MATH 137
Assignment 6
Due: 11 am, Friday, November 4
Your assignment consists of two parts.
Part 1 is available online at
http:/mapleta.uwaterloo.ca
and is due on-line by 4 pm on Thursday, November 3.
Part 2 consists of the problems below. Place your solu
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MATH 137
Tutorial #8 Problems
Wed. Nov. 7, 2014
Topics: Linear Approx, Differentials, Min/Max & MVT
2
1) A function f (x) has the linear approximation at a = 3 given by La (x) = 5 + (x 3).
7
Find the corresponding linear approximation for g(x) = f 1 (x) a
MATH 137
Tutorial #5 Problems
Wed. Oct. 8, 2014
Topics: Limits, Continuity & IVT
1) Evaluate:
tan x
a) lim
x0 x + sin x
1
Answer:
2
b) lim ( x2 + 2x x2 2x)
x
Answer: 2
2) If 5 2x2 f (x) 5 x2 for 1 x 1, find lim f (x). Is f (x) continuous at
x0
x = 0.
Answ
MATH 137
Tutorial #10 Problems
Wed. Nov. 19, 2014
Topics: Optimization, Newtons Method & Integration
1) Find the height of the right circular cylinder of maximum volume V which can be
placed inside a sphere of radius r as shown in the figure below. Recall
MATH 137
Tutorial #11 Problems
Wed. Nov. 26, 2014
Topic: Integration
Z
x
dt
for < x < . Find where f (x) is increasing and where
2
0 1+t+t
f (x) is concave up. Find the equation of the tangent line to f (x) at x = 0.
1
,
Answer: f is increasing on < x < ,
MATH 137
Tutorial #3 Problems
Wed. Sept. 24, 2014
Topic: Inverse Functions
1) For f (x) =
1
,
1 + ex
a) Find the domain and range of f (x).
Answer: D(f ) = R, R(f ) = cfw_y|0 < y < 1
1
1
1
b) Show that f 1 exists and
findf (x), f (2), f (1/2).
x
, f 1 (1
MATH 137
Tutorial #4 Problems
Wed. Oct. 1, 2014
Topic: Limits
1) Find a and b such that lim
x0
ax + b 2
x
!
= 1.
Answer: a = b = 4
2) Evaluate lim |x| arctan
1
.
x
x0
Answer: 0
3) If lim [f (x) + g(x)] = 2 and lim [f (x) g(x)] = 1 find lim [f (x)g(x)].
MATH 137
Tutorial #6 Problems
Wed. Oct. 15, 2014
Topic: Differentiation
1) Find the equations of all lines through the point (2, 3) that are tangent to the parabola
y = x2 + x. Answer: y = 11x 25 and y = x 1
2) Let:
(
g(x) =
2 sin x cos x for x 0
becx
for
MATH 137
Tutorial #7 Problems
Wed. Oct. 29, 2014
Topics: Differentiation & Related Rates
1) Differentiate the following:
a) g(x) = sin |x| Does g 0 (0) exist?
x
Answer: g 0 (x) =
cos |x| (x 6= 0) , g 0 (0) does not exist
|x|
b) f (x) = (ln x)ln x for x >
MATH 137
Tutorial #9 Problems
Wed. Nov. 12, 2014
Topics: Curve Sketching & lH
opitals Rule
1) Evaluate the following limits:
x sin x
1
a) lim
Answer:
x0 x tan x
2
1
1
1
b) lim
Answer:
x1 x 1
ln x
2
c) lim+ x
x0
x
Answer: 1
d) lim
x
1 + sin
x
3
x
Answer
MATH 137
Tutorial #2 Problems
Wed. Sept. 17, 2014
Topic: Functions
1) If |x 5| < 1 use the Triangle Inequality to show that |x2 5x + 1| < 7.
2
2) Find the domain of f (x) = ln( x2 3x + 2 x). Answer: x <
3
3) Sketch the functions:
a) y =
x2
1 + x4
1
x2 2x
MATH 137
Assignment 10
Do not hand in
The following problems are recommended for you to work on. Problems such as
these may appear on the nal exam.
1. Evaluate the Riemann sum for the function y = 1 + t3 taken over the
interval [0, 1] corresponding to a u
MATH 137
Assignment 9
Due: Friday, November 25 by 11 am
As usual, your assignment consists of two parts.
Part 1 is available online at http:/mapleta.uwaterloo.ca.
The usual due date
is postponed by one week just for Part 1. Part 1 is due on-line by Thursd
MATH 137
Assignment 8
Due: Friday, November 18 by 11 am
As usual, your assignment consists of two parts.
Part 1 is available online at http:/mapleta.uwaterloo.ca,
4 pm on Thursday, November 17.
and is due on-line by
Part 2 consists of the problems below.
MATH 137
Assignment 7
Due: Friday, November 11 by 11 am
As usual, your assignment consists of two parts.
Part 1 is available online at
http:/mapleta.uwaterloo.ca
and is due on-line by 4 pm on Thursday, November 10.
Part 2 consists of the problems below.
P
MATH 138
Assignment 6
Due at noon Friday March 6
Last Name :
First Name :
Student Id. #:
SECTION :
1- Determine whether sequence cfw_an converges or diverges. If it converges, nd the limit.
(a) an = n (1 cos(1/n).
7n
(b) an = 7 n .
n 5
(c) an =
2 arctan