
(x+h)-1
 = 1
x+h-1

Step 2.


f(x+h)-f(x) = 1
x+h-1
 - 1
x-1
 = x-1-(x+h-1)
(x+h-1)(x-1)
 = -h
(x+h-1)(x-1)

Step 3.


f(x+h)-f(x)
h
 = -1
(x+h-1)(x-1)

Step 4.


f¢(x) =
lim
h® 0 
 -1
(x+h-1)(x-1)
 = -1
(x-1)2

(b) The slope is f¢(-1) = -1/4. So an equation is


y -(-1/2) = -(1/4) (x+1)        or        y = -(1/4)x -3/4

3. [Exercise 34, page 172]

(a) Use the four step process and obtain C¢(x) = -20 x+300. Here are the details:

Step 1. C(x+h) = -10(x+h)2 +300(x+h) +130 = -10 x2 -20 xh -10 h2 +300x +300 h +130

Step 2. C(x+h)-C(x) = (-10 x2 -20 xh -10 h2 +300x +300 h +130) - (-10 x2 +300 x+130) = -20 xh -10 h2 +300 h = h(-20 x-10 h+300)

Step 3.


C(x+h)-C(x)
h
 = h(-20 x-10 h+300)
h
 = -20 x-10 h+300

Step 4.


C¢(x) =
lim
h® 0 
 ( -20 x-10 h+300) = -20 x+300

(b) The rate of change of the total cost when the level of production is ten surfboards/day is C'(10)= -20(10)+300= 100 or $100.

(c) The average cost is


C(10)-C(0)
10
 = (-1000+3000+130 ) -130
10
 =200
or $200.

4. [Exercise 42, page 173]


f(a+h)-f(a)
h

gives the average rate of change of the cost incurred in producing the commodity over the production level [a,a+h]



lim
h® 0 
 f(a+h)-f(a)
h

gives the instantaneous rate of change of the cost of producing the commodity at x=a.


5. [Exercise 44 and 46, page 173]

44(a) f has a limit at x=a.

44(b) f is continuous at x=a.

44(c) f is differentiable at x=a.

46(a) f does not have a limit at x=a because the left-hand and the right-hand limits are not equal.

46(b) f is not continuous at x=a because the limit does not exist there.

46(c) f is not differentiable at x=a because it is not continuous at x=a.