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Calculus and Its Applications

Marvin L. Bittinger, David J. Ellenbogen, Scott A. Surgent

Chapter 5

Applications of Integration - all with Video Answers

Educators

Section 1

An Economics Application: Consumer Surplus and Producer Surplus

08:08

Problem 1

In each $D(x)$ is the price, in dollars per unit, that consumers are willing to pay for $x$ units of an item, and $S(x)$ is the price, in dollars per unit, that producers are willing to accept for $x$ units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.
$$D(x)=-\frac{5}{6} x+9, \quad S(x)=\frac{1}{2} x+1$$

Ahmad Reda
Ahmad Reda
Numerade Educator
06:34

Problem 2

In each $D(x)$ is the price, in dollars per unit, that consumers are willing to pay for $x$ units of an item, and $S(x)$ is the price, in dollars per unit, that producers are willing to accept for $x$ units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.
$$D(x)=-3 x+7, \quad S(x)=2 x+2$$

Ahmad Reda
Ahmad Reda
Numerade Educator
06:58

Problem 3

In each $D(x)$ is the price, in dollars per unit, that consumers are willing to pay for $x$ units of an item, and $S(x)$ is the price, in dollars per unit, that producers are willing to accept for $x$ units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.
$$D(x)=(x-4)^{2}, \quad S(x)=x^{2}+2 x+6$$

Ahmad Reda
Ahmad Reda
Numerade Educator
06:09

Problem 4

In each $D(x)$ is the price, in dollars per unit, that consumers are willing to pay for $x$ units of an item, and $S(x)$ is the price, in dollars per unit, that producers are willing to accept for $x$ units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.
$$D(x)=(x-3)^{2}, \quad S(x)=x^{2}+2 x+1$$

Ahmad Reda
Ahmad Reda
Numerade Educator
05:17

Problem 5

In each $D(x)$ is the price, in dollars per unit, that consumers are willing to pay for $x$ units of an item, and $S(x)$ is the price, in dollars per unit, that producers are willing to accept for $x$ units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.
$$D(x)=(x-6)^{2}, \quad S(x)=x^{2}$$

Ahmad Reda
Ahmad Reda
Numerade Educator
05:09

Problem 6

In each $D(x)$ is the price, in dollars per unit, that consumers are willing to pay for $x$ units of an item, and $S(x)$ is the price, in dollars per unit, that producers are willing to accept for $x$ units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.
$$D(x)=(x-8)^{2}, \quad S(x)=x^{2}$$

Ahmad Reda
Ahmad Reda
Numerade Educator
03:59

Problem 7

In each $D(x)$ is the price, in dollars per unit, that consumers are willing to pay for $x$ units of an item, and $S(x)$ is the price, in dollars per unit, that producers are willing to accept for $x$ units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.
$$D(x)=1000-10 x, \quad s(x)=250+5 x$$

Ahmad Reda
Ahmad Reda
Numerade Educator
04:37

Problem 8

In each $D(x)$ is the price, in dollars per unit, that consumers are willing to pay for $x$ units of an item, and $S(x)$ is the price, in dollars per unit, that producers are willing to accept for $x$ units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.
$$D(x)=8800-30 x, \quad S(x)=7000+15 x$$

Ahmad Reda
Ahmad Reda
Numerade Educator
07:21

Problem 9

In each $D(x)$ is the price, in dollars per unit, that consumers are willing to pay for $x$ units of an item, and $S(x)$ is the price, in dollars per unit, that producers are willing to accept for $x$ units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.
$$D(x)=5-x, \text { for } 0 \leq x \leq 5 ; \quad S(x)=\sqrt{x+7}$$

Ahmad Reda
Ahmad Reda
Numerade Educator
06:55

Problem 10

In each $D(x)$ is the price, in dollars per unit, that consumers are willing to pay for $x$ units of an item, and $S(x)$ is the price, in dollars per unit, that producers are willing to accept for $x$ units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.
$$D(x)=7-x, \text { for } 0 \leq x \leq 7 ; \quad S(x)=2 \sqrt{x+1}$$

Ahmad Reda
Ahmad Reda
Numerade Educator
05:57

Problem 11

In each $D(x)$ is the price, in dollars per unit, that consumers are willing to pay for $x$ units of an item, and $S(x)$ is the price, in dollars per unit, that producers are willing to accept for $x$ units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.
$$D(x)=\frac{100}{\sqrt{x}}, \quad S(x)=\sqrt{x}$$

Ahmad Reda
Ahmad Reda
Numerade Educator
08:39

Problem 12

In each $D(x)$ is the price, in dollars per unit, that consumers are willing to pay for $x$ units of an item, and $S(x)$ is the price, in dollars per unit, that producers are willing to accept for $x$ units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.
$$D(x)=\frac{1800}{\sqrt{x+1}}, \quad S(x)=2 \sqrt{x+1}$$

Ahmad Reda
Ahmad Reda
Numerade Educator
06:34

Problem 13

In each $D(x)$ is the price, in dollars per unit, that consumers are willing to pay for $x$ units of an item, and $S(x)$ is the price, in dollars per unit, that producers are willing to accept for $x$ units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.
$$D(x)=(x-4)^{2}, \quad S(x)=x^{2}+2 x+8$$

Ahmad Reda
Ahmad Reda
Numerade Educator
07:01

Problem 14

In each $D(x)$ is the price, in dollars per unit, that consumers are willing to pay for $x$ units of an item, and $S(x)$ is the price, in dollars per unit, that producers are willing to accept for $x$ units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.
$$D(x)=13-x, \text { for } 0 \leq x \leq 13 ; \quad S(x)=\sqrt{x+17}$$

Ahmad Reda
Ahmad Reda
Numerade Educator
07:28

Problem 15

Follow the directions given for Exercises $1-14$.
$$D(x)=e^{-x+4.5}, \quad S(x)=e^{x-5.5}$$

Ahmad Reda
Ahmad Reda
Numerade Educator
06:40

Problem 16

Follow the directions given for Exercises $1-14$.
$$D(x)=\sqrt{56-x}, \quad S(x)=x$$

Ahmad Reda
Ahmad Reda
Numerade Educator
03:12

Problem 17

Explain why both consumers and producers feel good when consumer and producer surpluses exist.

Ahmad Reda
Ahmad Reda
Numerade Educator
03:02

Problem 18

Do some research on consumer and producer surpluses in an economics book. Write a brief description.

Ahmad Reda
Ahmad Reda
Numerade Educator
04:27

Problem 19

a) Find the equilibrium point using the INTERSECT feature or another feature that will allow you to find this point of
intersection.
b) Graph $$y=D\left(x_{E}\right)$$ and determine the regions of both consumer and producer surpluses.
c) Find the consumer surplus.
$$D(x)=\frac{x+8}{x+1}, \quad S(x)=\frac{x^{2}+4}{20}$$
d) Find the producer surplus.

Ahmad Reda
Ahmad Reda
Numerade Educator
03:50

Problem 20

Graph each pair of demand and supply functions. Then:
a) Find the equilibrium point using the INTERSECT feature or another feature that will allow you to find this point of
intersection.
b) Graph $$y=D\left(x_{E}\right)$$ and determine the regions of both consumer and producer surpluses.
c) Find the consumer surplus.
d) Find the producer surplus.
$$D(x)=15-\frac{1}{3} x, \quad S(x)=2 \sqrt[3]{x}$$

Ahmad Reda
Ahmad Reda
Numerade Educator
04:39

Problem 21

Regina loves to go bungee jumping. The table shows the number of half-hours that Regina is willing to go bungee jumping at various prices.
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(TABLE CANNOT COPY)
a) Make a scatterplot of the data, and determine the type of function that you think fits best.
b) Fit that function to the data using REGRESSION.
c) If Regina goes bungee jumping for 6 half-hours per month, what is her consumer surplus?
d) At a price of $\$ 11.50$ per half-hour, what is Regina's consumer surplus?

Ahmad Reda
Ahmad Reda
Numerade Educator