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# 2011 Calculus AB free response #5b

Using the second derivative to judge whether an approximation with the tangent line is an overestimate or underestimate. Created by Sal Khan.

## Want to join the conversation?

• Isn't it posssible that the second derivative switches back and we have an overestimate?
• That would a possibility, except that we know for sure that W(t) is concave up for the whole interval from 0 to 1/4. This is because W''(t) is always positive on this interval.
• When we used the 2nd derivative equation to check if it was an under/over-estimate, why did we plug in t as 0 instead of 1/4, because in the problem it says when t=1/4
• Strictly speaking, to rely on this test we need to be able to determine that the 2nd derivative is either positive for the entire interval or negative for the entire interval. If it changes sign during the interval we can't use it to determine whether the estimate is high or low. In this case, we can see that if W'' is positive at t = 0 it will always be positive because W is increasing and the only way for W'' to become negative would be for W to fall below 300.
• why do we plug in 0 for t and not 1/4?
• Because we are estimating what t=1/4 might be based on the tangent we found at t=0 because we only know (and are given) the initial conditions. It is enough that the second derivative at 0 tells us if the function's tangent is increasing or decreasing or 0, so as to tell us if our tangent estimate is over, under or about right.
• What is this wierd notation `d^2W/ dt^2` ? If it's a second derivative then why not `d^2W/ d^2t`?