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
MATH 151: Applied Calculus I
Study Guide 2
The test is 45 minutes in-class. You are allowed to bring one sheet of notes
(8.5 11). Cell phones, iPads, iPods, etc. cannot be used as calculators.
The test will cover all the topics listed below.
1. 2.3 Interp
Worksheet 1
1. The sporting goods company FootLocker sells mens Jordan 3D Wristband at $5.99 each. John wants to order a certain quantity of wristbands online and re-sell them at a higher price in order to earn money.
(a) For priority mail via USPS, the t