Name: TA: Math 20A Midterm Exam 2 V1 November 19, 2009 Sec. No:
PID: Sec. Time:
Turn o and put away your cell phone. No calculators or any other electronic devices are allowed during this exam. You ma
Section 6.5 Integrals of powers and indenite integrals
(3/20/08)
Overview: In this section we use Part I of the Fundamental Theorem of Calculus to derive a formula for integrals of powers y = xn with
Section 6.4 The Fundamental Theorem, Part II
(3/20/08)
Overview: We discussed Part I of the Fundamental Theorem of Calculus in the last section. We establish Part II of the theorem here. We also show
Section 6.3 The Fundamental Theorem, Part I
(3/20/08)
Overview: The Fundamental Theorem of Calculus shows that dierentiation and integration are, in a sense, inverse operations. It is presented in two
Section 6.2 The denite integral
(3/20/08)
Overview: We saw in Section 6.1 how the change of a continuous function over an interval can be calculated from its rate of change if the rate of change is a
CHAPTER 6 INTEGRALS AND APPLICATIONS
(3/20/08)
The derivative, which we studied in Chapters 2 through 5, is used to nd rates of change of functions. In this chapter we begin the study of the second ma
Section 2.R Review exercises
1.A 2.
A
(3/1908)
What is the constant rate of change of f (x) = 10x + 6 with respect to x? Give a formula for the linear function y = g (x) whose constant rate of change
Section 2.8 Linear approximations and dierentials
(3/19/08)
Overview: In this section we return to the main concept of this chapter: the approximations of graphs by tangent lines. We discuss additiona
Section 2.7 Derivatives of powers of functions
(3/19/08)
Overview: In this section we discuss the Chain Rule formula for the derivatives of composite functions that are formed by taking powers of othe
Section 6.6 Estimating denite integrals
(3/20/08)
In this section we discuss techniques for nding approximate values of denite integrals and work with applications where the data is given approximatel
Section 6.7 Integrals involving transcendental functions
(3/20/08)
In this section we derive integration formulas from formulas for derivatives of logarithms, exponential functions, hyperbolic functio
Section 6.R Review exercises
1.A 2.A
(3/20/08)
A cars velocity on a straight road at time t (hours) is 60 miles per hour for 1 < t < 4 and 75 miles per hour for 4 < t < 6. How far does it travel for 1
Chapter 0. Mathematical Models: Functions and Graphs
(5/17/07)
The story of calculus goes back thousands of years. Mathematicians of the ancient world, including Pythagoras (c. 580 BC), Euclid (c. 300
Section 14.R Review exercises
1.A 2.A 3.A 4.
A
(3.24/08)
What is the value of f (x, y ) = x2 ey + 2 cos y at (5, 0)? Find g (10, 2) for g (x, y ) = x2 10y 3 . Draw the level curves h = c of the linear
Section 14.6 Tangent planes and dierentials
(3/23/08)
Overview: In this section we study linear functions of two variables and equations of tangent planes to the graphs of functions of two variables.
Section 14.5 Directional derivatives and gradient vectors
(3/23/08)
Overview: The partial derivatives fx (x0 , y0 ) and fy (x0 , y0 ) are the rates of change of z = f (x, y ) at (x0 , y0 ) in the posi
Section 14.4 Chain Rules with two variables
(3/23/08)
Overview: In this section we discuss procedures for dierentiating composite functions with two variables. Then we consider second-order and higher
Section 14.3 Partial derivatives with two variables
(3/23/08)
Overview: In this section we begin our study of the calculus of functions with two variables. Their derivatives are called partial derivat
Section 14.2 Horizontal cross sections of graphs and level curves
(3/23/08)
Overview: In the last section we analyzed graphs of functions of two variables by studying their vertical cross sections. He
CHAPTER 14 Derivatives with Two or More Variables
(3/22/08)
Many mathematical models involve functions of two or more variables. The elevation of a point on a mountain, for example, is a function of t
Section 2.6 Derivatives of products and quotients
(3/19/08)
Overview: In this section, we derive formulas for derivatives of functions that are constructed by taking products and quotients of other fu
Section 2.5 Derivatives as functions and estimating derivatives
(3/19/08)
Overview: In this section we discuss more examples of derivatives as rates of change and show how approximate derivatives can
Section 2.4 Derivatives of power functions and linear combinations
(3/19/08)
Overview: In this section we derive rules for dierentiating power functions y = xn and linear combinations y = Af (x) + Bg
Section 0.4 Inverse functions and logarithms
(5/31/07)
Overview: Some applications require not only a function that converts a number x into a number y , but also its inverse, which converts y back in
Section 0.3 Power and exponential functions
(5/16/07)
Overview: As we will see in later chapters, many mathematical models use power functions y = xn and exponential functions y = bx . The denitions a
Section 0.2 Set notation and solving inequalities
(5/31/07)
Overview: Inequalities are almost as important as equations in calculus. Many functions domains are intervals, which are dened by inequaliti
Name: TA: Math 20A Midterm Exam 2 V1 November 20, 2008 Sec. No:
PID: Sec. Time:
Turn o and put away your cell phone. No calculators or any other electronic devices are allowed during this exam. You ma
Name: TA: Math 20A. Midterm Exam 1 October 23, 2008 Sec. No:
PID: Sec. Time:
Turn o and put away your cell phone. No calculators or any other electronic devices are allowed during this exam. You may u
Math 20A Final Examination. December 9, 2008 1. Find the following limits: (a) (2 points) lim 5x 1 , x 0 x 5x 1 H ln(5)5 x 0 = lim = ln(5) . x 0 x 0 x 1
By LHopitals Rule, lim (b) (2 points) lim+ x lo
Name: TA Name: Math 20A. Final Examination December 8, 2005
Section Number: Section Time:
Turn o and put away your cell phone. No calculators or any other devices are allowed on this exam. You may use