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# 2011 Calculus AB free response #5a

Using a tangent line to extrapolate a function from a known initial condition. Created by Sal Khan.

## Want to join the conversation?

• I have learned about the rules and techniques but I cannot answer these questions myself without Sal's help. What should I do? • instead of finding the numerical value of the tangent and then framing the equation can't we just analyse that since the differential equation tells us the rate of change of the garbage in the landfill on a yearly basis, so for 3 months the rate of change of the garbage in the landfills would be one fourth of that in one year??
that is,
the slope of the function at W is 44. so at the end of three months the slope would be one fourth of 44 i.e. 11.
so the amount of garbage at the end of three months would be 1411 tons.
(1 vote) • Yes, you can do almost that. You can realise that the slope is 44 at `t=0`, and since that means that the linear approximation would increase 44 in 1 year, then you can divide by 4 to get that it would increase by 11 in 1/4 of a year.

The only incorrect statement in your solution is that the slope will always be 44, it's the increase over 1/4 that's 11, the slope doesn't change.