Chapter 7
7.1 Antiderivatives
1)Find the antiderivative of
2)Find
3)Find
4)Find
5)Find
6)Find
7)Find
8)Find .
9)Find .
10)Find
11)Find
12)Find
13)Find .
14)Find .
15)Find the equation of the curve whose slope is +4 and goes through the point (1,−2).
16)Suppose a publishing company has found that the marginal cost at a level of production of x thousand books is given by , and the fixed cost is $25,000. Find the cost function .
17)Suppose the marginal revenue from a product is given by , where q is the number of products produced. Find the revenue function for the product and the demand function for the product.
7.2 Substitution
18) Find .
19) Find .
20) Find .
21) Find .
22) Find .
23) Find .
24)Find .
25) Find .
26) Find .
27) Find .
28) Find .
29) The research department for a hardware store has determined that the marginal price of x boxes per week for an item is given by . Find the demand equation if the weekly demand for this type of item is 10 boxes when the price of a box is $4.
7.3 Area and Definite Integral
30) Estimate R3 (Right Riemann Sum) and L3 (Left Riemann Sum), under the graph of f(x) = x2 +1 from x = 0 and x = 3 using 3 rectangles. Draw the graphs with the rectangles clearly indicating each area.
31) Estimate R4 (Right Riemann Sum), L4(Left Riemann Sum), and M4 (Midpoint Riemann Sum) under the graph of from x = 0 and x = 2 using 4 rectangles. Draw the graphs with the rectangles clearly indicating each area.
32) Evaluate by using formulas from geometry.
33) Evaluate by using formulas from geometry.
7.4 The Fundamental Theorem of Calculus
34) Evaluate .
35) Evaluate .
36) Evaluate .
37) Evaluate .
38) Evaluate .
39) Evaluate .
40) Evaluate .
41) Find the area of the region between the x-axis and the graph of from x =0 to
x=1.
42) Find the area of the region between the x-axis and the graph of from x =0 to
x=2.
43) Find the area of the region between the x-axis and the graph of in the interval [-1,2].
44) Find the area of the region between the x-axis and the graph of in the interval [e-1,e].
7.5 The Area Between Two Curves
24) Find the area bounded by the curves , on [-1,1].
25) Find the area bounded by the curves .
Consumers’ Surplus
The consumers’ surplus is the difference between the equilibrium price of an item and the higher prices than individuals are willing to pay. It is the difference between what consumers are willing to pay and what they actually pay. This is a savings realized by the consumers.
Producer’s Surplus
The producers’ surplus is the difference between the equilibrium price of an item and the lower prices at which manufacturers are willing to sell their product. It is the difference between the price producers are willing to sell their product and the price they actually sell. This is considered added income to the manufacturer.
26) Suppose the price (in dollars per ton) for oat bran is , when the demand for the product is q tons. Also, suppose the function gives the price (in dollars per ton) when the supply is q tons. Find the consumers’ and producers’ surplus.
27) Suppose the price (in dollars per ton) for oat bran is , when the demand for the product is q tons. Also, suppose the function gives the price (in dollars per ton) when the supply is q tons. Find the consumers’ and producers’ surplus.
28) Suppose that the supply function (in dollars) for some item is given by , and the demand function (in dollars) is given by . Graph both functions, find the equilibrium point, and find the consumers’ and producers’ function.