Math 181 Honors Quiz 1 Version A
1. Determine all intervals of numbers x satisfying the following inequalities.
(i) |2x + 1| < 5
(ii) x <
1
x
1
. In otherwords, for what numbers
x1
could you dene a function f by this formula?
2. Find the domain of the fun
4.4: Concavity and Curve Sketching
Let f (x) be dened on an open
interval I . Recall that if f > 0 on I
(that is, f (x) > 0 for all x in I ), then
f (x) is increasing on I and if f < 0
on I , then f (x) is decreasing on I .
Ex a: Discuss the concavity of
3.11: Linearization and Dierentials
If f (x) is dierentiable at a, then
Show 101 10.05 without a
m = f (a) is the slope of the tangent calculator.
at (a, f (a). Tangent line equation
Use () with f (x) = x, a = 100.
y = f (a) + f (a)(x a)
f (x) = x,
f (100
4.5: Indeterminate Forms and LHpitals Rule
o
ln x
.
1
lim ln x = 0 = lim x 2 1
Indeterminate Forms: limits of form
0 , , 0 , 00 . . .
0
Ex a: Find lim
LHpitals Rule 0 : Suppose
o
0
lim f (x) = 0 = lim g (x) then
x1
xa
xa
lim
xa
f (x)
f (x)
= lim
xa g (x)
4-6: Applied Optimization
Problem
A farmer wants to enclose a
rectangular plot with 60 m of fencing.
Find the dimensions of the plot that
maximize the area enclosed. Find the
resulting area.
then substitute into ()
A(x) = xy = x(30 x) = 30x x 2 .
Since bo
4.3: Monotonic Functions and the First Derivative Test
Important consequence of the MVT:
the sign of f (x) tells us where f (x)
is increasing and decreasing.
Denition
Let f (x) be dened on the interval I .
a. Say that f (x) is increasing on I ,
if f (x1 )
4-6: Applied Optimization, continued
Methodology for optimization
i. Read the problem carefully;
make a sketch and assign
variable names.
ii. What is to be optimized (max or
min)?
Let x, y denote the width and
height of the printed area.
constraint:
xy =
4.1: Extreme Values of Functions
Denition: Let f (x) be dened on D. Let f (x) = x 2 (see graphs)
a. f (x) has an absolute maximum
on D at c if
f (x) f (c)
for all x in D
b. f (x) has an absolute minimum
on D at c if
f (x) f (c)
for all x in D
A function m
4.8: Antiderivatives
Recall that if F (x) = f (x) on I , then
F (x) is an antiderivative of f (x) on I .
The most general antiderivative is
F (x) + C ,
C arbitrary constant
In Ch. 5 we use antiderivatives to
evaluate integrals.
Physical motivation: Given
5.3: The Denite Integral
Suppose f (x) is continuous on [a, b]
(but not necessarily positive). Then
the limits of Ln , Rn and Mn all exist
and are the same!
Fact If f (x) 0 on [a, b], then the
area under the curve is given by
b
f (x) dx = A
a
b
f (x) dx =
5.1: Area and Estimating with Finite (Riemann) Sums
Important application of calculus:
nd the area A under the curve
y = f (x) for a x b.
y
y = f (x)
Approximate the area under the
curve by adding the areas of the
rectangles; the i th rectangle has
height
4.7: Newtons Method
Newtons method is an algorithm for
nding roots of f (x) = 0 using the
tangent line. Need to start with a
reasonable guess x0 .
Set y = 0 and solve for x = x1 :
f (x0 )
x1 = x0
f (x0 )
To get a better approximation
repeat the procedure
5.4: The Fundamental Theorem of Calculus
There are two fundamental theorems: Note that F (x) is the antiderivative
The rst is more theoretical and the of f (x) with F (a) = 0 in FTC1.
x
second is more useful.
1
Ex a: ln x =
dt
Theorem: FTC1
1 t
x
d
2
2
Su
4.4 (ctd): Sketching graphs
Use all information provided by a sign
analysis of both rst and second
derivatives; also, include intercepts
and asymptotes to sketch y = f (x).
To begin plot points corresponding to
the local extrema and inection pts.
Let
If f
4.2: Mean Value Theorem
Theorem (MVT):
Suppose that f (x) is continuous on
[a, b] and dierentiable on (a, b).
Then there is a point c in (a, b) s. t.
f (b) f (a)
= f (c)
ba
( )
m = f (c)
y
m=
f (b)f (a)
ba
y = f (x)
Ex a: f (x) = x 2 2x + 3,
[a, b] = [1,
Math 181 Honors Quiz 9 Version A
1. Fill in the derivatives in the following table:
f ( x)
xr
sin x
cos x
tan x
arcsin x
arccos x
arctan x
x2 + x 6
x+1
x1
sin(x2 + 1)
x arctan x
sin2 x + cos2 x
f ( x)
Math 181 Honors Quiz 9 Version A
2. Suppose w(x) = f (
Math 181 Honors Quiz 2 Version A
1. Determine all intervals of numbers x satisfying the inequality x2 < x
2. State and prove the pythagorean theorem. State both the hypothesis and conclusion
of the theorem as well as giving a proof written using complete
Math 181 Honors Quiz 10 Version A
1. Show that if f (x) is dierentiable at x = a then f (x) is continuous at x = a.
2. [Extra Credit] Give an example of a function which is continuous at x = a but not
dierentiable at x = a.
Math 181 Honors Quiz 10 Version
Math 181 Honors Exam 2 Version B
1. Convert the repeating decimal 2.17 to a fraction.
2. Solve the inequality
(x 1)(x 3)
0.
(x 2)(x 4)
1
1
=.
x 3 x
3
3. Use the - denition of limit to verify lim
Math 181 Honors Exam 2 Version B
4. Use the limit laws to n
Math 181 Honors Exam 1 Version A
1. The Order Axioms are
(POS1) If a, b are positive, so is ab and a + b.
(POS2) If a is a number, then either a is positive, or a = 0,
or a is positive, and these possibilities are mutually exclusive.
Use the order axioms
Math 181 Honors Quiz 2 Version B
1. Determine all intervals of numbers x satisfying the inequality x < x2
2. State and prove the pythagorean theorem. State both the hypothesis and conclusion
of the theorem as well as giving a proof written using complete
Math 181 Honors Quiz 3 Version A
1. Let the sets A, B, C R be given by
A = cfw_ 1, 2, 3 ,
B = cfw_ 0, 2, 4
and
C=
sin 0, sin
, sin , sin
.
6
3
2
Write down the following sets in the form cfw_ elements in the set where each element
in the set is listed o
Math 181 Honors Quiz 7 Version A
1. State the denition of lim f (x) = L in terms of and .
x a
2. State the denition of lim f (x) = L in terms of and N .
x
3. State the denition of lim f (x) = in terms of M and .
x a
4. State the denition of lim f (x) = in
Math 181 Honors Quiz 8 Version A
1. Show that lim+
0
sin
= 1 using geometry and the - denition of limit.
(1, tan )
(cos sin )
,
(1, 0)
Math 181 Honors Quiz 8 Version A
2. Let f (x) = sin x. Use the limits
sin x
=1
x 0 x
lim
and
cos x 1
= 0,
x 0
x
lim
th
Math 181 Honors Quiz 6 Version A
1. State the denition of derivative in terms of limits.
2. Find all x such that x2 > 2x 1.
3. Convert the repeating decimal 1.524 to a fraction.
Math 181 Honors Quiz 6 Version A
4. The limit laws are
(0) lim c = c
x a
1
(
Math 181 Honors Quiz 5 Version A
1. Find the vertex of the parabola y = 7 + 3x2 + x 2.
1
1
=.
x 3 x
3
2. Use the - denition of limit to verify lim
Math 181 Honors Quiz 5 Version A
3. Find the center and radius of the ellipse 3x2 + 2y 2 + x + y = 5.
4. Use
5.5: Indenite Integrals and the Substitution Method
d 5x+7
e
= 5e 5x+7 ; so
dx
1
e 5x+7 dx = e 5x+7 + C .
5
d
cos x
sin x = ; so
b:
dx
2 x
cos x
dx = 2 sin x + C .
x
d
x
; so
c:
x2 + 3 =
2+3
dx
x
x
dx = x 2 + 3 + C .
2+3
x
Substitution Method
If F (x) =