Unit 4 The Denite Integral
The formal denition of the denite integral Suppose the function f is continuous on the interval [a,b]. We begin by partitioning the interval [a,b] into n subintervals of equ
Unit 5 Area Between Curves and Volumes of Revolution
We have learned how to nd the area beneath a positive function. In this section we will learn how to nd more general areas, those bounded between t
Unit 6 Further Applications of The Antiderivative
Continuous Money Streams Future Value and Present Value Theorem 6.1. If P dollars is invested now (P standing for Present Value) at an annual rate of
Unit 8 Lagranges Method and Calculus of Trigonometric Functions
Constrained Optimization: Lagranges Method
Old Problem: Minimize z = x2 + y 2 + 2 subject to the constraint x + y = 8. Last time we solv
Unit 7 Partial Derivatives and Optimization
We have learned some important applications of the ordinary derivative in nding maxima and minima. We now move on to a topic called partial derivatives whic
Unit 10 Applications of rst order Dierential Equations
We now look at three main types of question which involve the applications of rst order dierential equations, namely (i)Heating and Cooling (ii)
Unit 9: First Order Dierential Equations with Applications
First Order Dierential Equations By a dierential equation, we mean an equation which involves the derivative of some unknown functon, say y ,
Unit 11 Probability and Calculus I
Discrete Distributions
Terminology: Each problem will concern an experiment, which can result in any one of a number of outcomes. The set consisting of all possible
Unit 21 Subspaces of
n
Denition 23.1. A subspace of n is a nonempty set , S , of vectors from n such that both of the following conditions are met: 1. if v1 and v2 are in S then (v1 + v2 ) S (closed u
Unit 20 Independence and Basis in
n
The idea of dimension is fairly intuitive. Consider a vector in n , (a1 , a2 , a3 , ., an ). Each of the n components are independent i.e. choosing a single compone
Unit 19 Properties of Determinants
Theorem 20.1. Suppose A1 and A2 are identical n n matrices with the exception that one row ( or column ) of A2 is obtained by multiplying the corresponding row ( or
Unit 18 Determinants
Associated with each square matrix is a number called its determinant. We dene the determinant and look at some of its properties in this section. Denition 19.1. Given an (n n) ma
Unit 17 The Theory of Linear Systems
Theorem 18.1. Every system of linear equations Ax = b has either no solution, or exactly one solution, or or innitely many solutions (i.e. parametric family of sol
Unit 15 Matrix Operations
Matrices Recall:
A = (aij ) =
a11 a21 a31 . . .
a12 a22 a32 . . .
a13 a23 a33 . . .
am1 am2 am3
a1n a2n a3n . . . . . . amn
is said to be an m n matrix where m n is the
Unit 14 Matrices and Systems of Linear Equations
In this section we will learn a method known as row reduction for solving SLEs, which utilizes things called matrices. Denition 15.1. If m and n are po
Unit 13 Euclidean n-Space and Linear Equations
Euclidean n-Space: n Much of this half of the course will involve objects called vectors. In order to introduce this subject we begin with some denitions
56.
A fighter plane, which can only shoot bullets straight ahead, travels along the path
r ( t ) = 5 t , 21 t 2 ,3 t 3 / 27 . Show that there is precisely one time t at which the pilot can hit a targe
Name Michele Student ID TA/Section (circle): Andy 4pm-D03 6pm-D05 8pm-D07 5pm-D04 7pm-D06 9pm-D08
Math 20C, Winter 2010, Midterm Exam 1 Solutions
Show all of your work to receive full credit. Simplif
Unit 2: The Antiderivative
Denition 2.1. F (x) is said to be an antiderivative of f (x) on an interval d if F (x) = f (x) or equivalently dx (F (x) = f (x) for every value of x on the interval. Exampl