MATH 205-02H
Exam #1
This test is closed-book and closed-notes. No calculator is allowed for this test. For full credit show all of your work (legibly!), unless otherwise specified. For the purposes of this exam, all arithmetic results should be simplifie
MATH 20502H
Quiz #1
Any answers which require logarithms to be expressed should be put in terms of natural or common logarithms. Show all work. 1. (3 points) If f (x) = 2x2 - 3x, simplify the expression f (a + h) - f (a).
2. (3 points) Find the equation o
MATH 20502H
Quiz #1 Solutions
Any answers which require logarithms to be expressed should be put in terms of natural or common logarithms. Show all work. 1. (3 points) If f (x) = 2x2 - 3x, simplify the expression f (a + h) - f (a).
f (a + h) - f (a) = 2(a
MATH 20502H
Quiz #2
Name:
2x if
x3
a
if 3 < x < 8 continuous everywhere.
1. (4 pts) Find a and b to make the function f (x) =
x
4x + b if
x8
2. (6 pts) For each of the following limits, either determine its value or demonstrate that the limit
does not ex
MATH 20502H
Quiz #2
Name:
2x if x3 a if 3 < x < 8 continuous everywhere. 1. (4 pts) Find a and b to make the function f (x) = x 4x + b if x8 Each of the individual parts of this function can be easily observed to be continuous on its domain, so problems
MATH 20502H
Quiz #3
Name:
1. (6 points) Compute the following infinite limits, or, if the limit is uncomputable, briefly describe (in symbols or words) the relevant long-term behavior of the function. (a) (2 points) lim
4x3 +6x2 -2 3 x- 2x -4x+5
(b) (2 po
MATH 20502H
Quiz #3 Solutions
1. (6 points) Compute the following infinite limits, or, if the limit is uncomputable, briefly describe (in symbols or words) the relevant long-term behavior of the function. (a) (2 points) lim
4x3 +6x2 -2 . 3 x- 2x -4x+5
x-
MATH 20502H
Quiz #4
Name:
1. (8 points) Compute the following derivatives:
(a) (4 points) Given y = x2 esec x , nd
(b) (4 points) For f (t) = cos
(c) (4 points) Compute
d
d
et 2
arcsin t
dy
.
dx
, nd f (t).
sin ( + ln ).
2. (4 points) Given the equality x
MATH 20502H
Quiz #4 Solutions
1. (12 points) Compute the following derivatives:
dy
(a) (4 points) Given y = x2 esec x , nd dx .
This is a product rule stacked with a chain rule (since esec x is a composition). Let us preemptively assign u = sec x, so that
MATH 205-02H
Quiz #5
Name:
1. (4 points) Approximate the two following values with well-chosen linear approximations:
(a) 3 997.
(b) (3.04)4 .
2. (10 points) Let f (x) = 2 + 3x2 x3 . Answer the following questions about this function.
(a) (4 points) Where
MATH 205-02H
Quiz #5 Solutions
1. (4 points) Approximate the two following values with well-chosen linear approximations: 3 997. We consider the function f (x) = 3 x = x1/3 , whose derivative is f (x) = 1 x-2/3 . For x close 3 to 1000 (as 997 is), we can
MATH 205-02H
Quiz #6
Name:
1. (4 points) Using two steps of Newtons method with a starting value of x0 = 1, produce an
approximation x2 to a zero of the function f (x) = x3 x+1. Your answer need not be arithmetically
simplied.
2. (8 points) An open-topped
MATH 205-02H
Quiz #6 Solutions
1. (4 points) Using two steps of Newtons method with a starting value of x0 = 1, produce an
approximation x2 to a zero of the function f (x) = x3 x+1. Your answer need not be arithmetically
simplied.
Let us note that f (x) =