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mean_value_theorem.html
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mean_value_theorem.html
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<h1>Average of a function and the Mean Value Theorem for integrals</h1><p>Let $f(x)$ be a continuous function over the interval $[a,b]$ with $a < b$.</p><p>The average value of $f$ over $[a,b]$ is defined by:</p>$$~
\frac{1}{b-a} \int_a^b f(x) dx.
~$$<p>If $f$ is a constant, this is just the contant value, as would be expected. If $f$ is <em>piecewise</em> linear, then this is the weighted average of these constants.</p><h2>Examples</h2><h5>Average velocity</h5><p>The average velocity between times $a < b$, is simply the change in position during the time interval divided by the change in time. In notation, this would be $(x(b) - x(a)) / (b-a)$. If $v(t) = x'(t)$ is the velocity, then by the second part of the fundamental theorem of calculus, we have, in agreement with the definition above, that:</p>$$~
\text{average velocity} = \frac{x(b) - x(a)}{b-a} = \frac{1}{b-a} \int_a^b v(t) dt.
~$$<p>The average speed is the change in <em>total</em> distance over time, which is given by</p>$$~
\text{average speed} = \frac{1}{b-a} \int_a^b \lvert v(t)\rvert dt.
~$$<p>Let $\bar{v}$ be the average velocity. Then we have $\bar{v} \cdot(b-a) = x(b) - x(a)$, or the change in position can be written as a constant ($\bar{v}$) times the time, as though we had a constant velocity. This is an old intuition. <a href="http://www.math.harvard.edu/~knill/teaching/math1a_2011/exhibits/bressoud/">Bressoud</a> comments on the special case known to scholars at Merton College around 1350 that the distance traveled by an object under uniformly increasing velocity starting at $v_0$ and ending at $v_t$ is equal to the distance traveled by an object with constant velocity of $(v_0 + v_t)/2$.</p><h5>Example</h5><p>What is the average value of $f(x)=\sin(x)$ over $[0, \pi]$?</p>$$~
\text{average} = \frac{1}{\pi-0} \int_0^\pi \sin(x) dx = \frac{1}{\pi} (-\cos(x)) \big|_0^\pi = \frac{2}{\pi}
~$$<p>Visually, we have:</p><pre class="sourceCode julia">using Plots; gadfly()
plot(sin, 0, pi)
plot!([0, pi], 2/pi * [1,1])</pre>
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<script> <![CDATA[
(function (glob, factory) {
// AMD support
if (typeof define === "function" && define.amd) {
// Define as an anonymous module
define("Gadfly", ["Snap.svg"], function (Snap) {
return factory(Snap);
});
} else {
// Browser globals (glob is window)
// Snap adds itself to window
glob.Gadfly = factory(glob.Snap);
}
}(this, function (Snap) {
var Gadfly = {};
// Get an x/y coordinate value in pixels
var xPX = function(fig, x) {
var client_box = fig.node.getBoundingClientRect();
return x * fig.node.viewBox.baseVal.width / client_box.width;
};
var yPX = function(fig, y) {
var client_box = fig.node.getBoundingClientRect();
return y * fig.node.viewBox.baseVal.height / client_box.height;
};
Snap.plugin(function (Snap, Element, Paper, global) {
// Traverse upwards from a snap element to find and return the first
// note with the "plotroot" class.
Element.prototype.plotroot = function () {
var element = this;
while (!element.hasClass("plotroot") && element.parent() != null) {
element = element.parent();
}
return element;
};
Element.prototype.svgroot = function () {
var element = this;
while (element.node.nodeName != "svg" && element.parent() != null) {
element = element.parent();
}
return element;
};
Element.prototype.plotbounds = function () {
var root = this.plotroot()
var bbox = root.select(".guide.background").node.getBBox();
return {
x0: bbox.x,
x1: bbox.x + bbox.width,
y0: bbox.y,
y1: bbox.y + bbox.height
};
};
Element.prototype.plotcenter = function () {
var root = this.plotroot()
var bbox = root.select(".guide.background").node.getBBox();
return {
x: bbox.x + bbox.width / 2,
y: bbox.y + bbox.height / 2
};
};
// Emulate IE style mouseenter/mouseleave events, since Microsoft always
// does everything right.
// See: http://www.dynamic-tools.net/toolbox/isMouseLeaveOrEnter/
var events = ["mouseenter", "mouseleave"];
for (i in events) {
(function (event_name) {
var event_name = events[i];
Element.prototype[event_name] = function (fn, scope) {
if (Snap.is(fn, "function")) {
var fn2 = function (event) {
if (event.type != "mouseover" && event.type != "mouseout") {
return;
}
var reltg = event.relatedTarget ? event.relatedTarget :
event.type == "mouseout" ? event.toElement : event.fromElement;
while (reltg && reltg != this.node) reltg = reltg.parentNode;
if (reltg != this.node) {
return fn.apply(this, event);
}
};
if (event_name == "mouseenter") {
this.mouseover(fn2, scope);
} else {
this.mouseout(fn2, scope);
}
}
return this;
};
})(events[i]);
}
Element.prototype.mousewheel = function (fn, scope) {
if (Snap.is(fn, "function")) {
var el = this;
var fn2 = function (event) {
fn.apply(el, [event]);
};
}
this.node.addEventListener(
/Firefox/i.test(navigator.userAgent) ? "DOMMouseScroll" : "mousewheel",
fn2);
return this;
};
// Snap's attr function can be too slow for things like panning/zooming.
// This is a function to directly update element attributes without going
// through eve.
Element.prototype.attribute = function(key, val) {
if (val === undefined) {
return this.node.getAttribute(key);
} else {
this.node.setAttribute(key, val);
return this;
}
};
Element.prototype.init_gadfly = function() {
this.mouseenter(Gadfly.plot_mouseover)
.mouseleave(Gadfly.plot_mouseout)
.dblclick(Gadfly.plot_dblclick)
.mousewheel(Gadfly.guide_background_scroll)
.drag(Gadfly.guide_background_drag_onmove,
Gadfly.guide_background_drag_onstart,
Gadfly.guide_background_drag_onend);
this.mouseenter(function (event) {
init_pan_zoom(this.plotroot());
});
return this;
};
});
// When the plot is moused over, emphasize the grid lines.
Gadfly.plot_mouseover = function(event) {
var root = this.plotroot();
var keyboard_zoom = function(event) {
if (event.which == 187) { // plus
increase_zoom_by_position(root, 0.1, true);
} else if (event.which == 189) { // minus
increase_zoom_by_position(root, -0.1, true);
}
};
root.data("keyboard_zoom", keyboard_zoom);
window.addEventListener("keyup", keyboard_zoom);
var xgridlines = root.select(".xgridlines"),
ygridlines = root.select(".ygridlines");
xgridlines.data("unfocused_strokedash",
xgridlines.attribute("stroke-dasharray").replace(/(\d)(,|$)/g, "$1mm$2"));
ygridlines.data("unfocused_strokedash",
ygridlines.attribute("stroke-dasharray").replace(/(\d)(,|$)/g, "$1mm$2"));
// emphasize grid lines
var destcolor = root.data("focused_xgrid_color");
xgridlines.attribute("stroke-dasharray", "none")
.selectAll("path")
.animate({stroke: destcolor}, 250);
destcolor = root.data("focused_ygrid_color");
ygridlines.attribute("stroke-dasharray", "none")
.selectAll("path")
.animate({stroke: destcolor}, 250);
// reveal zoom slider
root.select(".zoomslider")
.animate({opacity: 1.0}, 250);
};
// Reset pan and zoom on double click
Gadfly.plot_dblclick = function(event) {
set_plot_pan_zoom(this.plotroot(), 0.0, 0.0, 1.0);
};
// Unemphasize grid lines on mouse out.
Gadfly.plot_mouseout = function(event) {
var root = this.plotroot();
window.removeEventListener("keyup", root.data("keyboard_zoom"));
root.data("keyboard_zoom", undefined);
var xgridlines = root.select(".xgridlines"),
ygridlines = root.select(".ygridlines");
var destcolor = root.data("unfocused_xgrid_color");
xgridlines.attribute("stroke-dasharray", xgridlines.data("unfocused_strokedash"))
.selectAll("path")
.animate({stroke: destcolor}, 250);
destcolor = root.data("unfocused_ygrid_color");
ygridlines.attribute("stroke-dasharray", ygridlines.data("unfocused_strokedash"))
.selectAll("path")
.animate({stroke: destcolor}, 250);
// hide zoom slider
root.select(".zoomslider")
.animate({opacity: 0.0}, 250);
};
var set_geometry_transform = function(root, tx, ty, scale) {
var xscalable = root.hasClass("xscalable"),
yscalable = root.hasClass("yscalable");
var old_scale = root.data("scale");
var xscale = xscalable ? scale : 1.0,
yscale = yscalable ? scale : 1.0;
tx = xscalable ? tx : 0.0;
ty = yscalable ? ty : 0.0;
var t = new Snap.Matrix().translate(tx, ty).scale(xscale, yscale);
root.selectAll(".geometry, image")
.forEach(function (element, i) {
element.transform(t);
});
bounds = root.plotbounds();
if (yscalable) {
var xfixed_t = new Snap.Matrix().translate(0, ty).scale(1.0, yscale);
root.selectAll(".xfixed")