Suppose that the number of hours of daylight en a given day in Seattle is modeled by the function − 3.75 cos ( π t 6 ) + 12.25 , with t given in months and t = 0 corresponding to the winter solstice. a. What is the average number of daylight hours in a year? b. At which times t 1 and t 2 , where 0 ≤ t 1 < t 2 < 12 , do the number of daylight hours equal the average number? c. Write an integral that expresses the total number of daylight hours in Seattle between t 1 and t 2 . d. Compute the mean hours of daylight in Seattle between t 1 and t 2 , where 0 ≤ t 1 < t 2 < 12 , and then between t 2 and t 1 , and show that the average of the two is equal to the average day length.
Suppose that the number of hours of daylight en a given day in Seattle is modeled by the function − 3.75 cos ( π t 6 ) + 12.25 , with t given in months and t = 0 corresponding to the winter solstice. a. What is the average number of daylight hours in a year? b. At which times t 1 and t 2 , where 0 ≤ t 1 < t 2 < 12 , do the number of daylight hours equal the average number? c. Write an integral that expresses the total number of daylight hours in Seattle between t 1 and t 2 . d. Compute the mean hours of daylight in Seattle between t 1 and t 2 , where 0 ≤ t 1 < t 2 < 12 , and then between t 2 and t 1 , and show that the average of the two is equal to the average day length.
Suppose that the number of hours of daylight en a given day in Seattle is modeled by the function
−
3.75
cos
(
π
t
6
)
+
12.25
, with t given in months and t = 0 corresponding to the winter solstice.
a. What is the average number of daylight hours in a year?
b. At which times t1 and t2, where
0
≤
t
1
<
t
2
<
12
, do the number of daylight hours equal the average number?
c. Write an integral that expresses the total number of daylight hours in Seattle between t1 and t2.
d. Compute the mean hours of daylight in Seattle between t1 and t2, where
0
≤
t
1
<
t
2
<
12
, and then between t2 and t1, and show that the average of the two is equal to the average day length.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
(^)
k
Recall that for numbers 0 ≤ k ≤ n the binomial coefficient (^) is defined as
n!
k! (n−k)!
Question 1.
(1) Prove the following identity: (22) + (1121) = (n+1).
(2) Use the identity above to prove the binomial theorem by induction. That
is, prove that for any a, b = R,
n
(a + b)" = Σ (^)
an-
n-kyk.
k=0
n
Recall that Σ0 x is short hand notation for the expression x0+x1+
+xn-
(3) Fix x = R, x > 0. Prove Bernoulli's inequality: (1+x)" ≥1+nx, by using
the binomial theorem.
-
Question 2. Prove that ||x| - |y|| ≤ |x − y| for any real numbers x, y.
Question 3. Assume (In) nEN is a sequence which is unbounded above. That is,
the set {xn|nЄN} is unbounded above. Prove that there are natural numbers
N] k for all k Є N.
be natural numbers (nk Є N). Prove that
Question content area top
Part 1
Find the measure of
ABC
for the congruent triangles ABC and
Upper A prime Upper B prime Upper C primeA′B′C′.
79 degrees79°
1533
2930
Part 1
m
ABCequals=enter your response heredegrees
Joy is making Christmas gifts. She has 6 1/12 feet of yarn and will need 4 1/4 to complete our project. How much yarn will she have left over compute this solution in two different ways 
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Definite Integral Calculus Examples, Integration - Basic Introduction, Practice Problems; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=rCWOdfQ3cwQ;License: Standard YouTube License, CC-BY