Lecture 11 - Business and Economics Optimization Problems and Asymptotes
11.1 More Economics Applications
Price Elasticity of Demand One way economists measure the responsiveness of consumers to a change in the price of a product is with what is called pr
MATH 1300B-MIDTERM1-2003
Multiple Choice Section-Question 1Which of the following functions is the inverse of f (x) = A) f 1 (x) = E) f 1 (x) =
4x+5 3x2 34x 5+2x 2x+5 ? 3x4
B) f 1 (x) =
3x4 2x+5
C) f 1 (x) =
52x 4+3x
D) f 1 (x) =
5x2 4x3
Solution: (A) In
6
6.1
Techniques of Integration
Integration by Parts
In section 5.2 we discussed solving integrals by making u-substitutions. It was explained then that this is the integration analogue of the Chain Rule for dierentiation. In this section we discuss the i
7
Functions of Several Variables
Suppose you run a company which produces two types of television. Let x1 be the quantity of type 1 and x2 be the quantity of type 2. Then the usual functions in which we are interested (prot, revenue, and cost) will depend
Lecture 1 - Precalculus Review
1.1 Real Line and Order
When discussing order on the real number line, we use the following symbols: < > less than less than or equal to greater than greater than or equal to
We use the following notation for intervals: x (a
Lecture 2 - Graphs in the Plane
2.1 The Cartesian Plane
In this course we will be dealing a lot with the Cartesian plane (also called the xy -plane ), so this section should serve as a review of it and its properties.
y -axis
T
Quadrant 2
Quadrant 1
'
E
x
Lecture 3 - Functions and Limits
3.1 Functions
In the expression y = f (x) = x2 + 3 we say that x is the independent variable and that y is the dependent variable. A function is a relationship between two variables such that to each value of the independe
Lecture 4 - Continuity and Exponential Functions
4.1 Continuity
Recall that if f (x) is a polynomial, then limxc f (x) = f (c). These types of limits are easy to calculate. This leads to the following denition. Denition 4.1 Let c (a, b) and f (x) a functi
Lecture 5 - Logarithms, Slope of a Function, Derivatives
5.1 Logarithms
Note the graph of ex
This graph passes the horizontal line test, so f (x) = ex is one-to-one and therefore has an inverse function. This is also true of f (x) = ax for any a > 0, a =
Lecture 6 - Rules of Dierentiation, Velocity and Marginals
6.1 Rules for Dierentiation
We would like to avoid the limit calculation for the derivative entirely, so we state some rules for nding the derivative directly. 1. Constant Rule: d (c) = 0 for any
Lecture 9 - Increasing and Decreasing Functions, Extrema, and the First Derivative Test
9.1 Increasing and Decreasing Functions
One of our goals is to be able to solve max/min problems, especially economics related examples. We start with the following de
Lecture 10 - Concavity, The Second Derivative Test, and Optimization Word Problems
10.1 Concavity and the Second-Derivative Test on I . It is concave
Intuition: a curve is concave up on an interval I if it looks like down on I if it looks like . We need a
Lecture 12 - Curve Sketching
12.1 Curve Sketching
This section is a summary of the information we have gained in this chapter and how to apply it to sketching the graph of a given function. When asked to graph a function, your graph should take into accou
Lecture 13 - Antiderivatives and the Denite Integral
13.1 Antiderivatives and Indenite Integrals
Denition 13.1 If f and g are two functions dened on an interval I , then we say that f is an antiderivative of g if f (x) = g (x) for all x in I . Example: Le
Lecture 14 - More on the General Power Rule; Exponential and Logarithmic Dierentiation
14.1 Antiderivatives and Indenite Integrals (continued) x dx. (x2 + 1)2
4. Evaluate
du solution: Once again we let u = x2 + 1, so du = 2x. In class, I then solved for d
Lecture 15 - Areas and the Denite Integral
15.1 Areas and the Denite Integral
Suppose that f is non-negative on a closed interval [a, b]. We wish to calculate the area underneath the curve but below the x-axis between a and b:
One approach is to use appro
Lecture 16 - The Area Between Two Functions
16.1 The Area Between Two Curves
We can use the denite integral to calculate the area between two curves. Its easy to see visually that if f and g are both positive and f (x) g (x) for all x in [a, b], then the
MAT1300 Midterm Review
Pieter Hofstra October 6, 2009
1
Material
Chapter 0: 0.1: Real line and Order 0.2: Absolute Value and Distance 0.3: Exponents and Radicals 0.4: Factoring Polynomials (you may omit the part on the rational zero theorem, pp. 23) 0.5:
MAT1300 Midterm Review
Pieter Hofstra November 10, 2009
1
Material
Chapter 2: 2.8: Related Rates Chapter 3: 3.1: Increasing and Decreasing Functions 3.2: Extrema and the First Derivative Test 3.3: Concavity and the Second Derivative Test (may skip dimini
MAT1300 Lecture 11
Concavity and Related Rates Pieter Hofstra
October 27, 2009
Overview Second Derivative Related Rates
1
Second Derivative Concavity Second Derivative Test
2
Related Rates
Pieter Hofstra
MAT1300 Lecture 11
Overview Second Derivative Relat
MAT1300 Lecture 18
Improper Integrals Pieter Hofstra
November 24, 2009
Overview Integration by Parts Improper Integrals
1
Integration by Parts
2
Improper Integrals
Pieter Hofstra
MAT1300 Lecture 18
Overview Integration by Parts Improper Integrals
Integrat