Math 263 Midterm II
Problem 1: Consider the function z = f (x, y) defined via intermediate variables u, v and
w:
z = ew sin(u + 2v), u = xy, v = x y, w = sin(x + y).
(a) [10 points] Use the chain rule
A EXAMPLE 1 Derivatives involving In x
"LI
Find —' for the- following functions.
d1
:1. y=ln=lx
l1. y=xlnx
4:. y=ln|secx|
inf"
d- Jr'= 1
If.
A EDLUHON
:1. Using the {Chain Rule=
a‘v d 1 1
—' =— ﬂn4x
The derivative of y = sin—2l 3: follows bv differentiaiing hem sides of x = sin y with respect to x, simplifying, and solving for
a" yin" Jr:
1 = sin y 3': sin ' z.» = sin 3'
— [Jr = — [sin v Differen
Prove that
Sum formula for sin
Sin x as a common factor
Apply the rules from slide 1
Theorem 3.13
The derivatives of csc, sec and cot can be determined by quotient rule. (Try)
n- EKAMPLE1 An—‘arm—up
Given the following information about the first and second derivatives of a function f: which is continuous on
E— on: on): summarize the information using a number line: and the
Department of Mathematics & Physics
Academic Bridge Program
Chapter: 12
2 (M6)
Class Quiz:
Students Name: -
Class Section:
Choose the Correct answer from given choices
Evaluate the limit
x
2
lim x 1
x
Department of Mathematics & Physics
Academic Bridge Program
Chapter: 12
2 (M6)
Students Name: -
Class Quiz:
Class Section:
Choose the Correct answer from given choices
Evaluate the limit
x
2
lim x 1
x
11-l EDRE M 4.1 3 L" Hﬁpital' 5 Rule
Suppose f and g are differentiable on an open interval I containing a with 3' Ex} at i] on I when x at a.
If limﬂx}= lim gEx}={l, then
I-H'I I-H'I
provided the lim
December 2006
Mathematics 263
Page 2 of 9 pages
Marks
[15]
1.
Suppose that the electrical potential V in space is given by the function
V (x, y, z) = x2 + 2y 2 + ez .
(a) What is the equation of the t
Math 263 Final Exam (December 11, 2007)
There are 6 Problems on both sides of the sheet.
Problem 1: [15 points] Given the equations of two planes 3x+2y+z = 4 and x2y+z = 1,
(a) Prove that the two plan
Suggested homework questions:
Section 3.4, Repeated roots, reduction of order (continued)
# 21. Suppose r1 and r2 are roots of ar2 + br + c = 0 and that r1 = r2 ; then er1 t and er2 t are solutions o
Suggested homework questions:
Section 6.5; Impulse Functions
In each of the following nd the solution of the initial value problem: # 1. y + 2y + 2y = (t ), # 7. y + y = (t 2 ) cos(t), # 9. y + y = u
Suggested homework questions:
1
Exact equations; section 2.6
# 2. dy =0 dx y (Note this one can be done (i think) using the method from the previous section; v := x .) (2x + 4y ) + (2x 2y ) # 6. ax b
Math 215/255
Prerequisite: Integral Calculus Corequisite: Multivariable Calculus and Linear algebra Textbook: Elementary Dierential Equations and Boundary Value Probems, 9th ed., W. E. Boyce and R. C
The University of British Columbia
Final Examination - December 2009
Mathematics 263
Section 101
Closed book examination
Last Name:
Time: 2.5 hours
First:
Signature
Student Number
Special Instructions
December 2010
[12] 1.
(a)
Mathematics 263
Page 2 of 11 pages
Find the scalar projection of the vector 1, , 2 along the vector 2, 6, 3 .
(b) Find the vector projection of 1, , 2 along the vector 2, 6,
The University of British Columbia
Final Examination - December 2009
Mathematics 263
Section 102
Closed book examination
Last Name:
Time: 2.5 hours
First:
Signature
Student Number
Special Instructions
Pre-Calculus MT132
Chapter 8
Applications of Trigonometric Functions
Objectives
Finding the value of the Trigonometric
Functions of Acute Angles Using Right
Triangles
Solve Right Triangles
Solve Ap