Skip to content

HTTPS clone URL

Subversion checkout URL

You can clone with HTTPS or Subversion.

Download ZIP
Fetching contributors…

Cannot retrieve contributors at this time

126 lines (118 sloc) 4.741 kb
<!DOCTYPE html>
<html data-require="math graphie expressions">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Derivatives 1</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div id="quadratic">
<div class="vars">
<div data-ensure="abs( C - ( B * B ) / ( 4 * A ) ) < 9">
<var id="A">randRangeNonZero( -2, 2 )</var>
<var id="B">randRangeNonZero( -4, 4 )</var>
<var id="C">( A > 0 ? -1 : 1 ) * randRange( 3, 8 )</var>
<var id="H">randFromArray([ "\\Delta x", "h" ])</var>
</div>
<var id="X" data-ensure="abs( ( A * X + B ) * X + C ) < 7">randRange( -6, 6 )</var>
</div>
<div class="question">
<p>What is the slope of the line tangent to <code>f(x) = <var>expr(["+", ["*", A, ["^", "x", 2]], ["*", B, "x"], C])</var></code> at <code>x = <var>X</var></code>?</p>
<div id="graph" class="graphie">
init({
range: [ [-10, 10], [-10, 10] ],
scale: [20, 20]
});
grid( [-10, 10], [-10, 10], {
stroke: "#e5e5e5"
});
style({
stroke: "#888",
strokeWidth: 2,
arrows: "->"
}, function() {
line( [-10, 0], [10, 0] );
line( [0, -10], [0, 10] );
});
plot( function( x ) {
return ( A * x + B ) * x + C;
}, [-10, 10], {
stroke: "#6495ED"
});
circle( [ X, ( A * X + B ) * X + C ], 3 / 20, {
fill: "black",
stroke: "none"
} );
</div>
</div>
<div class="solution"><var>2 * A * X + B</var></div>
</div>
</div>
<div class="hints">
<div>
<div class="graphie" data-update="graph">
plot( function( x ) {
return ( 2 * A * X + B ) * ( x - X ) + ( A * X + B ) * X + C;
}, [-10, 10], {
stroke: "black",
strokeWidth: 1
});
</div>
<p>The slope of the tangent line is <code>\displaystyle \lim_{<var>H</var> \to 0} \frac{f(x + <var>H</var>) - f(x)}{<var>H</var>}</code>.</p>
</div>
<p><code>\qquad = \displaystyle \lim_{<var>H</var> \to 0} \frac{(<var>expr(
["+", ["*", A, ["^", ["+", "x", H], 2]],
["*", B, ["+", "x", H]],
C]
)</var>) - (<var>expr(
["+", ["*", A, ["^", "x", 2]],
["*", B, "x"],
C]
)</var>)}{<var>H</var>}</code></p>
<p><code>\qquad = \displaystyle \lim_{<var>H</var> \to 0} \frac{(<var>expr(
["+", ["*", A, ["+", ["^", "x", 2],
"2x " + H,
["^", H, 2]]],
["*", B, ["+", "x", H]],
C]
)</var>) - (<var>expr(
["+", ["*", A, ["^", "x", 2]],
["*", B, "x"],
C]
)</var>)}{<var>H</var>}</code></p>
<p><code>\qquad = \displaystyle \lim_{<var>H</var> \to 0} \frac{<var>expr(
["+", ["*", A, ["^", "x", 2]],
["*", 2 * A, "x " + H],
["*", A, ["^", H, 2]],
["*", B, "x"],
["*", B, H],
C,
["*", -A, ["^", "x", 2]],
["*", -B, "x"],
-C]
)</var>}{<var>H</var>}</code></p>
<p><code>\qquad = \displaystyle \lim_{<var>H</var> \to 0} \frac{<var>expr(
["+", ["*", 2 * A, "x " + H],
["*", A, ["^", H, 2]],
["*", B, H]]
)</var>}{<var>H</var>}</code></p>
<p><code>\qquad = \displaystyle \lim_{<var>H</var> \to 0} <var>expr(
["+", ["*", 2 * A, "x"],
["*", A, H],
B]
)</var></code></p>
<p><code>\qquad = \displaystyle <var>expr(
["+", ["*", 2 * A, "x"],
B]
)</var></code></p>
<p><code>\qquad = \displaystyle <var>expr(
["+", ["*", 2 * A, X],
B]
)</var></code></p>
<p><code>\qquad = <var>2 * A * X + B</var></code></p>
</div>
</div>
</body>
</html>
Jump to Line
Something went wrong with that request. Please try again.