Sorry, dying here.

 Too busy to write, studying for APUSH, scribbled this out last week and meant to get this out earlier but forgot. 

I wrote myself a calc practice problem or two, all yarn related. How many can you solve?

1. The circumference C (in stitches) of a Pi Shawl is defined as a function of its radius r (in rounds); C = 2r. The area of a circle is pi*r^2. At the point where the circumference is 4, the radius is increasing by 3 rounds/hr. What is the rate of change of the area with respect to time? (Area is measured in rounds^2) 

2. The rate at which a knitter adds stitches to a blanket, in stitches per hour, can be modeled in terms of the number of stitches in the blanket (unrealistic, but still): dS/dt = 1/5(500 - S), for 0 ≤ t ≤ 25 . At time t=0, there are 10 stitches in the blanket. a) Find the rate at which the knitter adds stitches when there are 250 stitches in the blanket. b) Find S(t). c) Will the blanket be completely done (S=500) after 25 hours? How many stitches did the knitter leave undone, rounding to the nearest stitch? Explain your answer. (Bonus: How much sleep does the knitter get in this period?)

3. The amount of fiber (in ounces), f, that a spinner goes through to make a yarn of thickness x (in inches, and yes I am aware that that’s not how it’s measured) is a differentiable function for 0 ≤ x ≤ 3, with f’(x) = 3x+1. a) Find f’(1). Using correct units, explain its meaning in the context of the problem. b) Given that f(1) = 2, find f(3). 

Answers in the next paragraph, and don’t look until you actually finish. 

1. 12*pi rounds^2 per hour. 

2. a) 50 stitches/hour. b) S(t) = 500 - 490*e^(-t/5). c) no, S(25)=496. Bonus: no. The knitter won’t sleep. 

3. a) f’(1)= 4, so the amount of fiber needed is increasing by 4 ounces per inch of thickness at 1 inch of thickness. b) f(3) = 16.