This week in AP Calc we revisited u-substitution in order to accomplish solving definite integrals. Basically, we were looking for a way to re-write a definite integral equation from terms of x to terms of u which means the bounds would have to change. We first would assign a (u) to f(x) equation and then derive this in terms of x. Using u and d(u) you would re-write the equation and anti-derive it. Then, to find the bounds in terms of u, you would take the bounds in terms of x and plug them into the equation you have designated for u. The two answers will be the bounds of the new equation. After anti-deriving the new equation, plug in the two new bound values into F(b) - F(a) and the answer will be the definite integral.
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For solving differential equations, we looked at two separate methods. First, you can take a function and anti-derive it just by reasoning. However, there are some more in depth and conceptually challenging differential equations. So, we used a method in which we would plug in the x value for the a and the y value in for the c. This makes solving for c much simpler and thus makes solving for the entire equation much simpler.
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We also took a look at slope fields. Essentially, we look at what the derivative of y in terms of x will be in relation to a specific function pertaining to x and/or y. So, if you have a function dy/dx = 1.1x^2, each x-value will display a different slope but the y-values will not change since there is only an x-value present in the equation.