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2 years ago

HELP ASAP 40 POINTS REAL ANSWER

Mathematics

1 answer:

Jet001 [13]2 years ago

8 0

Answer:

b) number of flowers.

c) 2830

Step-by-step explanation:

We are given the following in the question:

The number of flowers bloomed is given by the function:

where s is the s is the number of seeds she planted.

The number of seeds planted per week is given by

where w represents the number of weeks.

a) composite function that represents how many flowers Lily can expect to bloom over a certain number of weeks.

The composite function can be written as:

where f(w) gives the number of flowers that bloomed in w weeks.

b) units of measurement for the composite function

The composite functions gives the number of flower that will bloom in w weeks. Thus, the unit of measurement is number of flowers.

c) Number of flowers in 35 weeks.

We put w = 35 in the composite function.

2830 flowers will bloom in 35 weeks.

Thanks

Step-by-step explanation:

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Part 1)

See Below.

Part 2)

$\displaystyle (-0.179, -0.178) \cup (-0.010, 0.012)$

Step-by-step explanation:

Part 1)

The linear approximation L for a function f at the point x = a is given by:

$\displaystyle L \approx f'(a)(x-a) + f(a)$

We want to verify that the expression:

$1-36x$

Is the linear approximation for the function:

$\displaystyle f(x) = \frac{1}{(1+9x)^4}$

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Simplify. Hence:

$\displaystyle f'(x) = -\frac{36}{(1+9x)^{5}}$

Then the slope of the linear approximation at x = 0 will be:

$\displaystyle f'(1) = -\frac{36}{(1+9(0))^5} = -36$

And the value of the function at x = 0 is:

$\displaystyle f(0) = \frac{1}{(1+9(0))^4} = 1$

Thus, the linear approximation will be:

$\displaystyle L = (-36)(x-(0)) + 1 = 1 - 36x$

Hence verified.

Part B)

We want to determine the values of x for which the linear approximation L is accurate to within 0.1.

In other words:

$\displaystyle \left| f(x) - L(x) \right | \leq 0.1$

By definition:

$\displaystyle -0.1\leq f(x) - L(x) \leq 0.1$

Therefore:

$\displaystyle -0.1 \leq \left(\frac{1}{(1+9x)^4} \right) - (1-36x) \leq 0.1$

We can solve this by using a graphing calculator. Please refer to the graph shown below.

We can see that the inequality is true (i.e. the graph is between y = 0.1 and y = -0.1) for x values between -0.179 and -0.178 as well as -0.010 and 0.012.

In interval notation:

$\displaystyle (-0.179, -0.178) \cup (-0.010, 0.012)$

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