Karan Patel, ID # 588379
Hirvi Dalal, ID # 597367
Christopher Varghese, ID # 592957
Mathematical Analysis Lab , MA 102, Edward Reimers
Problem 1/Equation 1:
a. Initial amount of medicine is 3000 mcg
b. At t=2 hour
Rushabh D. Lagdiwala ID#: 0621704
0.1 10.16568 1.016568
0.1 10.22558 1.022558
Rushabh D. Lagdiwala
a. Initial amount of medicine is 3000 mcg
Double Integrals over Rectangles
Let y = f (x) be a function dened
on an interval I
Let z = f (x, y) be a function dened
on a rectangle R in xy-plane consisting
of (x, y) points such that
a x b, and c y d.
a x b.
Suppose that f is positiv
Double Integrals over General Regions
Let z = f (x, y) be a function of two variables dened on a region D in xy-plane that consist of all
points (x, y) such that
a x b and c(x) y d(x)
Suppose that f is positive on the region D. The double
Constrained Optimization for functions of two variables. To nd the maximum and
minimum values of z = f (x, y), objective function, subject to a constraint g(x, y) = c :
1. Introduce a new variable , the Lagrange multi
Maximum and Minimum Values
Let z = f (x, y) be a function of two variables. To nd the local maximum and minimum values,
1. Find the rst partial derivatives fx and fy . Then nd all points (a, b) at which the partial
derivatives are zer
Let z = f (x, y) be a function of two variables. The partial derivative of z with respect to x
is obtained by regarding y as a constant and dierentiating z with respect to x. Notation:
This derivative at
Review of vectors. The dot and cross products
Review of vectors in two and three dimensions.
A two dimensional vector is an ordered pair
= a1 , a2 of real numbers.
A three dimensional vector is an ordered pair
= a1 , a2 , a3 of real n
The Chain Rule
Recall the chain rule for functions of single variable:
If y = f (x) and x = g(t), then y (t) = y (x) x (t) ( or
The chain rule for function z = f (x, y) with x = g(t) and y = h(t) is
z (t) = zx x (t)
Parametric Curves in two and three dimensional space.
Parametric curve in plane
Parametric curve in space
x = x(t)
y = y(t)
x = x(t)
y = y(t)
z = z(t)
To nd a tangent line to the curve
x = x(t), y = y(t) at t = t0 , use
Three Dimensional Coordinate System. Functions of Two
From two to three dimensions.
A point in the two dimensional
coordinate system is represented
by an ordered pair (x, y).
A point in the three dimensional
If P = (x, y) is a point in the xy-plane and O denotes the origin, let
r denote the distance from the origin O to the point P = (x, y). Thus, x2 + y 2 = r2 ;
be the angle between the vector OP and the positive part
In the past, we mostly worked with curves in the form y = f (x). However, this format does not
encompass all the curves one encounters in applications. For example, consider the circle x2 +y 2 = a2 .
Solving for y does
Recall that the line which approximates a function f (x) at a point (a, f (a) has the slope f (a).
By point-slope equation, the equation of this line is
y f (a) = f (a)(xa) y = f (a)+f (a)(xa).
The expression f (a)+f
Modeling with Dierential Equations
In most cases, it is equally important to be able to come up with a dierential equation that
accurately describes the problem you need to solve as to being able to solve the equation. The
process of wr
Linear Dierential Equation
A rst order dierential equation is linear if it can be written in the form a1 (x)y +a0 (x)y = b(x).
Note that if a1 (x) = 0, the equation is not dierential. So, let us assume that the function a1 (x)
is not ze
Review for Final Exam Solutions
1) Denite and Indenite Integrals.
1. 1/21 (3x + 5)7 + c
b ax2 +1
3. ln |x| + 1/x + c
3 ln 2
ln |ax + b| + c.
6. 1/5 sin(5x + 1) + c
7. 1/3 tan1 (3x) + c
8. 1/3 sin1 (3x) +
Review for Exam 3
a) General Solution. Find the general solutions of the following.
1. y = 2x 1 y 2
2. y = y 2 xe2x
3. y = x(y + 1)
4. xy + 2y = x3
5. xy + y = x cos x
6. Show that y = ce2x is a solution of dierential equation y 6y + 8y =
Review for Final Exam
1) Integrals. Evaluate the following integrals.
(3x + 5)6 dx
dx where a and b are arbitrary constants.
where a and b are arbitrary constants.
cos(5x + 1)
Review for Exam 3 Solutions
a) General Solution. 1. y = sin(x2 + c)
2. y =
xe2x + 4 e2x +c
3. y = cex
4. y = x /5 + c/x
5. y = sin x + (cos x)/x + c/x
6. Substitute y = 2ce , and
y = 4ce into the equation and show
Autonomous Dierential Equations and Population
If a dierential equation is of the form
y = f (y),
it is called autonomous. Note that an autonomous equation is a separable dierential equation.
If f is zero at a, then the horizont
Arc Length. Surface Area.
Arc Length. Suppose that y = f (x) is a continuous function with a continuous derivative on
[a, b]. The arc length L of f (x) for a x b can be obtained by integrating the length element
ds from a to b. The leng
Dierential Equations of First Order. Separable Dierential
Equations. Eulers Method
A dierential equation is an equation in unknown function that contains one or more derivatives of the unknown function.
A function y is a solution of a d
Approximate Integration. Trapezoidal and Simpsons sums.
Recall that the Left and the Right Sums approximate the area under a curve as the sum of certain
rectangles. On each subinterval, the left sum uses rectangles whose heights are obt
f (x) dx is improper if it is of one of the following three types:
1. At least one of the bounds is positive or negative innity.
2. The function f (x) is not dened or is discontinuous at at least one
Let us consider the integrals of the form
f (sin x) cos xdx
f (cos x) sin xdx
where f (x) is a function with antiderivative F (x). Using the substitution
u = sin x for the rst integral
u = cos x for the se
A rational function is a quotient of two polynomial functions. The method of partial fractions is
a general method for evaluating integrals of rational function. The idea is to write a rational function
as a sum of sim