Calculus
posted by Sarah .
I was wondering if you would be able to check if my answers are correct:
1. An oil tank is being drained for cleaning. After t minutes there are v litres of oil left in the tank, where v(t)=40(20t)^2, 0<=t<=20. Determine the rate of change at the time t=10.
For this question i got 400L/min by substituting 10 into the equation.
2. Consider the function f(x)=5/2+x.
a) Determine the slope at the point with xcoordinate 1
To answer this question I used the formula m=lim h>0 f(x+h)f(x)/h. Then I got lim h>0 (5/3+h)(5/3)/h and my answer was 1/9.
b) Determine the equation tangent to the curve f(x) at the point with xcoordinate 1
my answer for this is y=1/9x+14/9

The volume is
v(t) = 40(40040t+t^2)
the rate of change is the derivative, so
v'(t) = 40(40+2t)
v'(10) = 40(40+20) = 800
2a Since your solution used 5/3, I will assume a typo in the problem.
if you meant (5/3) + x, the slope is always 1
if you meant 5/(3+x) then the slope at x is 5/(3+x)^2, so at x=1, the slope is 5/16
If you used the definition of the limit, then you went wrong somewhere.
f(x+h) = 5/(3+x+h)
f(x) = 5/(3+x)
f(x+h)f(x) = [5(3+x)  5(3+x+h)]/[(3+x)(3+x+h)]
= 5h/[(3+x)(3+x+h)]
divide by h and you have
5/[(3+x)(3+x+h)]
the limit as h>0 is thus
5/(3+x)^2
So, at (1,5/4), the tangent line is
y  5/4 = 5/16 (x1)
see the graphs at
http://www.wolframalpha.com/input/?i=plot+y%3D5%2F%283%2Bx%29%2C+y+%3D+5%2F16+%28x1%29%2B5%2F4 
Thanks Steve! For 2a, I meant 5/2+x, then i got 5/2+1 and then 5/3
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