Hey guys! Quick Calculus question... How do you find ? I know that you can take the derivative of a natural log, but I can't fathom doing the integral of a natural log.

Thanks in advance!

the input value is the number of times you fold it. the input value is also known as the x-value. 

the output value is its thickness. the output value is the y-value.

when you don't fold it all, the thickness will remain the same, which is 1. the ordered pair would be (0,1).

when you fold it once, the thickness will double from 1 to  2. the ordered pair would be (1,2).

when you fold it a second time, the thickness will double from 2 to  4. the ordered pair would be (2,4).

fold it one more time, and the thickness will be 8. the ordered pair is (3,8)

fold it the fourth time, and the thickness doubles. 8 x 2 = 16. the ordered pair is (4,16)

fold it the fifth time, and the thickness goes from 16 to 32. the ordered pair is (5,32)

the ordered pairs would be:

(0,1)

(1,2)

(2,4)

(3,8)

(4,16)

(5,32)

(6,64)

(7,128)

and so on : p

the question asks for 6 ordered pairs, so i bolded the first six.

now, the equation for this would be 

if we graph that, and we do the vertical line test, then we know this is a function.

it is a function because each x-value has a different y-value. there are no y-values that have the same x-values.

i have attached the graph of that line.

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step-by-step explanation:

14

step-by-step explanation:

the ratio of grapes to blueberries she should use is 12: 3

step-by-step explanation:

if mandie is making 15 cups of fruit salad, the ratio of grapes to blueberries she should use is 12: 3.

its an acute right triangle

step-by-step explanation: