Chapter 2
Sec,on 2.1 and 2.2
Intro and Frequency distribu,ons
Preview
Characteris,cs of Data
1. Center: A representative value that indicates
where the middle of the data set is located.
2. Variation: A measure of the amount that the da
Worksheet 1 Solution
1. At a selling price of $17 per book, a publisher is willing to publish 5
million copies of a book while the demand is 5.7 million copies. When
the selling price is increased to $19, the supply and demand change to
6 million and 5.3
Worksheet 5 Solutions
1. The demand equation for a quantity q of a product at p dollars is
p = 4000 5q. Companies producing the product report the cost, C,
in dollars, to produce a quantity q is C = 6q + 5. What production
level earns the company the larg
Worksheet 6 Solutions
The total cost C, in thousands of dollars, of a certain production is given
by the short-run Cobb-Douglas cost curve
C(q) = 2q 3/2 + 1 ,
where 0 q 10 is the number of items in hundreds.
1. What is the xed cost of the production?
Solu
Worksheet 4 Solutions
Consider the function f (x) = 1/(x2 1) in the interval 1/2 x 1/2.
1. Find all local maxima and minima of f (x).
f (x) =
2x
= 0 2x = 0 x = 0
1)2
(x2
Using the rst or second derivative test, you can conrm that this is a
local maximum.
Worksheet 2: Exponential functions
1. The half-life of radioactive carbon-14 is about 5730 years. A fossil is
found that has 35% carbon-14 compared to the living sample. How old
is the fossil? Solve using the base e and base a exponential functions.
Solut
Worksheet 3 Solutions
1. The Bay of Fundy in Canada has the largest tides in the world. The
dierence between low and high water levels is 15 meters (nearly 50
feet). At a particular point the depth of the water, y meters, is given
as a function of time, t
Math 162 Spring 2014
Test 1 Study Sheet
You may bring 1 page of handwritten notes to the test. You may NOT use a calculator!
1. Sketch the graph of (a) a function that has an inverse, and (b) a function that doesnt
have an inverse.
2. If f (a) = b, what i
Math 162 Spring 2014
Test 2 Study Sheet
You may bring 1 page (front and back) of handwritten notes to the test. You may use a
standard scientic calculator but you may NOT use a graphing calculator!
1. Compute the integrals:
x sec2 x dx
(a)
(b)
(x2
1
dx
2
Worksheet 2 Solutions
Solve the following equations:
1. y + 4y + 3y = 0 with y(0) = 2 and y (0) = 1.
The characteristic equation is r 2 +4r+3 = 0, and the two roots are r =
1 and r = 3. Therefore the general solution is y = c1 ex + c2 e3x .
Initial condit
Worksheet 3: Method of undetermined coecients
Consider the motion of a harmonic oscillator governed by the equation
x + 2x + 2x = F (t)
where x(t) is the displacement from the rest position of oscillator. F (t)
represents the external driving force applie
Worksheet 3 Solutions
Consider the motion of a harmonic oscillator governed by the equation
x + 2x + 2x = F (t)
where x(t) is the displacement from the rest position of oscillator. F (t)
represents the external driving force applied to the oscillator.
1.
Worksheet 2: Homogeneous, linear, second order
equations with constant coecients
Solve the following equations:
1. y + 4y + 3y = 0 with y(0) = 2 and y (0) = 1.
2. y + 2y + 8y = 0 with y(0) = 1 and y (0) = 0.
3. 9y 12y + 4y = 0 with y(0) = 2 and y (0) = 1.
Worksheet 1: First order equations
For the following equations, determine whether they are linear or nonlinear
and then solve each of them.
1. xy + 2y = sin x.
2. y = x2 /y.
3. 2xy 2 + 2y + 2x2 y + 2x y = 0.
1
Worksheet 4 solutions
Solve y + 2 y = cos 2t where 2 = 4 with Laplace transform. The initial
conditions are y(0) = 1 and y (0) = 0.
Transforming the equation and applying the initial conditions gives
s
s+4
s
2
2
+s
(s + s)Y = 2
s +4
1
s
s
2 5
Y = 2
+ 2
2
Worksheet 5: Delta function
Consider the harmonic oscillator with an impulsive force y + y = (t 1).
The initial conditions are y(0) = 0 and y (0) = 0.
Solve this problem using the Laplace transform.
The homogeneous solution of the problem is yh = c1 cos
Worksheet 6: Eulers equation
1. Solve x2 y + 4xy + 2y = 0 for x > 0. What happens to y1 and y2 when
x approaches zero?
2. Solve 2x2 y 4xy + 6y = 0 for x > 0. What happens to y1 and y2
when x approaches zero?
3. Solve x2 y 3xy + 4y = 0 for x > 0. What happ
Consider the cost function C(q) = 0.3q 3 0.8q 2 + 4q + 8 and revenue
function R(q) = 6q 2q 2 .
1. Sketch the graph of y = f (x).
2. Estimate f (1) by using a point at x = 1 and a point at x = 1.2.
3. Estimate f (1) by using a point at x = 1 and a point at
Find the rst and second derivatives of the following functions.
1. y = 5x3 + 7x2 3x + 1
2. y = 6x2 + 3x3 4x1/2
3. y = 8 ln(2x + 1)
4. y =
e2x
x2 +1
5. y =
x2 +3x+2
x+1
6. y =
x2 +2
3
2
7. y = ln(sin x + cos x)
8. y =
x3
9 (3 ln x
9. y =
x2 + x+1
x3/2
1)
Keplers third law of planetary motion states that P 2 = kd3 , where P
represents the time, in earth days, it takes a planet to orbit the sun once, d
is the planets average distance, in miles, from the sun, and k is a constant.
1. If k = 1.65864 1019 , wri
1. Locate the local maxima, local minima, and the points of inection of
2
the Gaussian function f (x) = ex . Sketch the graph of this function.
4
2. Now consider the super Gaussian function f (x) = ex . Locate the
local maxima, local minima, and the point
Consider the function f (x) =
1
.
x2 +1
1. Sketch the graph of y = f (x).
2. Estimate f (1) by using a point at x = 1 and a point at x = 1.2.
3. Estimate f (1) by using a point at x = 1 and a point at x = 0.8.
4. Estimate f (1) by taking the average of th
1. A person breathes in and out every three seconds. The volume of air in
the persons lungs varies between a minimum of 2 liters and a maximum
of 4 liters. Express the volume of air in the persons lungs in the form
y = A sin t + C.
2. Spring-mass system
T