Chapter 3: Applications of Differentiation
Section 3.1: Tangent and Normal Lines
Which point on the graph of gx=7 x+9−x2 is closest to the point 3,10? What is that minimal distance?
Context Panel: Assign Function
gx=7 x+9−x2→assign as functiong
Define dx, the distance from 3,10 to x,gx
dx=3−x2+10− gx2→assign as functiond
Figure 3.1.2(a) contains a graph of dx, the function giving the distance from the fixed point 3,10 to x,gx, the generic point on the graph of gx.
The smallest value for dx occurs in the interval 0,1/2.
The minimum distance is slightly less than 3.
Figure 3.1.2(a) Graph of dx
The animation in Figure 3.1.2(b) is controlled by the slider that sets the value of the x-coordinate of the moving green dot in the figure. This dot marks the point on the graph of gx corresponding to that value of x. The black dot marks the fixed point 3,10.
The corresponding y-coordinate is
and the length of the red segment (giving d, the distance between the points) is
Figure 3.1.2(b) Slider-controlled animation
Move the slider to determine the minimum value of d.
The line from the fixed point 3,10 to x,gx on the graph of g must be orthogonal to the graph, that is, orthogonal to the tangent line at x,gx. Hence, this line must be coincident with the normal line through x,gx.
Alternatively, the slope of the segment connecting 3,10 with x,gx and the slope of the normal line at x,gx) must agree.
Solve for xmin
Assuming gx and dx have already been defined, write the equation expressing the equality of slopes.
Press the Enter key.
Context Panel: Solve≻
Numerically Solve from point≻x=1
Evaluate dx at xmin.
Context Panel: Evaluate and Display Inline
d = 2.822319482
<< Previous Example Section 3.1
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2023. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document