73) After a new firm starts in business, it finds that its rate of profits (in hundreds of dollars per year) after x years of operation is given by P'(x) = 3x^2 + 6x + 6. Find the profit in year 6 of the operation.
the answer is $13,000 but how?
53) what can you say about the inflection points of the quartic curve y= ax^4 + bx^3 + cx^2 + dx + e, a doesn't equal 0? give reasons.
I've never even seen a quartic curve before..
the answer is $13,000 but how?
53) what can you say about the inflection points of the quartic curve y= ax^4 + bx^3 + cx^2 + dx + e, a doesn't equal 0? give reasons.
I've never even seen a quartic curve before..
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73) You take an integral of your rate of profits, P'(x) = 3x² + 6x + 6, for the 6th year of operation (from yrs 5 to 6):
∫P'(x) from x = 5 to x = 6
= ∫(3x² + 6x + 6)dx from x = 5 to x = 6
Use reverse power rule to integrate and you get:
x³ + 3x² + 6x evaluated from x = 5 to x = 6:
= [(6)³ + 3(6)² + 6(6)] - [(5)³ + 3(5)² + 6(5)]
= [216 + 108 + 36] - [125 + 75 + 30]
= 360 - 230 = 130
And since profits are in hundreds of dollars, 130 * 100 = $13000
53) Points of inflection occur when the second derivative changes sign (we'll try to find the zeros first, then determine whether y'' changes sign around these zeros):
y = ax⁴ + bx³ + cx² + dx + e
y' = 4ax³ + 3bx² + 2cx + d
y'' = 12ax² + 6bx + 2c
To find zeros, we would need to use the quadratic formula. However, we DO NOT know whether there are any zeros or not because we don't have enough information to determine whether the discriminant, b² - 4ac, is positive, negative or 0:
Our quadratic is in the form y = ax² + bx + c
y'' = 12ax² + 6bx + 2c --> a = 12a, b = 6b, c = 2c
b² - 4ac = (6b)² - 4(12a)(2c) = 36b² - 96ac
We have no idea whether 36b² - 96ac is positive, negative, or zero, so unfortunately, there isn't enough information to determine anything about the inflection points.
∫P'(x) from x = 5 to x = 6
= ∫(3x² + 6x + 6)dx from x = 5 to x = 6
Use reverse power rule to integrate and you get:
x³ + 3x² + 6x evaluated from x = 5 to x = 6:
= [(6)³ + 3(6)² + 6(6)] - [(5)³ + 3(5)² + 6(5)]
= [216 + 108 + 36] - [125 + 75 + 30]
= 360 - 230 = 130
And since profits are in hundreds of dollars, 130 * 100 = $13000
53) Points of inflection occur when the second derivative changes sign (we'll try to find the zeros first, then determine whether y'' changes sign around these zeros):
y = ax⁴ + bx³ + cx² + dx + e
y' = 4ax³ + 3bx² + 2cx + d
y'' = 12ax² + 6bx + 2c
To find zeros, we would need to use the quadratic formula. However, we DO NOT know whether there are any zeros or not because we don't have enough information to determine whether the discriminant, b² - 4ac, is positive, negative or 0:
Our quadratic is in the form y = ax² + bx + c
y'' = 12ax² + 6bx + 2c --> a = 12a, b = 6b, c = 2c
b² - 4ac = (6b)² - 4(12a)(2c) = 36b² - 96ac
We have no idea whether 36b² - 96ac is positive, negative, or zero, so unfortunately, there isn't enough information to determine anything about the inflection points.