diff --git a/ptx/sec_par_calc.ptx b/ptx/sec_par_calc.ptx index 5f91f4226..98fa296f5 100644 --- a/ptx/sec_par_calc.ptx +++ b/ptx/sec_par_calc.ptx @@ -371,7 +371,7 @@ any line that passes through the center of a circle intersects the circle at right angles.

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Illustrating how a circle's normal lines pass through its center @@ -661,7 +661,7 @@

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Graphing the parametric equations in to demonstrate concavity @@ -789,38 +789,12 @@ .

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- The points of inflection are found by setting \frac{d^2y}{dx^2}=0. - This is not trivial, - as equations that mix polynomials and trigonometric functions generally do not have nice solutions. -

- -

- In we see a plot of the second derivative. - It shows that it has zeros at approximately t=0.5,\,3.5,\,6.5,\,9.5,\,12.5 and 16. - These approximations are not very good, - made only by looking at the graph. - Newton's Method provides more accurate approximations. - Accurate to 2 decimal places, we have: - - t=0.65,\,3.29,\,6.36,\,9.48,\,12.61\,\text{ and } \,15.74 - . -

- -

- The corresponding points have been plotted on the graph of the parametric equations in . - Note how most occur near the x-axis, - but not exactly on the axis. -

- -
- In (a), a graph of \frac{d^2y}{dx^2}, showing where it is approximately 0. In (b), graph of the parametric equations in along with the points of inflection - -
- +
+ Graphing \frac{d^2y}{dx^2} in , showing where it is approximately 0 + + - - Graph of the second derivative is a sinusoid with increasing amplitude. + Graph of the second derivative is a sinusoid with increasing amplitude.

The image shows the graph y = 2\cos(t)-4t\sin(t), which is a graph of \frac{d^2y}{dx^2}. @@ -848,15 +822,40 @@ \end{tikzpicture} - +

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- +

+ The points of inflection are found by setting \frac{d^2y}{dx^2}=0. + This is not trivial, + as equations that mix polynomials and trigonometric functions generally do not have nice solutions. +

+ +

+ In we see a plot of the second derivative. + It shows that it has zeros at approximately t=0.5,\,3.5,\,6.5,\,9.5,\,12.5 and 16. + These approximations are not very good, + made only by looking at the graph. + Newton's Method provides more accurate approximations. + Accurate to 2 decimal places, we have: + + t=0.65,\,3.29,\,6.36,\,9.48,\,12.61\,\text{ and } \,15.74 + . +

+ +

+ The corresponding points have been plotted on the graph of the parametric equations in . + Note how most occur near the x-axis, + but not exactly on the axis. +

+ +
+ A graph of the parametric equations in along with the points of inflection + + - - Graph of the parametric curve in this example, with points of inflection marked. + Graph of the parametric curve in this example, with points of inflection marked.

The graph shows a curve that appears to be sinusoidal, but with a frequency that increases with x. @@ -892,9 +891,6 @@

- - -
@@ -1281,7 +1277,7 @@ Find the surface area if this shape is rotated about the x-axis, as shown in .

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Rotating a teardrop shape about the x-axis in diff --git a/ptx/sec_param_eqs.ptx b/ptx/sec_param_eqs.ptx index 7e221ffa2..10f5e078f 100644 --- a/ptx/sec_param_eqs.ptx +++ b/ptx/sec_param_eqs.ptx @@ -187,7 +187,7 @@ A table of values of the parametric equations in along with a sketch of their graph
- + t x @@ -306,7 +306,7 @@ A table of values of the parametric equations in along with a sketch of their graph
- + t x @@ -1072,7 +1072,7 @@

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Graphing the parametric equations x=4\cos(t) +3, y=2\sin(t) +1 in @@ -1538,7 +1538,7 @@ illustrating the cusp at (1,4).

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Graphing the curve in ; note it is not smooth at (1,4) diff --git a/ptx/sec_polar.ptx b/ptx/sec_polar.ptx index 9a61b34a8..6ebf1e159 100644 --- a/ptx/sec_polar.ptx +++ b/ptx/sec_polar.ptx @@ -48,7 +48,7 @@ coordinatespolar

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Illustrating polar coordinates Illustration of polar coordinates relative to a pole and initial ray. @@ -706,7 +706,7 @@ This graph is also plotted in .

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Plotting standard polar plots diff --git a/ptx/sec_polarcalc.ptx b/ptx/sec_polarcalc.ptx index 3403e0cf5..da80b6812 100644 --- a/ptx/sec_polarcalc.ptx +++ b/ptx/sec_polarcalc.ptx @@ -126,7 +126,7 @@ .

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The limaçon in with its tangent line at \theta=\pi/4 and points of vertical and horizontal tangency @@ -940,7 +940,7 @@ Find the area bounded between the polar curves r=1 and r=2\cos(2\theta), as shown in .

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The region bounded by the functions in A zoomed in view of a region bounded by a circle, a rose curve, and the x axis. @@ -995,7 +995,7 @@ .

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Breaking the region bounded by the functions in into its component parts A zoomed in view of a polar region, showing it divided into two parts. @@ -1157,7 +1157,7 @@ .

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The limaçon in whose arc length is measured