![]() Kind of a placeholder variable to help us keep this x up here. And so the integral from zero to seven, if this was a t-axis, and, once again, t is just When x is equal to seven, orĪnother way to think about it, g of seven is going to be the integral from zero to seven of f of t dt. Here we're gonna think a little bit deeper about what it means toīe this definite integral from zero to x. Positive on the interval from the closed interval from seven to 12? So the positive on the closed What is a appropriateĬalculus-based justification for the fact that g is Have a different f and g here, and we see it every time with the graph. Well, this would be a justificationįor a relative minimum, but it is not calculus-based. Graph of g around x equals eight where g of eight is the smallest value. f is negative before x equals eight and positive after x equals eight. And the choice that describes that, this is starting to get there, but this alone isn't enoughįor a relative minimum point. Your derivative goes from being negative to positive, that means your original function goes from decreasing to increasing. Why is that valuable? Because think about if In order to have a relative minimum point, our derivative has to crossįrom being negative to positive. But that alone does not tell us we have a relative minimum point. Slope of the tangent line of g at that point is zero. To zero at x equals eight, that tells us that the That y is equal to zero, that the derivative is equal Have a relative minimum at x equals eight? Well, the fact that we cross, that we're at the x-axis, Graph of the derivative, how do we know that we What is an appropriateĬalculus-based justification for the fact that g has a relative minimum at x equals eight? So once again, they've graphed f here, which is the same thingĪs the derivative of g. Have the exact same setup, which actually all of Well, if we had the graph of g, this would be a justification, but it wouldn't be aĬalculus-based justification. Over much of that interval, then actually on this part our original function Interval right over here, if this was our graph of f or Up over that interval, but for much of that In fact, you could haveĪ situation like this where you're concave Well, just because yourĭerivative is concave up doesn't mean that your originalįunction is concave up. It doesn't tell you that your original function is concave up. Because if your derivative is positive, that just means your originalįunction is increasing. That's not a sufficientĬalculus-based justification. Increasing on that interval, which means that the originalįunction is concave up. So our calculus-based justification that we'd wanna use is that, look, f, which is g prime, is Graph of the derivative, and it is indeed increasing Increasing over an interval, then you're concave up on that interval. Or another way to think about it, if your derivative is Or another way of thinking about it, your derivative is increasing. Slope of tangent line, of tangent, slope of So what does it mean to be concave up? Well, that means that your ![]() And we wanna know aĬalculus-based justification from this graph that lets us And so we're thinking about the interval, the open interval from five to 10, and we have g's derivative graphed here. If this is x, this would be g prime of x. Graph of our function f, which you could also viewĪs the graph of g prime. If this is the t axis, then this is y is equal to f of t. But when you take theĭerivative of both sides, you realize that the functionį, which is graphed here. It could be alpha, it could be gamma, it could be a, b, orĬ, whatever we choose, but this is still, right over So we just had to pick kind ofĪnother placeholder variable. We had x as an upper bound, or at least confusing, and we were also integrating This thing right over here is actually a function of x 'cause x is this upper bound. Reason why we introduced this variable t here is The derivative of this with respect to x would just be f of x. If we took the derivative of both sides of this equation, we would get that g prime Understand this relationship between g and f. What it means to be concave up, let's just make sure we Up on the open interval from five to 10? So concave up. What is an appropriateĬalculus-based justification for the fact that g is concave Equal to the definite integral from zero to x of f of t dt. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |