Calculus Refresher, version 2008.4 (2008)(en)(78s) by Garrett P.

By Garrett P.

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Additional info for Calculus Refresher, version 2008.4 (2008)(en)(78s)

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131 cos x sin5 x dx =? 132 sec2 x tan7 x dx =? 2 6 dx =? dx =? 133 (3 cos x + x) e6 sin x+x dx =? 134 ex ex + 1 dx =? 49 1 sin7 (3x + 1) + C 21 35. Area and definite integrals The actual definition of ‘integral’ is as a limit of sums, which might easily be viewed as having to do with area. One of the original issues integrals were intended to address was computation of area. First we need more notation. Suppose that we have a function f whose integral is another function F : f (x) dx = F (x) + C Let a, b be two numbers.

Well, the usual scenario is that some story problem will give you information in a way that requires you to take some trouble in order to determine the constants c, k. And, in case you were wondering where you get to take a derivative here, the answer is that you don’t really: all the ‘calculus work’ was done at the point where we granted ourselves that all solutions to that differential equation are given in the form f (t) = cekt . First to look at some general ideas about determining the constants before getting embroiled in story problems: One simple observation is that c = f (0) that is, that the constant c is the value of the function at time t = 0.

The second derivative of a function y = f (x) is written as y = f (x) = d2 d2 f d2 y f= = 2 2 dx dx dx 2 The third derivative is d3 d3 f d3 y f = = dx 3 dx 3 dx 3 And, generally, we can put on a ‘prime’ for each derivative taken. Or write y = f (x) = dn dn f dn y f= = n n dx dx dx n for the nth derivative. There is yet another notation for high order derivatives where the number of ‘primes’ would become unwieldy: dn f = f (n) (x) dx n as well. The geometric interpretation of the higher derivatives is subtler than that of the first derivative, and we won’t do much in this direction, except for the next little section.

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