I can visualize the exponential function with the graph y = e x , but I can do that for almost any function.In addition to its graph, the function f ( x ) = x n can be visualized as the volume of a box with sides of length x in n-dimensional space, and the trigonometric functions can be interpreted as side lengths of certain right triangles.Is there a similar geometric interpretation of the exponential function?

Question

I can visualize the exponential function with the graph   y  =      e    x  , but I can do that for almost any function.In addition to its graph, the function   f  (  x  )  =      x    n   can be visualized as the volume of a box with sides of length x in n-dimensional space, and the trigonometric functions can be interpreted as side lengths of certain right triangles.Is there a similar geometric interpretation of the exponential function?

Regan Holloway · Accepted Answer

There&#039;s a geometric interpretation of the natural log. From the definition  log  ⁡  x  =      ∫    1    x              1      t          d  twe see that the area between the &quot;standard&quot; hyperbola xy=1 and the horizontal axis between 1 and x is logx.So, turning this around, the line   x  =      e    t   is the vertical line such that the area between x=1 and   x  =      e    t  , between this hyperbola and the x-axis, is t.

inurbandojoa · Accepted Answer

This answer is not about       e    x  , but about the closely connected hyperbolic functions  cosh  ⁡  x  =                    e        x            +              e                  −          x                      2    ,    sinh  ⁡  x  =                    e        x            −              e                  −          x                      2    .A little algebra shows that       cosh    2    ⁡  x  −      sinh    2    ⁡  x  =  1. Thus, (coshx,sinhx) are points in the hyperbola       x    2    −      y    2    =  1; then coshx and sinhx are the legs of a right triangle whose hypotenuse is the segment joining the origin and the point with coordinates (coshx,sinhx).A different interpretation is the following: Imagine that you are running away fron some fixed point O. If your speed at each moment is equal to the distance to the point O, then your speed will be   C        e    x   for some constant C&amp;gt;0.

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