Chapter 2
The limit does not exist if there is an open hole on the graph. (True or False)
False
A plastic straw manufacturer determines that its weekly cost, in dollars, of producing x plastic straws is given by C(x)=0.02x2+0.46x+6.31, 0≤ x≤ 200.
Find C(13)-C(6)/13-6, and interpret this result.
On average, each additional straw produced between 6 and 13 costs about $0.84.
Draw a graph example of each of the following:
a) Concave of and concave down
b) Increase and decrease
c) inflection point
Drawing
A function is continuous at x=a if...
1. f(a) exists
2. lim x->a f(x) exists
3. f(a) = lim x->a f(x)
For f(x)=x2+2x-5, find f'(x) using the difference quotient. Additionally, find the equation of the tangent line at x=2.
f'(x) = 2x+2
The tangent line at x=2 is y=2x-3.
For the following demand function x=D(p)=√(150-4p) state the type of elasticity at p=75. Should the price be lowered or raised in order to increase revenue? Additionally, at what price is the total revenue maximum?
Inelastic at x=75. The price should be raised. Lastly, the total revenue is maximized at $25.
F(x)= x-5 if x≤8, ∛x if x>8
What is limx->8-F(x) and limx->8+F(x)?
Is the function continuous?
limx->8-F(x)=3
limx->8+F(x)=2
The function is not continuous at x = 8 because the limit does not exist.
Find the second derivative of f(x)=2x8+xe-2ex.
f''(x) = 112x6+x(e-1)ee-2-2ex
How do you find the inflection point and critical numbers?
Inflection point: First, find f''(x)=0 for x values, then plug in the values for the y value. Build a coordinate
Critical Numbers: Find f'(x)=0 and where f'(x)=DNE.
Find limx->2 √(x+7).
3
Find the points on the curve f(x)=2x3+3x2-12x+1 where the tangent line is horizontal.
(1, 6) and (-2, 21)
At a price of $40 for a two-hour harbor tour, a water-tower company attracts 200 customers. For every $1 decrease in price, an additional 5 customers are attracted. The cost of carrying x customers in the tour is c(x)=3000+5x+0.01x2.
Find the # of customers the company should attract in order to maxprofit. Find the price for a two-hour harbor tour the company should change in order to max profit.
179 customers at the price of $44.20 would max profit for the company.
Find the limit, if it exists: limx->0 ((1/(x+h))-1/x)/h
-1/(X2)
Differentiate the following
a) y=4x1/2ex+2/x
b) y=xe+3xe6x-4
c) y=ln(2x3-x2)2
a) y'=2exx-1/2+4x1/2ex
b) y'=exe-1+3e6x-4+18xe6x-4
c) y'=(2(2x3-x2)(6x2-2x))/(2x3-x2)2
A motel owner observes that when a room is priced at $60 per day, all 80 rooms of the motel are occupied. For every $3 rise in the charge per room per day, one more room is vacant. Each occupied room costs $15 per day to maintain. Find the price per room per day the motel should charge in order to maximize its profit.
48 rooms to maximize, thus the motel should charge $6,768 per room per day to maximize profits.