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Problem A.1
The graph below is made of three line segments:
-1 1 2 3 4 5 6 7 8 9 10 11 12
1
2
3
4
y
x
f(x)
g(x)
h(x)
The segments correspond to the following three functions:
f(x) = x − 2, g(x) = p
4 − (x − 6)2 + 2, h(x) = x − 6
Find the total length L of the graph between x = 2 and x = 10.
Problem A.1
The graph below is made of three line segments:
-1 1 2 3 4 5 6 7 8 9 10 11 12
1
2
3
4
y
x
f(x)
g(x)
h(x)
The segments correspond to the following three functions:
f(x) = x − 2, g(x) = p
4 − (x − 6)2 + 2, h(x) = x − 6
Find the total length L of the graph between x = 2 and x = 10.
Problem A.1
The graph below is made of three line segments:
-1 1 2 3 4 5 6 7 8 9 10 11 12
1
2
3
4
y
x
f(x)
g(x)
h(x)
The segments correspond to the following three functions:
f(x) = x − 2, g(x) = p
4 − (x − 6)2 + 2, h(x) = x − 6
Find the total length L of the graph between x = 2 and x = 10.
Find the smallest positive integer N that satisfies all of the following conditions:
• N is a square.
• N is a cube.
• N is an odd number.
• N is divisible by twelve prime numbers.
How many digits does this number N have?
Find the derivative of f(x)=(x4-3x2+7x).ex /ln(x2-3)
(1) Find and classify the extremes value of f(x) = 6x4 - 4x6 over the interval [ -2, 2].
(2) A retailer determines that the cost of ordering and storing units of a product can be modelled by C(x) = 3x + (2000/x), 0 ≤ x ≤ 200. Find the order size that will minimize ordering and storage cost.
The demand curve of a firm is p = 1200 - 21q and its total cost is C(q) =2q3-66q2+ 600q + 1000 where q is the output of the firm (in thousands).
(i) Derive an expression R (q ) , for the firm' s revenue function.
(ii) Derive an expression π(q) for the firm' s profit function.
(iii) Is the rate of change of profit increasing or decreasing when the output level of the firm is 10,000 units?
(iv) Determine the level of output at which profit is maximized.
Determine the y-intercept, the zeros, the number of turning points, and the behavior of the graph and sketch the graph of the following polynomial functions.
1. P(x)=-2x³-22x²-60x
2. P(x)=6x³+5x²-2x-1
4. P(x)=x⁴+x³-10x²+8
The formula for calculating the sum of all natural integers from 1 to n is well-known:
Similary, we know about the formula for calculating the sum of the first n squares:
Now, we reduce one of the two multipliers of each product by one to get the following sum:
Mn = 0 · 1 + 1 · 2 + 2 · 3 + 3 · 4 + ... + (n − 1) · n
Find an explicit formula for calculating the sum Mn.
The well-known formula for calculating the sum Sn of the positive integers from 1 to n was already part of Problem A.3. For this problem, we consider the following rollercoaster sum:
Here, we multiple the summands successively with 1, 2, 1, 2, 1, 2, ...
(a) Find an explicit formula to calculate this sum SN(2). (Assume that n is a multiple of 2.) Now, we consider the sum:
Here, we multiple the summands successively with 1, 2, 3, 1, 2, 3, ...
(b) Again, find an explicite formula for the sum . (Assume that n is a multiple of 3.)
(c) Express in the form of
where Sn is the formula from Problem A.3 and I, Y are rational constants.
(d) Find a formula for the general case of . (That means we multiple the summands successively with 1, 2, 3, ..., m, 1, 2, 3, ..., m, ...; Assume that n is a multiple of m.)
(e) Now, express the general formula as
and find explicit equations to calculate Im and Ym for a given m.
Determine the growth behaviour by expressing Im and Ym with the big O notation
