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erica [24]
4 years ago
10

Determine where the function is increasing and where it is decreasing. (Enter your answers using interval notation. If an answer

does not exist, enter DNE.)
f(t) = (t3 + 3t2)3
increasing    

decreasing    
Mathematics
1 answer:
tankabanditka [31]4 years ago
3 0
<span>The function can only change from increasing to decreasing, and visa-versa at those points where the slope of the function is 0. And the slope of the function is determined by the first derivative of the function. So let's calculate the first derivative. f(t) = (t^3 + 3t^2)^3 f'(t) = d/dt[ (t^3 + 3t^2)^3 ] f'(t) = 3(t^3 + 3t^2)^2 * d/dt[ t^3 + 3t^2 ] f'(t) = 3(d/dt[ t^3 ] + 3 * d/dt[ t^2 ])(t^3 + 3t^2)^2 f'(t) = 3(3t^2 + 3 * 2t)(t^3 + 3t^2)^2 f'(t) = 3(3t^2 + 6t)(t^3 + 3t^2)^2 Simplify f'(t) = 3(3t^2 + 6t)(t^3 + 3t^2)^2 f'(t) = 3 * 3t(t + 2)(t^3 + 3t^2)^2 f'(t) = 9t(t + 2)(t^2(t + 3))^2 f'(t) = 9t(t + 2)t^4(t + 3)^2 f'(t) = 9t^5(t + 2)(t + 3)^2 And looking at the function, it becomes obvious that the roots (or inflection points) are at t = 0, t = -2, and t = -3. Now the only places where f(t) can switch directions is at those 3 inflection points. And at exactly those inflection points the curve is neither increasing, nor decreasing. If the slope of the function is positive, then its value is increasing, and if the slope is negative, then the function is decreasing. So all we need to do is calculate the value of the first derivative for any value between each inflection point plus one value smaller than the smallest inflection point and another value higher than the highest inflection point. Range from [-infinity, -3) f'(-4) = 18432 Since the value is positive, the function is increasing from [-infinity, -3) Range from (-3, -2) f'(-2.5) = 30.51758 Since the value is positive, the function is increasing from (-3, -2) Range from (-2, 0) f(-1) = -36 Since the value is negative, the function is decreasing from (-2, 0) Range from (0, infinity) f(1) = 64 Since the value is positive, the function is increasing from (0, +infinity) To summarize: increasing from [-infinity, -3) increasing from (-3,-2) decreasing from (-2,0) increasing from (0,infinity]</span>
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185

Step-by-step explanation:

All you have to do is multiply 200 by 92.5/100 (because it is 92.5%). This gives you 185.

Hope this helps!

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Question 6(Multiple Choice Worth 5 points)
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Answer:

Product 3

Step-by-step explanation:

You can easily find it out based on the last column (Year  3)... where product 3 has the highest market value.

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4 years ago
Scientists have found a relationship between the temperature and the height above a distant planet's surface. , given below, is
Liula [17]

Answer:

1) Option B is correct.

The inverse of the function, T⁻¹(x), represents the The height above the surface (in kilometers) when the temperature is x degrees Celsius.

2) T⁻¹(x) = 12.2 - 0.4x

3) T⁻¹(15) = 6.2 m

Step-by-step explanation:

1) The inverse of a function is a function that reverses the effects of the original function on the variable that determines the original function's value.

T(h) = 30.5 - 2.5h

The original function takes the height in kilometres and converts it to temperature at that point in degree Celsius, So, the inverse function will take the temperature in degree Celsius and produce the corresponding height in kilometres.

So, it is the The height above the surface (in kilometers) when the temperature is x degrees Celsius.

The inverse functuon is given as T⁻¹ (x)

2) To obtain T⁻¹(x)

T(h) = 30.5 - 2.5h

We make h the subject of formula

2.5h = 30.5 - T

h = (30.5 - T)/2.5

h = 12.2 - 0.4T

T⁻¹(x) = 12.2 - 0.4x

3) T⁻¹(x) = 12.2 - 0.4x

when x = 15°C

T⁻¹(15) = 12.2 - 0.4(15) = 6.2 m

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4 years ago
What is the maximum integer value that satisfies the inequality 5x-11≤ 43?
Nitella [24]

Answer:

10

Step-by-step explanation:

We start by getting the value of x

We have this as follows:

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x ≤ 54/5

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As we can see, the maximum integer closest to this decimal is the value 10

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