theangelsanchez
30.07.2021 •
Mathematics
Use calculus to find the absolute maximum and minimum values of the function. f(x) = 5x − 10 cos(x), −2 ≤ x ≤ 0 (a) Use a graph to find the absolute maximum and minimum values of the function to two decimal places.
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Ответ:
The absolute maximum is about -5.84 at x = -2.
And the absolute minimum is about -11.28 at x = -π/6.
Step-by-step explanation:
We want to find the absolute maximum and minimum values of the function:
First, we should evaluate the endpoints of the interval:
And:
Recall that extrema of a function occurs at its critical points. The critical points of a function are whenever its derivative is zero or undefined.
So, find the derivative of the function:
Differentiate:
Set the function equal to zero:
And solve for x:
Using the unit circle, our solutions are:
There is only one solution in the interval [-2, 0]:
Thus, we only have one critical point on the interval.
Substituting this back into the function yields:
In conclusion, the absolute maximum value of f on the interval [-2, 0] is about -5.8385 at x = -2 and the absolute minimum value of f is about -11.2782 at x = -π/6.
We can see this from the graph below as well.
Ответ:
To make a gallons,
Ms. O'Brien needs
12 8/10 cups of water or 12.8 cups of water
3 1/5 cups of water or 3.2 cups of vinegar
Step-by-step explanation:
Ms. O'Brien makes her own cleaning solution to clean her floors! (yet, she can't clean her measuring cups?)
She mixes 8 cups of water with 2 cups of vinegar.
Hence: The total number of cups for the cleaning mixture = 8 cups of water + 2 cups of vinegar = 10 cups
If she wants to make a 1 gallon mixture of her cleaning solution, how much water and how much vinegar should she add?
Note that: There are 16 cups in a gallon
Hence:
For number of cups of water:
10 cups of cleaning mixture = 8 cups of water
16 cups of cleaning mixture = x
Cross Multiply
10x = 16 × 8
x = 16 × 8/10
x = 12 8/10 cups of water or 12.8 cups of water
For the number of cups of vinegar
10 cups of cleaning mixture = 2 cups of water
16 cups of cleaning mixture = x
Cross Multiply
10x = 16 × 2
x = 16 × 2/10
x = 3 1/5 cups of water or 3.2 cups of vinegar
Therefore, to make a gallon
Ms. O'Brien needs
12 8/10 cups of water or 12.8 cups of water
3 1/5 cups of water or 3.2 cups of vinegar