Calculus Problems

Question

A yeast culture is growing at a rate of  grams per hour where t is time in hours since the culture was initiated. Use Riemann Sums with 50 left-point rectangles to compute and then interpret the area under Y'(t) on the interval from t= 0 to t=7. Round two decimal places.

Answer 1

Divide up the space into 50 rectangles by taking (7 - 0)/50 = 0.14 so each rectangle is 0.14 units wide. Since it is a Left Riemann Sum, the height of the first rectangle will be Y'(0) = 0.3. The height of the 2nd rectangle will be Y'(0.14) ≈0.3042. The height of the 3rd rectangle will be Y'(0.28) ≈ 0.3085 and then keep going until you get to the 50th rectangle which will be Y'(6.86) ≈ 0.5957. So the approximate given by the LRS will be 0.14(0.3 + 0.3042+0.3085+ . . . 5957)

BTW, Excel is a super useful tool for this. I did the problem in like 30 seconds.

The actual answer is about 3.0413 (using the integral). The Left Riemann Sum you calculate for this answer will be a bit less than that (the LRS is an underestimate).

The answer is the mass of the yeast culture that grew (in grams) between 0 and 7 hours