Calculus
Algebra
Analysis
High school Math
Probability
100
Compute
limx->3 5x2-8x-13/x2-5
11/4
100
Slove sqrt(x-8)=3
IN 10 SECONDS COUNT DOWN
GO!!!
x=17
100
Let H ⊆ R be a subset. Formalize the following statements and their negations. Is there a set with the given property?
1. H has at most 3 elements.
2. H has no least element.
3. Between any two elements of H there is a third one in H.
4. For any real number there is a greater one in H.
1. ∀ x, y, z, w ∈ H x = y ∨ x = z ∨ x = w ∨ y = z ∨ y = w ∨ z = w
2. ∀ x ∈ H ∃ y ∈ H y < x
3. ∀ x, y ∈ H x < y ∃ z ∈ H x < z < y
4. ∀ x ∈ R ∃ y ∈ H x < y
100
The triangle bounded by the lines y = 0, y = 2x and y = -0.5x + k, with k positive, is equal to 80 square units. Find k.
K=10
100
A gambler playing with 3 playing cubes wants to know weather to bet on sum 11 or 12. Which of the sums will occur more probably?
extra Credit +200 points
The gambler should bet on 11, as P(11) > P(12).
200
DIFFERENTIATE
4ln(ln(ln(sec x)))
4tan x/ln(ln(sec x))ln(sec x)
200
1/(x-3)+1/(x+3)=10/(x2-9)
x=5
200
Let A be a set and let P(A) denote the set of all subsets of A (i.e., the power set of A). Prove that A and P(A) do not have the same cardinality. (The term cardinality is used in mathematics to refer to the size of a set.)
This is a proof by contradiction. Suppose they have the same cardinality, then there exists a bijection T : A → P(A). Let K = {x ∈ A : x ∈ T(x)}. Since T is onto, there exists y ∈ A such that T(y) = K. If y ∈ K, then by the definition of K we can conclude y ∈ T(y) = K, so y ∈ K. Similarly, if y ∈ K, y ∈ T(y) so we conclude that y ∈ K. In both cases we have reached a contradiction.
200
If 3x−y=12, what is the value of 8x/2y?
212
200
Two coins are tossed, find the probability that two heads are obtained.
P(E) = 1 / 4
300
Integrate
sqrt(x)/(x-1) dx
2sqrt(x)+ln(sqrt(x)-1/(sqrt(x)+1))+C
300
At t = 0 there are 50 grams of a radioactive isotope. The isotope has a half-life of 16 minutes. Use the exponential decay model to write the amount A as a function of time t.
A = A0e −rt = 50e^( −0.04332t), where A is in grams and t in minutes.
300
Let f : R → R be differentiable and assume there is no x ∈ R such that f(x) = f 0 (x) = 0. Show that S = {x | 0 ≤ x ≤ 1, f(x) = 0} is finite
Bonus Question: extra 600 points
Consider f −1 ({0}). Since {0} is closed and f continuous, f −1 ({0}) is closed. Therefore S = [0, 1] ∩ f −1 ({0}) is a closed and bounded subset of R. Hence, S is compact.
Assume, by way of contradiction, that S is infinite. Then (by theorem A.1) there is a limit point x ∈ S; i.e., there is a sequence {xn} of distinct points in S which converges to x.
Also, as all points are in S, f(xn) = f(x) = 0 for all n ∈ N. We now show that f 0 (x) = 0, which will give us our desired contradiction. Since |xn − x| → 0, we can write the derivative of f as follows:
f 0 (x) = limn→∞ f(x + (xn − x)) − f(x) xn − x = limn→∞ f(xn) − f(x) xn − x = 0.
The last equality holds since f(x) = f(xn) = 0 holds for all n ∈ N.
300
C=5/9(F−32)
The equation above shows how temperature F, measured in degrees Fahrenheit, relates to a temperature C, measured in degrees Celsius. Based on the equation, which of the following must be true?
I.A temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of
| 5 |
| 9 |
degree Celsius.
II.A temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit.
III.A temperature increase of
| 5 |
| 9 |
degree Fahrenheit is equivalent to a temperature increase of 1 degree Celsius.
A) I only
B) II only
C) III only
D) I and II only
ANSWER IS D
300
Which of these numbers cannot be a probability?
a) -0.00001
b) 0.5
c) 1.001
d) 0
e) 1
f) 20%
a & c
400
A thin sheet of ice is in the form of a circle. If the ice is melting in such a way that the area of the sheet is decreasing at a rate of 0.5 m2/sec at what rate is the radius decreasing when the area of the sheet is 12 m2
rt=-0.040717
400
Slove 3tan(x/2)+3=0
x=2arctan(-1)
400
If A and B are sets, then show that
a) P(A) ∪ P(B) ⊆ P(A ∪ B)
b) P(A) ∩ P(B) = P(A ∩ B)
a) X ∈ P(A) ∪ P(B) ⇒ X ⊆ A or X ⊆ B
⇒ X ⊆ A ∪ B
⇒ X ∈ P(A ∪ B).
b) X ∈ P(A) ∩ P(B) ⇔ X ⊆ A and X ⊆ B
⇔ X ⊆ A ∩ B
⇔ X ∈ P(A ∩ B).
400
Four students about to purchase concert tickets for $18.50 for each ticket discover that they may purchase a block of 5 tickets for $80.00. How much would each of the 4 save if they can get a fifth person to join them and the 5 people equally divide the price of the 5-ticket block?
$2.50
400
The blood groups of 200 people is distributed as follows: 50 have type A blood, 65 have B blood type, 70 have O blood type and 15 have type AB blood. If a person from this group is selected at random, what is the probability that this person has O blood type?
P(E)=70/200=.35
500
Determine the area below 
and above the x-axis.
f(x)=3+2x-x2
32/3
500
Write down the equation of a circle with radius 8 and center (-4,7).
(x+4)2+(y-7)2=64
500
Show that
1. N ∼ E.
2. N ∼ Z.
3. (−1, 1) ∼ R.
1. Let f : N → E given by f(n)=2n. Clearly this map is one-to-one and onto.
2. This can be shown by defining a bijection f : N → Z as f(n) = 1−n 2 if n is odd, n 2 if n is even.
3. Using calculus one can show that the function f : (−1, 1) → R defined by f(x) = x x2 − 1 is one-to-one and onto. In fact, (a, b) ∼ R for any interval (a, b).
500
The owner of the Good Deals Store opens a new store across town. For the new store, the owner estimates that, during business hours, an average of 90 shoppers per hour enter the store and each of them stays an average of 12 minutes. The average number of shoppers in the new store at any time is what percent less than the average number of shoppers in the original store at any time? (Note: Ignore the percent symbol when entering your answer. For example, if the answer is 42.1%, enter 42.1)
60
500
A jar contains 3 red marbles, 7 green marbles and 10 white marbles. If a marble is drawn from the jar at random, what is the probability that this marble is white?
P(E)=1/2