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 to evaluate the partial derivatives fx , fy at (x, y)
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}=—-4=—.
d1 d1 41 x
An alternative method uses a proper
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 Differentiate with res e“: ".0
d1 ' l d] ' v} p L
m.
l = [cos
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 then sketch a possible graph of f.
f'ctL penance-3:0} fen:
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 4
x5
4
15
4
15
4
3
solution
(4)
(4) 2 1
lim
x4
(4) 5
4
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 4
x5
4
15
4
15
4
3
4
3
Evaluate the limit
x 2 x 12
lim
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 limit on the right side exists (or is i: no). The rule als
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 tangent plane to the equipotential surface V (x, y, z) =
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 planes are perpendicular to each other.
(b) Find the parame
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 of the dierential equation ay + by + cy = 0. Show that e
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 (t) + 3 (t 2 with y (0) = y (0) = 0. REMARK: Make sure
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 by dy = , dx bx cy (here there might be some restriction
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. DiPrima, Published by Wiley. The 8th edition is okay,
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:
- Be sure that this examination has 12 pages. Write y
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, 3 .
(c) Find the distance of the point (2, + 14, 2 27)
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:
- Be sure that this examination has 12 pages. Write y
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 Applied Problems
Acute Angles
o
An angle which is more t