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 equal length by choosing points, say x0 , x1 , .xn between
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 two curves. For example, suppose we wish to nd the area
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 r , compounded continuously then its future value (V) a
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 solved a similar problem by using the constraint to elimina
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 which may be used to nd local maxima and minima of surfaces
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) Mixing problems, and (iii) Falling objects We will disc
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 , and the task at hand is to nd an explicit expression f
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 outcomes of an experiment is called the sample space, S
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 under addition) 2. if v1 S and c is a scalar then cv1 S
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 component in no way eects the choice for the other components.
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 column ) of A1 by some nonzero constant c. Then det(A2
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) matrix A we dene the submatrix Aij of A as the matrix obt
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 solutions) Proof: We have seen examples of each kind, so w
Unit 16 Matrix Equations and Inverses
Denition 17.1. A set of m linear equations in n variables: a11 x1 a21 x1 . . . +a12 x2 + a1n xn +a22 x2 + a2n xn . . . = b1 = b2 . . .
am1 x1 +am2 x2 + amn xn = bm may be re-expressed in its matrix form as a11 a12 a21
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 dimension. Each entry is designated by aij where i is t
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 positive integers then an m n matrix (read m by n) is a r
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 and notation:
Denition 14.1. By we will mean the set o
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 target located at the origin.
Since the bullets will only fl
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. Simplify your answers. You will not be penalized for a mistake
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. Example 1. Find an antiderivative of x3 Solution: The exponen