M ATH 150 FALL 2010
T HIRD E XAM
N AME :
J ANUARY 13, 2011
12:302:00 PM
C ODE :
P ROBLEMS
(3x 2)20 dx
1. (10 points) Evaluate the integral
2
2. (10 points) Evaluate the integral
(2x ex ) dx
1
ex 1 + ex dx.
3. (10 points) Evaluate the integral
4. (10 point
M ATH 150 FALL 2010
S ECOND E XAM
N AME :
D ECEMBER 16, 2010
12:302:00 PM
C ODE :
P ROBLEMS
1. (10 points) Find the all points on the graph of the function y = 2 sin x + sin2 x at which tangent line is
horizontal.
2. (12 points) If f (x) + x2 [f (x)]3 = 1
MATH 150 FIRST MIDTERM EXAM
November 9, 2010
SSST, MLH, 2:003:30 pm
NAME:
Show all your work!
1. (10 points) Find the point on the y-axis that is equidistant from(5, 5) and (1, 1).
2. (10 points) An open rectangular box with volume 2m3 has a square base.
Lecture 22: The Denite Integral (Sections 5.2, 5.3)
Denition
If f is dened and continuous for a x b , we divide [a, b ] into n
subintervals of equal width x = (b a)/n. Let
x0 (= a), x1 , x2 , . . . , xn (= b ) be the endpoints of these subintervals and
le
Lecture 29: Integration by Parts (Section 8.1)
Recall the Product Rule for dierentiation
Product Rule
If f and g are dierentiable functions, then
(fg ) = f g + fg
This implies
f (x )g (x ) dx +
g (x )f (x ) dx = f (x )g (x )
that is, we get the formula fo
Lecture 24: Inverse Functions (Section 7.1)
Denition
A function f is called one-to-one if it never takes on the same value
twice, that is,
f (x1 ) = f (x2 )
whenever x1 = x2
Example 1: Is f (x ) = x 2 2 one-to-one?
1 / 38
Horizontal Line Test
A function i
Lecture 15: Maximum and Minimum (Section 4.1)
Denition
A function f has an absolute maximum at x = c if
f (c ) f (x )
for all x D . f (c ) is the maximum value of f on D.
Denition
A function f has an absolute minimum at x = c if
f (c ) f (x )
for all x D
Lecture 8: Derivatives (Sections 3.1, 3.2)
Denition
The tangent line to the curve y = f (x ) at the point P (a, f (a) is the
line through P with slope
f (x ) f (a)
x a
x a
m = lim
if this limit exists.
m is called the slope of the curve y = f (x ) at the
Lecture 5: Limits
Sections 2.1 & 2.2
Example 1: Find an equation of the tangent line to y = x 2 + 1 at the
point P (1, 2).
We know that the equation of this line is
y 2 = m(x 1)
where m is its slope.
As we are not given two points on the tangent line, we
Lecture 3: Elementary Functions
Denition
A function f is a rule that assigns to each element x in a set D exactly
one element, called f (x ) in a set E .
D = the domain of f
E = the range of f
We can dene functions in several ways:
algebraically - by usin