Calculus I (AB): First Order Differential Equations

If y changes at a rate exactly three times its value and , find y(5).

Difficulty:

Solution:

If the rate of change of y is three times y, this is easily reflected by the differential equation . To solve it, divide both sides by y, multiply both by dt, and integrate both sides separately.

Note that each C is not necessarily equal. Instead, C just represents some number we don't know. That is, we don't know until now. We can use the fact that y(1) = 4e to find the specific C that works for this problem. Plug 1 in for t , and set y = 4e:

Now we have the solution to our differential equation, complete with C. Our final goal is to calculate y(5), which means we need to plug in 5 for t:

Estimate the value of cos(0.3) using a sixth degree Maclaurin series. Note: 0.3 is measured in radians.

Difficulty:

Solution:

If you don't know this Maclaurin series by heart (which you should, along with sine and the natural exponential function), you can always generate it if you know that the generic Maclaurin term has form . Therefore, you should calculate six derivatives of cosine, and plug 0 into each to get started:

Now we can write out the terms of the series, noting that all the terms of odd degree will vanish, since they'll have a coefficient of 0, which we determined by our above work:

There's the Maclaurin polynomial of degree 6 for cos x. Now, plug 0.3 into each x to get the Maclaurin polynomial approximation of cos (0.3):

That's a dang good approximation of cos (0.3)! The actual value is .9553364891 (and some change).