With traditional integrals, our "path" was straight and linear (most of the …
With traditional integrals, our "path" was straight and linear (most of the time, we traversed the x-axis). Now we can explore taking integrals over any line or curve (called line integrals).
You've done some work with line integral with scalar functions and you …
You've done some work with line integral with scalar functions and you know something about parameterizing position-vector valued functions. In that case, welcome! You are now ready to explore a core tool math and physics: the line integral for vector fields. Need to know the work done as a mass is moved through a gravitational field. No sweat with line integrals.
If over the last hour on the highway, you averaged 60 miles …
If over the last hour on the highway, you averaged 60 miles per hour, then you must have been going exactly 60 miles per hour at some point. This is the gist of the mean value theorem (which generalizes the idea for any continuous, differentiable function).
Can calculus be used to figure out when a function takes on …
Can calculus be used to figure out when a function takes on a local or global maximum value? Absolutely. Not only that, but derivatives and second derivatives can also help us understand the shape of the function (whether they are concave upward or downward). If you have a basic conceptual understanding of derivatives, then you can start applying that knowledge here to identify critical points, extrema, inflections points and even to graph functions.
No restrictions on your remixing, redistributing, or making derivative works. Give credit to the author, as required.
Your remixing, redistributing, or making derivatives works comes with some restrictions, including how it is shared.
Your redistributing comes with some restrictions. Do not remix or make derivative works.
Most restrictive license type. Prohibits most uses, sharing, and any changes.
Copyrighted materials, available under Fair Use and the TEACH Act for US-based educators, or other custom arrangements. Go to the resource provider to see their individual restrictions.